Integrand size = 28, antiderivative size = 506 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c (a+b \text {arcsinh}(c x))}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 (a+b \text {arcsinh}(c x))^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {16 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {32 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {32 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}} \]
-1/3*(a+b*arcsinh(c*x))^2/d/x^3/(c^2*d*x^2+d)^(3/2)+2*c^2*(a+b*arcsinh(c*x ))^2/d/x/(c^2*d*x^2+d)^(3/2)+8/3*c^4*x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d )^(3/2)-1/3*b^2*c^2/d^2/x/(c^2*d*x^2+d)^(1/2)-2/3*b^2*c^4*x/d^2/(c^2*d*x^2 +d)^(1/2)+16/3*c^4*x*(a+b*arcsinh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*c* (a+b*arcsinh(c*x))/d^2/x^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+16/3*c^3* (a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+32/3*b*c^3* (a+b*arcsinh(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d^ 2/(c^2*d*x^2+d)^(1/2)-32/3*b*c^3*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^ (1/2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-8/3*b^2*c^3*polylog(2, -(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-8/3* b^2*c^3*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d* x^2+d)^(1/2)
Time = 2.60 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 \left (-1+6 c^2 x^2+24 c^4 x^4+16 c^6 x^6\right )}{x^3}-\frac {a b \left (-2 \left (-1+6 c^2 x^2+24 c^4 x^4+16 c^6 x^6\right ) \text {arcsinh}(c x)+c x \sqrt {1+c^2 x^2} \left (1+16 \left (c^2 x^2+c^4 x^4\right ) \log (c x)+8 \left (c^2 x^2+c^4 x^4\right ) \log \left (1+c^2 x^2\right )\right )\right )}{x^3}+b^2 c^3 \left (1+c^2 x^2\right )^{3/2} \left (-\frac {c x}{\sqrt {1+c^2 x^2}}-\frac {\sqrt {1+c^2 x^2}}{c x}-\frac {\text {arcsinh}(c x)}{c^2 x^2}+\frac {\text {arcsinh}(c x)}{1+c^2 x^2}-16 \text {arcsinh}(c x)^2+\frac {c x \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^{3/2}}+\frac {8 c x \text {arcsinh}(c x)^2}{\sqrt {1+c^2 x^2}}-\frac {\sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2}{c^3 x^3}+\frac {8 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2}{c x}-16 \text {arcsinh}(c x) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-16 \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+8 \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+8 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}} \]
((a^2*(-1 + 6*c^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6))/x^3 - (a*b*(-2*(-1 + 6*c ^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x] + c*x*Sqrt[1 + c^2*x^2]*(1 + 16*(c^2*x^2 + c^4*x^4)*Log[c*x] + 8*(c^2*x^2 + c^4*x^4)*Log[1 + c^2*x^2] )))/x^3 + b^2*c^3*(1 + c^2*x^2)^(3/2)*(-((c*x)/Sqrt[1 + c^2*x^2]) - Sqrt[1 + c^2*x^2]/(c*x) - ArcSinh[c*x]/(c^2*x^2) + ArcSinh[c*x]/(1 + c^2*x^2) - 16*ArcSinh[c*x]^2 + (c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^(3/2) + (8*c*x*ArcS inh[c*x]^2)/Sqrt[1 + c^2*x^2] - (Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(c^3*x^ 3) + (8*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(c*x) - 16*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 16*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] + 8*Po lyLog[2, -E^(-2*ArcSinh[c*x])] + 8*PolyLog[2, E^(-2*ArcSinh[c*x])]))/(3*d* (d + c^2*d*x^2)^(3/2))
Result contains complex when optimal does not.
