Integrand size = 26, antiderivative size = 275 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x (a+b \text {arcsinh}(c x))}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \text {arcsinh}(c x))}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}-\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \] Output:
-1/12*b^2/d^3/(c^2*x^2+1)-1/6*b*c*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^(3/ 2)-4/3*b*c*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^(1/2)+1/4*(a+b*arcsinh(c*x ))^2/d^3/(c^2*x^2+1)^2+1/2*(a+b*arcsinh(c*x))^2/d^3/(c^2*x^2+1)-2*(a+b*arc sinh(c*x))^2*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^3+2/3*b^2*ln(c^2*x^2+1)/ d^3-b*(a+b*arcsinh(c*x))*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^3+b*(a+b* arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)/d^3+1/2*b^2*polylog(3,- (c*x+(c^2*x^2+1)^(1/2))^2)/d^3-1/2*b^2*polylog(3,(c*x+(c^2*x^2+1)^(1/2))^2 )/d^3
Result contains complex when optimal does not.
Time = 2.92 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\frac {\frac {6 a^2}{\left (1+c^2 x^2\right )^2}+\frac {12 a^2}{1+c^2 x^2}+24 a^2 \log (c x)-12 a^2 \log \left (1+c^2 x^2\right )+a b \left (-\frac {15 \left (\sqrt {1+c^2 x^2}-i \text {arcsinh}(c x)\right )}{i+c x}-\frac {15 \left (\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)\right )}{-i+c x}-24 \text {arcsinh}(c x)^2-\frac {(-2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)}{(-i+c x)^2}-\frac {(2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)}{(i+c x)^2}+48 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+12 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )+12 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+24 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+b^2 \left (i \pi ^3-\frac {2}{1+c^2 x^2}-\frac {4 c x \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}-\frac {32 c x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {6 \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {12 \text {arcsinh}(c x)^2}{1+c^2 x^2}-16 \text {arcsinh}(c x)^3-24 \text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+24 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+16 \log \left (1+c^2 x^2\right )+24 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+24 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+12 \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )}{24 d^3} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^3),x]
Output:
((6*a^2)/(1 + c^2*x^2)^2 + (12*a^2)/(1 + c^2*x^2) + 24*a^2*Log[c*x] - 12*a ^2*Log[1 + c^2*x^2] + a*b*((-15*(Sqrt[1 + c^2*x^2] - I*ArcSinh[c*x]))/(I + c*x) - (15*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(-I + c*x) - 24*ArcSinh[ c*x]^2 - ((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x])/(-I + c*x)^2 - ((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x])/(I + c*x)^2 + 48*ArcSinh[ c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 12*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[ 1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + 12*(ArcSinh[ c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSi nh[c*x]]) + 24*PolyLog[2, E^(2*ArcSinh[c*x])]) + b^2*(I*Pi^3 - 2/(1 + c^2* x^2) - (4*c*x*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - (32*c*x*ArcSinh[c*x])/Sq rt[1 + c^2*x^2] + (6*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (12*ArcSinh[c*x]^2) /(1 + c^2*x^2) - 16*ArcSinh[c*x]^3 - 24*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSi nh[c*x])] + 24*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 16*Log[1 + c^2 *x^2] + 24*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 24*ArcSinh[c*x] *PolyLog[2, E^(2*ArcSinh[c*x])] + 12*PolyLog[3, -E^(-2*ArcSinh[c*x])] - 12 *PolyLog[3, E^(2*ArcSinh[c*x])]))/(24*d^3)
Result contains complex when optimal does not.
