Optimal. Leaf size=687 \[ -\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{5 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{5 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{5 a b c^3 d^2 x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{2 b c^5 d^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}+\frac{b c^3 d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}+\frac{5}{2} c^2 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b c d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{c^2 x^2+1}}-\frac{5 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+\frac{5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{40}{9} b^2 c^2 d^2 \sqrt{c^2 d x^2+d}+\frac{2}{27} b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}-\frac{5 b^2 c^3 d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{\sqrt{c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.988336, antiderivative size = 687, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 20, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {5739, 5744, 5742, 5760, 4182, 2531, 2282, 6589, 5653, 261, 5679, 444, 43, 270, 5730, 12, 1251, 897, 1153, 208} \[ -\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{5 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{5 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{5 a b c^3 d^2 x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{2 b c^5 d^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{c^2 x^2+1}}+\frac{b c^3 d^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{c^2 x^2+1}}+\frac{5}{2} c^2 d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{b c d^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{c^2 x^2+1}}-\frac{5 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+\frac{5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{40}{9} b^2 c^2 d^2 \sqrt{c^2 d x^2+d}+\frac{2}{27} b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d}-\frac{5 b^2 c^3 d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{\sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5739
Rule 5744
Rule 5742
Rule 5760
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5653
Rule 261
Rule 5679
Rule 444
Rule 43
Rule 270
Rule 5730
Rule 12
Rule 1251
Rule 897
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{1}{2} \left (5 c^2 d\right ) \int \frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\frac{\left (b c d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{2 b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{1}{2} \left (5 c^2 d^2\right ) \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{-3+6 c^2 x^2+c^4 x^4}{3 x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (5 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{5}{2} c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{\left (5 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{-3+6 c^2 x^2+c^4 x^4}{x \sqrt{1+c^2 x^2}} \, dx}{3 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x \left (1+\frac{c^2 x^2}{3}\right )}{\sqrt{1+c^2 x^2}} \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{5 a b c^3 d^2 x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{5}{2} c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{\left (5 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{-3+6 c^2 x+c^4 x^2}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 c^3 d^2 \sqrt{d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{c^2 x}{3}}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt{1+c^2 x^2}}\\ &=-\frac{5 a b c^3 d^2 x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}-\frac{5 b^2 c^3 d^2 x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{5}{2} c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{5 c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{-8+4 x^2+x^4}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{3 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (5 b c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1+c^2 x}}+\frac{1}{3} \sqrt{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{55}{9} b^2 c^2 d^2 \sqrt{d+c^2 d x^2}-\frac{5 a b c^3 d^2 x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{5}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{5 b^2 c^3 d^2 x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{5}{2} c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{5 c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{5 b c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{5 b c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (5 c^2+c^2 x^2-\frac{3}{-\frac{1}{c^2}+\frac{x^2}{c^2}}\right ) \, dx,x,\sqrt{1+c^2 x^2}\right )}{3 \sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=\frac{40}{9} b^2 c^2 d^2 \sqrt{d+c^2 d x^2}-\frac{5 a b c^3 d^2 x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{5 b^2 c^3 d^2 x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{5}{2} c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{5 c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{5 b c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{5 b c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=\frac{40}{9} b^2 c^2 d^2 \sqrt{d+c^2 d x^2}-\frac{5 a b c^3 d^2 x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2}-\frac{5 b^2 c^3 d^2 x \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}-\frac{b c d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt{1+c^2 x^2}}-\frac{2 b c^5 d^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt{1+c^2 x^2}}+\frac{5}{2} c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{5 c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b^2 c^2 d^2 \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{\sqrt{1+c^2 x^2}}-\frac{5 b c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{5 b c^2 d^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2} \text{Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{5 b^2 c^2 d^2 \sqrt{d+c^2 d x^2} \text{Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 8.00153, size = 990, normalized size = 1.44 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.415, size = 1404, normalized size = 2. \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{{\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]