Optimal. Leaf size=287 \[ \frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{c^2 d x^2+d}}-\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{c^2 d x^2+d}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{c^2 d x^2+d}}+\frac{3 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{2 d x \sqrt{c^2 d x^2+d}}+\frac{b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{d \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.430531, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5747, 5755, 5764, 5760, 4182, 2279, 2391, 203, 325} \[ \frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{c^2 d x^2+d}}-\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{c^2 d x^2+d}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{c^2 d x^2+d}}+\frac{3 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{2 d x \sqrt{c^2 d x^2+d}}+\frac{b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{d \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5755
Rule 5764
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 203
Rule 325
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}-\frac{1}{2} \left (3 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 d x \sqrt{d+c^2 d x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}-\frac{\left (3 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{d+c^2 d x^2}} \, dx}{2 d}-\frac{\left (b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d \sqrt{d+c^2 d x^2}}+\frac{\left (3 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 d x \sqrt{d+c^2 d x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}+\frac{b c^2 \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt{d+c^2 d x^2}}-\frac{\left (3 c^2 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 d x \sqrt{d+c^2 d x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}+\frac{b c^2 \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt{d+c^2 d x^2}}-\frac{\left (3 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 d x \sqrt{d+c^2 d x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}+\frac{b c^2 \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt{d+c^2 d x^2}}+\frac{3 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (3 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{\left (3 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 d x \sqrt{d+c^2 d x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}+\frac{b c^2 \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt{d+c^2 d x^2}}+\frac{3 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{\left (3 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{\left (3 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 d x \sqrt{d+c^2 d x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{d+c^2 d x^2}}+\frac{b c^2 \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{d \sqrt{d+c^2 d x^2}}+\frac{3 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d \sqrt{d+c^2 d x^2}}+\frac{3 b c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{d+c^2 d x^2}}-\frac{3 b c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 d \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 6.36783, size = 381, normalized size = 1.33 \[ \frac{b c^2 \left (-12 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+12 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \sqrt{c^2 x^2+1} \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \sqrt{c^2 x^2+1} \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )+16 \sqrt{c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-8 \sinh ^{-1}(c x)\right )}{8 d \sqrt{d \left (c^2 x^2+1\right )}}+\sqrt{d \left (c^2 x^2+1\right )} \left (-\frac{a c^2}{d^2 \left (c^2 x^2+1\right )}-\frac{a}{2 d^2 x^2}\right )+\frac{3 a c^2 \log \left (\sqrt{d} \sqrt{d \left (c^2 x^2+1\right )}+d\right )}{2 d^{3/2}}-\frac{3 a c^2 \log (x)}{2 d^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.172, size = 389, normalized size = 1.4 \begin{align*} -{\frac{a}{2\,d{x}^{2}}{\frac{1}{\sqrt{{c}^{2}d{x}^{2}+d}}}}-{\frac{3\,a{c}^{2}}{2\,d}{\frac{1}{\sqrt{{c}^{2}d{x}^{2}+d}}}}+{\frac{3\,a{c}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{{c}^{2}d{x}^{2}+d} \right ) } \right ){d}^{-{\frac{3}{2}}}}-{\frac{3\,b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{bc}{2\,{d}^{2}x}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{2\,{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ){x}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{b\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\arctan \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){c}^{2}}{\sqrt{{c}^{2}{x}^{2}+1}{d}^{2}}}+{\frac{3\,b{c}^{2}}{2\,{d}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it dilog} \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{3\,b{c}^{2}}{2\,{d}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it dilog} \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{3\,b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,{d}^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \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 \operatorname{arsinh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{7} + 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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