Time = 3.19 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.18, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {6226, 27, 6203, 241, 6202, 240, 6226, 6202, 240, 6214, 5984, 3042, 26, 4670, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 d x^2+d\right )^3} \, dx\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{5/2}}dx}{2 d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (c^2 x^2+1\right )^2}dx}{d}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{5/2}}dx}{2 d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {b c \left (\frac {2}{3} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {b c \left (\frac {2}{3} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -\frac {b c \left (\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-b c \int \frac {x}{c^2 x^2+1}dx\right )+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle \frac {-b c \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle \frac {-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-b c \int \frac {x}{c^2 x^2+1}dx\right )+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {2 \int (a+b \text {arcsinh}(c x))^2 \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int i (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 i \int (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 i \left (i b \int (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 i \left (-i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 i \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{d^3}+\frac {(a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 d^3}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^3),x]
Output:
(a + b*ArcSinh[c*x])^2/(4*d^3*(1 + c^2*x^2)^2) - (b*c*(b/(6*c*(1 + c^2*x^2 )) + (x*(a + b*ArcSinh[c*x]))/(3*(1 + c^2*x^2)^(3/2)) + (2*((x*(a + b*ArcS inh[c*x]))/Sqrt[1 + c^2*x^2] - (b*Log[1 + c^2*x^2])/(2*c)))/3))/(2*d^3) + ((a + b*ArcSinh[c*x])^2/(2*(1 + c^2*x^2)) - b*c*((x*(a + b*ArcSinh[c*x]))/ Sqrt[1 + c^2*x^2] - (b*Log[1 + c^2*x^2])/(2*c)) + (2*I)*(I*(a + b*ArcSinh[ c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])] - I*b*(-1/2*((a + b*ArcSinh[c*x])*Poly Log[2, -E^(2*ArcSinh[c*x])]) + (b*PolyLog[3, -E^(2*ArcSinh[c*x])])/4) + I* b*(-1/2*((a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])]) + (b*PolyLog [3, E^(2*ArcSinh[c*x])])/4)))/d^3
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[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/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/(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])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(675\) vs. \(2(300)=600\).
Time = 1.79 (sec) , antiderivative size = 676, normalized size of antiderivative = 2.46
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {1}{2 c^{2} x^{2}+2}+\ln \left (x c \right )\right )}{d^{3}}+\frac {b^{2} \left (\frac {-16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+16 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-18 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +32 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+9 \operatorname {arcsinh}\left (x c \right )^{2}-c^{2} x^{2}+16 \,\operatorname {arcsinh}\left (x c \right )-1}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{3}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}+\frac {2 a b \left (\frac {-8 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+8 c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +16 c^{2} x^{2}+9 \,\operatorname {arcsinh}\left (x c \right )+8}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(676\) |
| default | \(\frac {a^{2} \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {1}{2 c^{2} x^{2}+2}+\ln \left (x c \right )\right )}{d^{3}}+\frac {b^{2} \left (\frac {-16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+16 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-18 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +32 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+9 \operatorname {arcsinh}\left (x c \right )^{2}-c^{2} x^{2}+16 \,\operatorname {arcsinh}\left (x c \right )-1}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{3}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}+\frac {2 a b \left (\frac {-8 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+8 c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +16 c^{2} x^{2}+9 \,\operatorname {arcsinh}\left (x c \right )+8}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(676\) |
