Optimal. Leaf size=146 \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (c^2 x^2+1\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac{4 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{b c}{2 d^2 x \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.259758, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5747, 5755, 5720, 5461, 4182, 2279, 2391, 191, 271} \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (c^2 x^2+1\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac{4 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac{b c}{2 d^2 x \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5755
Rule 5720
Rule 5461
Rule 4182
Rule 2279
Rule 2391
Rule 191
Rule 271
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\left (2 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx+\frac{(b c) \int \frac{1}{x^2 \left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac{\left (2 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac{b c}{2 d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}-\frac{\left (4 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{\left (2 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}-\frac{\left (2 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2}\\ &=-\frac{b c}{2 d^2 x \sqrt{1+c^2 x^2}}-\frac{c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \left (1+c^2 x^2\right )}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left (1+c^2 x^2\right )}+\frac{4 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}+\frac{b c^2 \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac{b c^2 \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}\\ \end{align*}
Mathematica [B] time = 0.481177, size = 326, normalized size = 2.23 \[ \frac{4 b c^2 \text{PolyLog}\left (2,\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right )+4 b c^2 \text{PolyLog}\left (2,\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-2 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+\frac{2 a^2 c^2}{b}+\frac{a}{c^2 x^4+x^2}+2 a c^2 \log \left (c^2 x^2+1\right )+4 a c^2 \sinh ^{-1}(c x)-4 a c^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac{2 a}{x^2}+\frac{2 b c^3 x}{\sqrt{c^2 x^2+1}}-\frac{2 b c \sqrt{c^2 x^2+1}}{x}+\frac{b c}{x \sqrt{c^2 x^2+1}}+\frac{b \sinh ^{-1}(c x)}{c^2 x^4+x^2}+4 b c^2 \sinh ^{-1}(c x) \log \left (\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right )+4 b c^2 \sinh ^{-1}(c x) \log \left (\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-4 b c^2 \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac{2 b \sinh ^{-1}(c x)}{x^2}}{2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.112, size = 311, normalized size = 2.1 \begin{align*} -{\frac{a}{2\,{d}^{2}{x}^{2}}}-2\,{\frac{{c}^{2}a\ln \left ( cx \right ) }{{d}^{2}}}-{\frac{{c}^{2}a}{2\,{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{{c}^{2}a\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{d}^{2}}}-{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) }{{d}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{bc}{2\,{d}^{2}x}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{2\,{d}^{2}{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{{d}^{2}}}+{\frac{{c}^{2}b}{{d}^{2}}{\it polylog} \left ( 2,- \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-2\,{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{2}}}-2\,{\frac{{c}^{2}b{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{2}}}-2\,{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{2}}}-2\,{\frac{{c}^{2}b{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{2}}} \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*} \frac{1}{2} \, a{\left (\frac{2 \, c^{2} \log \left (c^{2} x^{2} + 1\right )}{d^{2}} - \frac{4 \, c^{2} \log \left (x\right )}{d^{2}} - \frac{2 \, c^{2} x^{2} + 1}{c^{2} d^{2} x^{4} + d^{2} x^{2}}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{7} + 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}\,{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{b \operatorname{arsinh}\left (c x\right ) + a}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \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 )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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