Optimal. Leaf size=262 \[ -\frac{b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{b c x}{6 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{7 b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{6 d^2 \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 0.415274, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5755, 5764, 5760, 4182, 2279, 2391, 203, 199} \[ -\frac{b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{b c x}{6 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{7 b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{6 d^2 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5755
Rule 5764
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 203
Rule 199
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx &=\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx}{d}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c x}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{d+c^2 d x^2}} \, dx}{d^2}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{1}{1+c^2 x^2} \, dx}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c x}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{d+c^2 d x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\sqrt{1+c^2 x^2} \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c x}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{d+c^2 d x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\sqrt{1+c^2 x^2} \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c x}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{d+c^2 d x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c x}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{d+c^2 d x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{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^2 \sqrt{d+c^2 d x^2}}-\frac{\left (b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c x}{6 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{d+c^2 d x^2}}-\frac{7 b \sqrt{1+c^2 x^2} \tan ^{-1}(c x)}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{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^2 \sqrt{d+c^2 d x^2}}-\frac{b \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{b \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.2341, size = 247, normalized size = 0.94 \[ \frac{\frac{b d^2 \left (c^2 x^2+1\right )^{3/2} \left (6 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-6 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-\frac{c x}{c^2 x^2+1}+\frac{6 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{2 \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}+6 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-6 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-14 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{\left (c^2 d x^2+d\right )^{3/2}}+\frac{2 a \left (3 c^2 x^2+4\right ) \sqrt{c^2 d x^2+d}}{\left (c^2 x^2+1\right )^2}-6 a \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+6 a \sqrt{d} \log (x)}{6 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.146, size = 364, normalized size = 1.4 \begin{align*}{\frac{a}{3\,d} \left ({c}^{2}d{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}+{\frac{a}{{d}^{2}}{\frac{1}{\sqrt{{c}^{2}d{x}^{2}+d}}}}-{a\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{{c}^{2}d{x}^{2}+d} \right ) } \right ){d}^{-{\frac{5}{2}}}}+{\frac{b{\it Arcsinh} \left ( cx \right ){x}^{2}{c}^{2}}{{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{bcx}{6\,{d}^{3}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,b{\it Arcsinh} \left ( cx \right ) }{3\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{7\,b}{3\,{d}^{3}}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\arctan \left ( cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{b}{{d}^{3}}\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{b}{{d}^{3}}\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{b{\it Arcsinh} \left ( cx \right ) }{{d}^{3}}\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^{6} d^{3} x^{7} + 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} + d^{3} x}, 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{5}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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