Optimal. Leaf size=363 \[ -\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.328811, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {5690, 5687, 5714, 3718, 2190, 2531, 2282, 6589, 5717, 260} \[ -\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5690
Rule 5687
Rule 5714
Rule 3718
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 260
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^2} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \int \frac{\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \tanh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{1+a^2 x^2} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.535376, size = 195, normalized size = 0.54 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \left (12 \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a x)}\right )+3 \log \left (a^2 x^2+1\right )+\frac{4 a x \sinh ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{2 a x \sinh ^{-1}(a x)^3}{\left (a^2 x^2+1\right )^{3/2}}+\frac{3 \sinh ^{-1}(a x)^2}{a^2 x^2+1}-\frac{6 a x \sinh ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-4 \sinh ^{-1}(a x)^3-12 \sinh ^{-1}(a x)^2 \log \left (e^{-2 \sinh ^{-1}(a x)}+1\right )\right )}{6 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.184, size = 550, normalized size = 1.5 \begin{align*}{\frac{{\it Arcsinh} \left ( ax \right ) }{ \left ( 18\,{x}^{6}{a}^{6}+60\,{x}^{4}{a}^{4}+66\,{a}^{2}{x}^{2}+24 \right ) a{c}^{3}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ( 2\,{x}^{3}{a}^{3}-2\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}+3\,ax-2\,\sqrt{{a}^{2}{x}^{2}+1} \right ) \left ( -6\,{a}^{4}{x}^{4}{\it Arcsinh} \left ( ax \right ) -6\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}{a}^{3}{x}^{3}-6\,{x}^{4}{a}^{4}-6\,{a}^{3}{x}^{3}\sqrt{{a}^{2}{x}^{2}+1}+6\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}-12\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) -9\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax-18\,{a}^{2}{x}^{2}-6\,ax\sqrt{{a}^{2}{x}^{2}+1}+8\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}-6\,{\it Arcsinh} \left ( ax \right ) -12 \right ) }-2\,{\frac{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }\ln \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) }{\sqrt{{a}^{2}{x}^{2}+1}a{c}^{3}}}+{\frac{1}{a{c}^{3}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+ \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{3\,a{c}^{3}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-2\,{\frac{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+ \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{\sqrt{{a}^{2}{x}^{2}+1}a{c}^{3}}}-2\,{\frac{\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 2,- \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{\sqrt{{a}^{2}{x}^{2}+1}a{c}^{3}}}+{\frac{1}{a{c}^{3}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 3,- \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{3}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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