Optimal. Leaf size=203 \[ \frac{b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 d x^2+d}}-\frac{b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}+\frac{c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{2 x \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.291936, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5747, 5764, 5760, 4182, 2279, 2391, 30} \[ \frac{b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 d x^2+d}}-\frac{b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}+\frac{c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{2 x \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5764
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \sqrt{d+c^2 d x^2}} \, dx &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}-\frac{1}{2} c^2 \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{d+c^2 d x^2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^2} \, dx}{2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}-\frac{\left (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 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}-\frac{\left (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 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}+\frac{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 )}{\sqrt{d+c^2 d x^2}}+\frac{\left (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 \sqrt{d+c^2 d x^2}}-\frac{\left (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 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}+\frac{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 )}{\sqrt{d+c^2 d x^2}}+\frac{\left (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 \sqrt{d+c^2 d x^2}}-\frac{\left (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 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{2 x \sqrt{d+c^2 d x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 d x^2}+\frac{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 )}{\sqrt{d+c^2 d x^2}}+\frac{b c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{d+c^2 d x^2}}-\frac{b c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 3.02521, size = 229, normalized size = 1.13 \[ \frac{\frac{b c^2 d^2 \left (c^2 x^2+1\right )^{3/2} \left (-4 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+4 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{\left (c^2 d x^2+d\right )^{3/2}}-\frac{4 a \sqrt{c^2 d x^2+d}}{x^2}+4 a c^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )-4 a c^2 \sqrt{d} \log (x)}{8 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.187, size = 380, normalized size = 1.9 \begin{align*} -{\frac{a}{2\,d{x}^{2}}\sqrt{{c}^{2}d{x}^{2}+d}}+{\frac{a{c}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{{c}^{2}d{x}^{2}+d} \right ) } \right ){\frac{1}{\sqrt{d}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,d \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{bc}{2\,dx}\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{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,d}\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}}}}+{\frac{b{c}^{2}}{2\,d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ){c}^{2}}{2\,d}\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}}}}-{\frac{b{c}^{2}}{2\,d}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,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^{2} d x^{5} + d x^{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{a + b \operatorname{asinh}{\left (c x \right )}}{x^{3} \sqrt{d \left (c^{2} x^{2} + 1\right )}}\, 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{b \operatorname{arsinh}\left (c x\right ) + a}{\sqrt{c^{2} d x^{2} + d} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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