| parts | \(\frac {a^{2} \left (\ln \left (x \right )-\frac {c^{2} \left (-\frac {1}{2 c^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (c^{2} x^{2}+1\right )}{c^{2}}\right )}{2}\right )}{d^{3}}+\frac {b^{2} \left (\frac {-16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+16 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-18 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +32 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+9 \operatorname {arcsinh}\left (x c \right )^{2}-c^{2} x^{2}+16 \,\operatorname {arcsinh}\left (x c \right )-1}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{3}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}+\frac {2 a b \left (\frac {-8 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+8 c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +16 c^{2} x^{2}+9 \,\operatorname {arcsinh}\left (x c \right )+8}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(688\) |
Input:
int((a+b*arcsinh(x*c))^2/x/(c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
a^2/d^3*(1/4/(c^2*x^2+1)^2-1/2*ln(c^2*x^2+1)+1/2/(c^2*x^2+1)+ln(x*c))+b^2/ d^3*(1/12*(-16*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3+16*arcsinh(x*c)*c^4* x^4+6*arcsinh(x*c)^2*x^2*c^2-18*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+32*arcs inh(x*c)*c^2*x^2+9*arcsinh(x*c)^2-c^2*x^2+16*arcsinh(x*c)-1)/(c^4*x^4+2*c^ 2*x^2+1)+4/3*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-8/3*ln(x*c+(c^2*x^2+1)^(1/2)) +arcsinh(x*c)^2*ln(1+x*c+(c^2*x^2+1)^(1/2))+2*arcsinh(x*c)*polylog(2,-x*c- (c^2*x^2+1)^(1/2))-2*polylog(3,-x*c-(c^2*x^2+1)^(1/2))-arcsinh(x*c)^2*ln(1 +(x*c+(c^2*x^2+1)^(1/2))^2)-arcsinh(x*c)*polylog(2,-(x*c+(c^2*x^2+1)^(1/2) )^2)+1/2*polylog(3,-(x*c+(c^2*x^2+1)^(1/2))^2)+arcsinh(x*c)^2*ln(1-x*c-(c^ 2*x^2+1)^(1/2))+2*arcsinh(x*c)*polylog(2,x*c+(c^2*x^2+1)^(1/2))-2*polylog( 3,x*c+(c^2*x^2+1)^(1/2)))+2*a*b/d^3*(1/12*(-8*(c^2*x^2+1)^(1/2)*c^3*x^3+8* c^4*x^4+6*arcsinh(x*c)*c^2*x^2-9*(c^2*x^2+1)^(1/2)*x*c+16*c^2*x^2+9*arcsin h(x*c)+8)/(c^4*x^4+2*c^2*x^2+1)+arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))+p olylog(2,-x*c-(c^2*x^2+1)^(1/2))-arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2)) ^2)-1/2*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)+arcsinh(x*c)*ln(1-x*c-(c^2*x ^2+1)^(1/2))+polylog(2,x*c+(c^2*x^2+1)^(1/2)))
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x/(c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^6*d^3*x^7 + 3* c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx}{d^{3}} \] Input:
integrate((a+b*asinh(c*x))**2/x/(c**2*d*x**2+d)**3,x)
Output:
(Integral(a**2/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x), x) + Integral( b**2*asinh(c*x)**2/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x), x) + Integ ral(2*a*b*asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x), x))/d**3
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x/(c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
1/4*a^2*((2*c^2*x^2 + 3)/(c^4*d^3*x^4 + 2*c^2*d^3*x^2 + d^3) - 2*log(c^2*x ^2 + 1)/d^3 + 4*log(x)/d^3) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2 /(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x) + 2*a*b*log(c*x + s qrt(c^2*x^2 + 1))/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 + d^3*x), x )
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x/(c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^3*x), x)
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \] Input:
int((a + b*asinh(c*x))^2/(x*(d + c^2*d*x^2)^3),x)
Output:
int((a + b*asinh(c*x))^2/(x*(d + c^2*d*x^2)^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (d+c^2 d x^2\right )^3} \, dx=\frac {8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) a b \,c^{4} x^{4}+16 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) a b \,c^{2} x^{2}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) a b +4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) b^{2} c^{4} x^{4}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) b^{2} c^{2} x^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) b^{2}-2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2} c^{4} x^{4}-4 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2} c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2}+4 \,\mathrm {log}\left (x \right ) a^{2} c^{4} x^{4}+8 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a^{2}-a^{2} c^{4} x^{4}+2 a^{2}}{4 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/x/(c^2*d*x^2+d)^3,x)
Output:
(8*int(asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x),x)*a*b*c**4* x**4 + 16*int(asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x),x)*a* b*c**2*x**2 + 8*int(asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x) ,x)*a*b + 4*int(asinh(c*x)**2/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x), x)*b**2*c**4*x**4 + 8*int(asinh(c*x)**2/(c**6*x**7 + 3*c**4*x**5 + 3*c**2* x**3 + x),x)*b**2*c**2*x**2 + 4*int(asinh(c*x)**2/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x),x)*b**2 - 2*log(c**2*x**2 + 1)*a**2*c**4*x**4 - 4*log( c**2*x**2 + 1)*a**2*c**2*x**2 - 2*log(c**2*x**2 + 1)*a**2 + 4*log(x)*a**2* c**4*x**4 + 8*log(x)*a**2*c**2*x**2 + 4*log(x)*a**2 - a**2*c**4*x**4 + 2*a **2)/(4*d**3*(c**4*x**4 + 2*c**2*x**2 + 1))