Optimal. Leaf size=412 \[ -\frac{2 b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{c^2 d x^2+d}}-\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.59453, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {5755, 5764, 5760, 4182, 2531, 2282, 6589, 5693, 4180, 2279, 2391} \[ -\frac{2 b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 b^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{c^2 d x^2+d}}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{c^2 d x^2+d}}-\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5755
Rule 5764
Rule 5760
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5693
Rule 4180
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}+\frac{\int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{d+c^2 d x^2}} \, dx}{d}-\frac{\left (2 b c \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}+\frac{\sqrt{1+c^2 x^2} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{1+c^2 x^2}} \, dx}{d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}-\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}-\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{\left (2 i b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}-\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}-\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{\left (2 b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}\\ &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d \sqrt{d+c^2 d x^2}}-\frac{4 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 i b^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{2 b^2 \sqrt{1+c^2 x^2} \text{Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}-\frac{2 b^2 \sqrt{1+c^2 x^2} \text{Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.56519, size = 568, normalized size = 1.38 \[ \frac{2 a b d \left (\sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-2 \sqrt{c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+\sinh ^{-1}(c x)\right )+b^2 d \left (2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+2 i \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-2 i \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )-2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2 \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2 \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 i \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-2 i \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x)^2\right )+a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log (c x)-a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+a^2 d}{d^2 \sqrt{c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x} \left ({c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}, 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{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{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{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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