Integrand size = 28, antiderivative size = 421 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c (a+b \text {arcsinh}(c x))}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {16 b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \] Output:
1/3*b^2*c^2*x/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b*arcsinh(c*x))/d^2/(c^2* x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-(a+b*arcsinh(c*x))^2/d/x/(c^2*d*x^2+d)^(3 /2)-4/3*c^2*x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(3/2)-8/3*c^2*x*(a+b*ar csinh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)-8/3*c*(c^2*x^2+1)^(1/2)*(a+b*arcsinh (c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)-4*b*c*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x) )*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^2/(c^2*d*x^2+d)^(1/2)+16/3*b*c*(c^2 *x^2+1)^(1/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/d^2/(c^2* d*x^2+d)^(1/2)+5/3*b^2*c*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/ 2))^2)/d^2/(c^2*d*x^2+d)^(1/2)+b^2*c*(c^2*x^2+1)^(1/2)*polylog(2,(c*x+(c^2 *x^2+1)^(1/2))^2)/d^2/(c^2*d*x^2+d)^(1/2)
Time = 1.95 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {3 a^2+12 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4-b^2 c^4 x^4+a b c x \sqrt {1+c^2 x^2}+6 a b \text {arcsinh}(c x)+24 a b c^2 x^2 \text {arcsinh}(c x)+16 a b c^4 x^4 \text {arcsinh}(c x)+b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+3 b^2 \text {arcsinh}(c x)^2+12 b^2 c^2 x^2 \text {arcsinh}(c x)^2+8 b^2 c^4 x^4 \text {arcsinh}(c x)^2-8 b^2 c x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2-6 b^2 c x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-10 b^2 c x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-6 a b c x \left (1+c^2 x^2\right )^{3/2} \log (c x)-5 a b c x \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )+5 b^2 c x \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+3 b^2 c x \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 d x \left (d+c^2 d x^2\right )^{3/2}} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(5/2)),x]
Output:
-1/3*(3*a^2 + 12*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 - b^2*c^4*x^4 + a*b*c*x*Sqrt[1 + c^2*x^2] + 6*a*b*ArcSinh[c*x] + 24*a*b*c^2*x^2*ArcSinh[c *x] + 16*a*b*c^4*x^4*ArcSinh[c*x] + b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 3*b^2*ArcSinh[c*x]^2 + 12*b^2*c^2*x^2*ArcSinh[c*x]^2 + 8*b^2*c^4*x^4*Ar cSinh[c*x]^2 - 8*b^2*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - 6*b^2*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 10*b^2*c*x*( 1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 6*a*b*c*x*( 1 + c^2*x^2)^(3/2)*Log[c*x] - 5*a*b*c*x*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^ 2] + 5*b^2*c*x*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 3*b^ 2*c*x*(1 + c^2*x^2)^(3/2)*PolyLog[2, E^(-2*ArcSinh[c*x])])/(d*x*(d + c^2*d *x^2)^(3/2))
Result contains complex when optimal does not.
Time = 4.18 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.04, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6224, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 d x^2+d\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6212 |
| \(\displaystyle -4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \int \frac {c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \int \frac {e^{2 \text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{1+e^{2 \text {arcsinh}(c x)}}d\text {arcsinh}(c x)-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )+\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx-\frac {1}{2} b c \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (\int \frac {a+b \text {arcsinh}(c x)}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (2 \int (a+b \text {arcsinh}(c x)) \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (2 \int i (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \int (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \left (2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}\right )}{d^2 \sqrt {c^2 d x^2+d}}-4 c^2 \left (-\frac {2 b c \sqrt {c^2 x^2+1} \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\right )}{3 d}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 d x^2+d\right )^{3/2}}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(5/2)),x]
Output:
-((a + b*ArcSinh[c*x])^2/(d*x*(d + c^2*d*x^2)^(3/2))) - 4*c^2*((x*(a + b*A rcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) - (2*b*c*Sqrt[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*Pol yLog[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])*ArcTanh[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])
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[(((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. \(3512\) vs. \(2(411)=822\).
