Optimal. Leaf size=309 \[ -\frac{3 i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3}+\frac{3 i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3}+\frac{3 i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{3 i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{c^2 x^2+1}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (c^2 x^2+1\right )}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (c^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c d^3}-\frac{b^2 x}{12 d^3 \left (c^2 x^2+1\right )}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.364411, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5690, 5693, 4180, 2531, 2282, 6589, 5717, 203, 199} \[ -\frac{3 i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3}+\frac{3 i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3}+\frac{3 i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{3 i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{c^2 x^2+1}}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (c^2 x^2+1\right )}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (c^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c d^3}-\frac{b^2 x}{12 d^3 \left (c^2 x^2+1\right )}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5690
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 203
Rule 199
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^3} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{(b c) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac{3 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}-\frac{b^2 \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{6 d^3}-\frac{(3 b c) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac{3 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac{b^2 x}{12 d^3 \left (1+c^2 x^2\right )}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{12 d^3}-\frac{\left (3 b^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 d^3}+\frac{3 \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c d^3}\\ &=-\frac{b^2 x}{12 d^3 \left (1+c^2 x^2\right )}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3}-\frac{(3 i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c d^3}+\frac{(3 i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c d^3}\\ &=-\frac{b^2 x}{12 d^3 \left (1+c^2 x^2\right )}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3}-\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c d^3}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c d^3}\\ &=-\frac{b^2 x}{12 d^3 \left (1+c^2 x^2\right )}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3}-\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}\\ &=-\frac{b^2 x}{12 d^3 \left (1+c^2 x^2\right )}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{3 b \left (a+b \sinh ^{-1}(c x)\right )}{4 c d^3 \sqrt{1+c^2 x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3}-\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}+\frac{3 i b^2 \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}-\frac{3 i b^2 \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{4 c d^3}\\ \end{align*}
Mathematica [A] time = 2.23292, size = 546, normalized size = 1.77 \[ \frac{\frac{a b \left (\frac{9}{2} i \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )\right )-\frac{9}{2} i \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )\right )+\frac{9 \left (\sinh ^{-1}(c x)-i \sqrt{c^2 x^2+1}\right )}{c x-i}+\frac{9 \left (\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right )}{c x+i}-\frac{i \left (3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x-2 i)\right )}{(c x-i)^2}+\frac{i \left (3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right )}{(c x+i)^2}\right )}{c}+\frac{b^2 \left (-18 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+18 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )-18 i \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )+18 i \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )-\frac{2 c x}{c^2 x^2+1}+\frac{9 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{6 c x \sinh ^{-1}(c x)^2}{\left (c^2 x^2+1\right )^2}+\frac{18 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}-9 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+9 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-40 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{c}+\frac{9 a^2 x}{c^2 x^2+1}+\frac{6 a^2 x}{\left (c^2 x^2+1\right )^2}+\frac{9 a^2 \tan ^{-1}(c x)}{c}}{24 d^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{ \left ({c}^{2}d{x}^{2}+d \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, a^{2}{\left (\frac{3 \, c^{2} x^{3} + 5 \, x}{c^{4} d^{3} x^{4} + 2 \, c^{2} d^{3} x^{2} + d^{3}} + \frac{3 \, \arctan \left (c x\right )}{c d^{3}}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]