Optimal. Leaf size=148 \[ -\frac{b \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}+\frac{b \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi c^2 x^2+\pi }}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac{2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{5/2}}-\frac{b c x}{6 \pi ^{5/2} \left (c^2 x^2+1\right )}-\frac{7 b \tan ^{-1}(c x)}{6 \pi ^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.339061, antiderivative size = 187, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5755, 5760, 4182, 2279, 2391, 203, 199} \[ -\frac{b \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}+\frac{b \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi c^2 x^2+\pi }}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac{2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{5/2}}-\frac{b c x}{6 \pi ^2 \sqrt{c^2 x^2+1} \sqrt{\pi c^2 x^2+\pi }}-\frac{7 b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{6 \pi ^2 \sqrt{\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5755
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 203
Rule 199
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{\pi }-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{3 \pi ^2 \sqrt{\pi +c^2 \pi x^2}}\\ &=-\frac{b c x}{6 \pi ^2 \sqrt{1+c^2 x^2} \sqrt{\pi +c^2 \pi x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi +c^2 \pi x^2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{\pi +c^2 \pi x^2}} \, dx}{\pi ^2}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{6 \pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{\pi ^2 \sqrt{\pi +c^2 \pi x^2}}\\ &=-\frac{b c x}{6 \pi ^2 \sqrt{1+c^2 x^2} \sqrt{\pi +c^2 \pi x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 \pi ^2 \sqrt{\pi +c^2 \pi x^2}}+\frac{\operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{5/2}}\\ &=-\frac{b c x}{6 \pi ^2 \sqrt{1+c^2 x^2} \sqrt{\pi +c^2 \pi x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 \pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{5/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{5/2}}\\ &=-\frac{b c x}{6 \pi ^2 \sqrt{1+c^2 x^2} \sqrt{\pi +c^2 \pi x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 \pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}\\ &=-\frac{b c x}{6 \pi ^2 \sqrt{1+c^2 x^2} \sqrt{\pi +c^2 \pi x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 \pi ^2 \sqrt{\pi +c^2 \pi x^2}}-\frac{2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}-\frac{b \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}+\frac{b \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.822381, size = 209, normalized size = 1.41 \[ \frac{6 b \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-6 b \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\frac{6 a}{\sqrt{c^2 x^2+1}}+\frac{2 a}{\left (c^2 x^2+1\right )^{3/2}}-6 a \log \left (\pi \left (\sqrt{c^2 x^2+1}+1\right )\right )+6 a \log (x)-\frac{b c x}{c^2 x^2+1}+\frac{6 b c^2 x^2 \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}+\frac{8 b \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}+6 b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-6 b \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-14 b \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{6 \pi ^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.171, size = 220, normalized size = 1.5 \begin{align*}{\frac{a}{3\,\pi } \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{-{\frac{3}{2}}}}+{\frac{a}{{\pi }^{2}}{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}}-{\frac{a}{{\pi }^{{\frac{5}{2}}}}{\it Artanh} \left ({\sqrt{\pi }{\frac{1}{\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ) }+{\frac{b{\it Arcsinh} \left ( cx \right ){x}^{2}{c}^{2}}{{\pi }^{{\frac{5}{2}}}} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{bcx}{6\,{\pi }^{5/2} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{4\,b{\it Arcsinh} \left ( cx \right ) }{3\,{\pi }^{5/2}} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,b}{3\,{\pi }^{5/2}}\arctan \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b}{{\pi }^{{\frac{5}{2}}}}{\it dilog} \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b}{{\pi }^{{\frac{5}{2}}}}{\it dilog} \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{b{\it Arcsinh} \left ( cx \right ) }{{\pi }^{{\frac{5}{2}}}}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) } \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}{3} \, a{\left (\frac{3 \, \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right )}{\pi ^{\frac{5}{2}}} - \frac{1}{\pi{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}} - \frac{3}{\pi ^{2} \sqrt{\pi + \pi c^{2} x^{2}}}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{5}{2}} x}\,{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{\sqrt{\pi + \pi c^{2} x^{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{\pi ^{3} c^{6} x^{7} + 3 \, \pi ^{3} c^{4} x^{5} + 3 \, \pi ^{3} c^{2} x^{3} + \pi ^{3} x}, 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{b \operatorname{arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]