Time = 1.35 (sec) , antiderivative size = 3513, normalized size of antiderivative = 8.34
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3513\) |
| parts | \(\text {Expression too large to display}\) | \(3513\) |
Input:
int((a+b*arcsinh(x*c))^2/x^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*polylog(2,x*c+(c^2*x^2+1 )^(1/2))*c-16/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*arcsinh(x* c)^2*c+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*polylog(2,-x*c-(c ^2*x^2+1)^(1/2))*c-3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^ 2*x^2+9)/d^3*(c^2*x^2+1)^(1/2)*c-5*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25 *c^4*x^4+26*c^2*x^2+9)/d^3*x*c^2-9*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25 *c^4*x^4+26*c^2*x^2+9)/d^3/x*arcsinh(x*c)^2-160/3*b^2*(d*(c^2*x^2+1))^(1/2 )/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^5*c^6-29*b^2*(d*(c^2*x^2+1))^( 1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*c^4+5/3*b^2*(d*(c^2*x^2+1 ))^(1/2)/(c^2*x^2+1)^(1/2)/d^3*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)*c-32/ 3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^9*c^ 10-40*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^ 7*c^8-48*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3 *x^3*c^4-8*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d ^3*x*c^2-3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d ^3*(c^2*x^2+1)^(1/2)*c-18*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+ 26*c^2*x^2+9)/d^3/x*arcsinh(x*c)+10/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1 )^(1/2)/d^3*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*c+2*a*b*(d*(c^2*x^2+1))^(1/2)/ (c^2*x^2+1)^(1/2)/d^3*ln((x*c+(c^2*x^2+1)^(1/2))^2-1)*c+128/3*a*b*(d*(c^2* x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^4*(c^2*x^2+1)^(...
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
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^8 + 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 + d^3*x^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*asinh(c*x))**2/x**2/(c**2*d*x**2+d)**(5/2),x)
Output:
Integral((a + b*asinh(c*x))**2/(x**2*(d*(c**2*x**2 + 1))**(5/2)), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
-1/3*a^2*(8*c^2*x/(sqrt(c^2*d*x^2 + d)*d^2) + 4*c^2*x/((c^2*d*x^2 + d)^(3/ 2)*d) + 3/((c^2*d*x^2 + d)^(3/2)*d*x)) + integrate(b^2*log(c*x + sqrt(c^2* x^2 + 1))^2/((c^2*d*x^2 + d)^(5/2)*x^2) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1 ))/((c^2*d*x^2 + d)^(5/2)*x^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x, algorithm="giac" )
Output:
integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^(5/2)*x^2), x)
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:
int((a + b*asinh(c*x))^2/(x^2*(d + c^2*d*x^2)^(5/2)),x)
Output:
int((a + b*asinh(c*x))^2/(x^2*(d + c^2*d*x^2)^(5/2)), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-8 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-12 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{6}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) a b \,c^{4} x^{5}+12 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{6}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) a b \,c^{2} x^{3}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{6}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) a b x +3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{6}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} c^{4} x^{5}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{6}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} c^{2} x^{3}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{6}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) b^{2} x +8 a^{2} c^{5} x^{5}+16 a^{2} c^{3} x^{3}+8 a^{2} c x}{3 \sqrt {d}\, d^{2} x \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x)
Output:
( - 8*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 - 12*sqrt(c**2*x**2 + 1)*a**2*c** 2*x**2 - 3*sqrt(c**2*x**2 + 1)*a**2 + 6*int(asinh(c*x)/(sqrt(c**2*x**2 + 1 )*c**4*x**6 + 2*sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1)*x**2), x)*a*b*c**4*x**5 + 12*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**6 + 2*sq rt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1)*x**2),x)*a*b*c**2*x**3 + 6*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**4*x**6 + 2*sqrt(c**2*x**2 + 1)*c **2*x**4 + sqrt(c**2*x**2 + 1)*x**2),x)*a*b*x + 3*int(asinh(c*x)**2/(sqrt( c**2*x**2 + 1)*c**4*x**6 + 2*sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x** 2 + 1)*x**2),x)*b**2*c**4*x**5 + 6*int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)* c**4*x**6 + 2*sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1)*x**2),x) *b**2*c**2*x**3 + 3*int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**4*x**6 + 2*s qrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1)*x**2),x)*b**2*x + 8*a** 2*c**5*x**5 + 16*a**2*c**3*x**3 + 8*a**2*c*x)/(3*sqrt(d)*d**2*x*(c**4*x**4 + 2*c**2*x**2 + 1))