Optimal. Leaf size=349 \[ \frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.60894, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5801, 5831, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b^2 c^3 d \text{PolyLog}\left (2,-\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5801
Rule 5831
Rule 3324
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{(b c) \int \frac{a+b \sinh ^{-1}(c x)}{(d+e x)^2 \sqrt{1+c^2 x^2}} \, dx}{e}\\ &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{a+b x}{(c d+e \sinh (x))^2} \, dx,x,\sinh ^{-1}(c x)\right )}{e}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{c d+e \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 d^2+e^2}+\frac{\left (b c^3 d\right ) \operatorname{Subst}\left (\int \frac{a+b x}{c d+e \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{c d+x} \, dx,x,c e x\right )}{e \left (c^2 d^2+e^2\right )}+\frac{\left (2 b c^3 d\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{-e+2 c d e^x+e e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac{\left (2 b c^3 d\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c d-2 \sqrt{c^2 d^2+e^2}+2 e e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^{3/2}}-\frac{\left (2 b c^3 d\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c d+2 \sqrt{c^2 d^2+e^2}+2 e e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^{3/2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}-\frac{\left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e e^x}{2 c d-2 \sqrt{c^2 d^2+e^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac{\left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e e^x}{2 c d+2 \sqrt{c^2 d^2+e^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}-\frac{\left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 e x}{2 c d-2 \sqrt{c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac{\left (b^2 c^3 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 e x}{2 c d+2 \sqrt{c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b c^3 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac{b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac{b^2 c^3 d \text{Li}_2\left (-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac{b^2 c^3 d \text{Li}_2\left (-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.874181, size = 270, normalized size = 0.77 \[ \frac{\frac{2 b c^3 d \left (b \text{PolyLog}\left (2,\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right )-b \text{PolyLog}\left (2,-\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}\right )+\left (a+b \sinh ^{-1}(c x)\right ) \left (\log \left (\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right )-\log \left (\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right )\right )\right )}{\left (c^2 d^2+e^2\right )^{3/2}}-\frac{2 b c e \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{(d+e x)^2}+\frac{2 b^2 c^2 \log (d+e x)}{c^2 d^2+e^2}}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.262, size = 1013, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + 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*} \int \frac{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \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 (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]