Optimal. Leaf size=232 \[ \frac{3 b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (c^2 x^2+1\right )}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac{6 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}+\frac{2 b c^3 x}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 x}{12 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c}{2 d^3 x \left (c^2 x^2+1\right )^{3/2}} \]
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Rubi [A] time = 0.343745, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5747, 5755, 5720, 5461, 4182, 2279, 2391, 191, 192, 271} \[ \frac{3 b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (c^2 x^2+1\right )}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac{6 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}+\frac{2 b c^3 x}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 x}{12 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c}{2 d^3 x \left (c^2 x^2+1\right )^{3/2}} \]
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 192
Rule 271
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\left (3 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x^2 \left (1+c^2 x^2\right )^{5/2}} \, dx}{2 d^3}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}+\frac{\left (3 b c^3\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}-\frac{\left (2 b c^3\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{d^3}-\frac{\left (3 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (1+c^2 x^2\right )}+\frac{\left (b c^3\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^3}-\frac{\left (4 b c^3\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^3}+\frac{\left (3 b c^3\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^3}-\frac{\left (3 c^2\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d^2}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 b c^3 x}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (1+c^2 x^2\right )}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 b c^3 x}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (1+c^2 x^2\right )}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 b c^3 x}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}-\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 b c^3 x}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}\\ &=-\frac{b c}{2 d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 b c^3 x}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{3 b c^2 \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 b c^2 \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 0.846377, size = 353, normalized size = 1.52 \[ \frac{-18 c^2 \left (b \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )\right )+36 b c^2 \text{PolyLog}\left (2,\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right )+36 b c^2 \text{PolyLog}\left (2,\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+\frac{9 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 x^4+x^2}+\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 x^3+x\right )^2}+\frac{18 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{b}-\frac{18 \left (a+b \sinh ^{-1}(c x)\right )}{x^2}+18 a c^2 \log \left (c^2 x^2+1\right )+\frac{9 b c \left (2 c^2 x^2+1\right )}{x \sqrt{c^2 x^2+1}}+\frac{b c \left (8 c^4 x^4+12 c^2 x^2+3\right )}{x \left (c^2 x^2+1\right )^{3/2}}-\frac{18 b c \sqrt{c^2 x^2+1}}{x}-18 b c^2 \sinh ^{-1}(c x)^2+36 b c^2 \sinh ^{-1}(c x) \log \left (\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right )+36 b c^2 \sinh ^{-1}(c x) \log \left (\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )}{12 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.194, size = 575, normalized size = 2.5 \begin{align*} -{\frac{a}{2\,{d}^{3}{x}^{2}}}-3\,{\frac{{c}^{2}a\ln \left ( cx \right ) }{{d}^{3}}}-{\frac{{c}^{2}a}{4\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{{c}^{2}a}{{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) }}+{\frac{3\,{c}^{2}a\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{d}^{3}}}+{\frac{2\,{c}^{5}b{x}^{3}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{6}b{x}^{4}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{3\,{c}^{4}b{\it Arcsinh} \left ( cx \right ){x}^{2}}{2\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{b{c}^{3}x}{4\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{4}b{x}^{2}}{3\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{9\,{c}^{2}b{\it Arcsinh} \left ( cx \right ) }{4\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{bc}{2\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) x}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{2}b}{3\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{2\,{d}^{3} \left ({c}^{4}{x}^{4}+2\,{c}^{2}{x}^{2}+1 \right ){x}^{2}}}+3\,{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+ \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }{{d}^{3}}}+{\frac{3\,{c}^{2}b}{2\,{d}^{3}}{\it polylog} \left ( 2,- \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-3\,{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{3}}}-3\,{\frac{{c}^{2}b{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{3}}}-3\,{\frac{{c}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{3}}}-3\,{\frac{{c}^{2}b{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }{{d}^{3}}} \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}{4} \, a{\left (\frac{6 \, c^{4} x^{4} + 9 \, c^{2} x^{2} + 2}{c^{4} d^{3} x^{6} + 2 \, c^{2} d^{3} x^{4} + d^{3} x^{2}} - \frac{6 \, c^{2} \log \left (c^{2} x^{2} + 1\right )}{d^{3}} + \frac{12 \, c^{2} \log \left (x\right )}{d^{3}}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} 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^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} 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 )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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