Time = 5.09 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.37, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6224, 6224, 245, 208, 6203, 6202, 6212, 3042, 26, 4201, 2620, 2715, 2838, 6213, 208, 6226, 208, 6214, 5984, 3042, 26, 4670, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (c^2 d x^2+d\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}-2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \left (-2 c^2 \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{x \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -2 c^2 \left (-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -2 c^2 \left (-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6212 |
| \(\displaystyle -2 c^2 \left (-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \int \frac {c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -2 c^2 \left (-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \int \frac {e^{2 \text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{1+e^{2 \text {arcsinh}(c x)}}d\text {arcsinh}(c x)-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 c^2 \left (-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -2 c^2 \left (-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx-\frac {1}{2} b c \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx-\frac {1}{2} b c \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (\int \frac {a+b \text {arcsinh}(c x)}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (2 \int (a+b \text {arcsinh}(c x)) \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (2 \int (a+b \text {arcsinh}(c x)) \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (2 \int i (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (2 \int i (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \int (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (2 i \int (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (2 i \left (\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (-2 c^2 \left (2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (-\frac {2 c^2 x}{\sqrt {c^2 x^2+1}}-\frac {1}{x \sqrt {c^2 x^2+1}}\right )\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-2 c^2 \left (\frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}\) |
-1/3*(a + b*ArcSinh[c*x])^2/(d*x^3*(d + c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[1 + c^2*x^2]*((b*c*(-(1/(x*Sqrt[1 + c^2*x^2])) - (2*c^2*x)/Sqrt[1 + c^2*x^2] ))/2 - (a + b*ArcSinh[c*x])/(2*x^2*(1 + c^2*x^2)) - 2*c^2*(-1/2*(b*c*x)/Sq rt[1 + c^2*x^2] + (a + b*ArcSinh[c*x])/(2*(1 + c^2*x^2)) + (2*I)*(I*(a + b *ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyLog[2, -E^(2*ArcS inh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x])]))))/(3*d^2*Sqrt[d + c^ 2*d*x^2]) - 2*c^2*(-((a + b*ArcSinh[c*x])^2/(d*x*(d + c^2*d*x^2)^(3/2))) - 4*c^2*((x*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) - (2*b*c*Sq rt[1 + c^2*x^2]*((b*x)/(2*c*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(2*c ^2*(1 + c^2*x^2))))/(3*d^2*Sqrt[d + c^2*d*x^2]) + (2*((x*(a + b*ArcSinh[c* x])^2)/(d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b*Sqrt[1 + c^2*x^2]*(((-1/2*I)*(a + b*ArcSinh[c*x])^2)/b + (2*I)*(((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh [c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4)))/(c*d*Sqrt[d + c^2*d* x^2])))/(3*d)) + (2*b*c*Sqrt[1 + c^2*x^2]*(-1/2*(b*c*x)/Sqrt[1 + c^2*x^2] + (a + b*ArcSinh[c*x])/(2*(1 + c^2*x^2)) + (2*I)*(I*(a + b*ArcSinh[c*x])*A rcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyLog[2, -E^(2*ArcSinh[c*x])] - (I/ 4)*b*PolyLog[2, E^(2*ArcSinh[c*x])])))/(d^2*Sqrt[d + c^2*d*x^2]))
3.4.19.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp [b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSinh[ c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x ], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x] ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar cSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(3505\) vs. \(2(484)=968\).
Time = 0.37 (sec) , antiderivative size = 3506, normalized size of antiderivative = 6.93
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3506\) |
| parts | \(\text {Expression too large to display}\) | \(3506\) |
344/3*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2*x ^2-1)/d^3*x^3*arcsinh(c*x)^2*c^6+64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c^8*x^ 8+36*c^6*x^6+35*c^4*x^4+10*c^2*x^2-1)/d^3*x^3*arcsinh(c*x)*c^6+22/3*b^2*(d *(c^2*x^2+1))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2*x^2-1)/d^3*x^ 2*c^5*(c^2*x^2+1)^(1/2)+12*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c^8*x^8+36*c^6*x^ 6+35*c^4*x^4+10*c^2*x^2-1)/d^3*x*arcsinh(c*x)^2*c^4-16/3*b^2*(d*(c^2*x^2+1 ))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2*x^2-1)/d^3*x*arcsinh(c*x )*c^4-6*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2 *x^2-1)/d^3/x*arcsinh(c*x)^2*c^2-16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1 )^(1/2)/d^3*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c^3-16/3*b^2*(d*(c^2* x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/ 2))^2)*c^3+16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^ 4+10*c^2*x^2-1)/d^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3-4*b^2*(d*(c^2*x^2 +1))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2*x^2-1)/d^3*(c^2*x^2+1) ^(1/2)*arcsinh(c*x)*c^3-16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d ^3*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c^3+256/3*b^2*(d*(c^2*x^2+1))^ (1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2*x^2-1)/d^3*x^11*arcsinh(c*x )*c^14-64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c^8*x^8+36*c^6*x^6+35*c^4*x^4+10 *c^2*x^2-1)/d^3*x^9*(c^2*x^2+1)*c^12+896/3*b^2*(d*(c^2*x^2+1))^(1/2)/(12*c ^8*x^8+36*c^6*x^6+35*c^4*x^4+10*c^2*x^2-1)/d^3*x^9*arcsinh(c*x)*c^12+1/...
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^ 2)/(c^6*d^3*x^10 + 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 + d^3*x^4), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
-1/3*a*b*c*(8*c^2*log(c^2*x^2 + 1)/d^(5/2) + 16*c^2*log(x)/d^(5/2) + 1/(c^ 2*d^(5/2)*x^4 + d^(5/2)*x^2)) + 2/3*(16*c^4*x/(sqrt(c^2*d*x^2 + d)*d^2) + 8*c^4*x/((c^2*d*x^2 + d)^(3/2)*d) + 6*c^2/((c^2*d*x^2 + d)^(3/2)*d*x) - 1/ ((c^2*d*x^2 + d)^(3/2)*d*x^3))*a*b*arcsinh(c*x) + 1/3*(16*c^4*x/(sqrt(c^2* d*x^2 + d)*d^2) + 8*c^4*x/((c^2*d*x^2 + d)^(3/2)*d) + 6*c^2/((c^2*d*x^2 + d)^(3/2)*d*x) - 1/((c^2*d*x^2 + d)^(3/2)*d*x^3))*a^2 + b^2*integrate(log(c *x + sqrt(c^2*x^2 + 1))^2/((c^2*d*x^2 + d)^(5/2)*x^4), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]