Optimal. Leaf size=358 \[ -\frac{b c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{b c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{\sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.382681, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5737, 5661, 266, 63, 208, 5760, 4182, 2531, 2282, 6589} \[ -\frac{b c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{b c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{\sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5737
Rule 5661
Rule 266
Rule 63
Rule 208
Rule 5760
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^2} \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (c^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac{\left (c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{x \sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b^2 c^2 \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{\sqrt{1+c^2 x^2}}-\frac{b c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^2 \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt{1+c^2 x^2}}-\frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac{c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b^2 c^2 \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{\sqrt{1+c^2 x^2}}-\frac{b c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b c^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b^2 c^2 \sqrt{d+c^2 d x^2} \text{Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{b^2 c^2 \sqrt{d+c^2 d x^2} \text{Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 5.06922, size = 446, normalized size = 1.25 \[ \frac{1}{8} \left (\frac{2 a b c^2 \sqrt{c^2 d x^2+d} \left (4 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-4 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{\sqrt{c^2 x^2+1}}+\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-8 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+8 \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )-8 \text{PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x)^2 \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x)^2 \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+4 \sinh ^{-1}(c x) \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-4 \sinh ^{-1}(c x) \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+\sinh ^{-1}(c x)^2 \left (-\text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-\sinh ^{-1}(c x)^2 \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )+8 \log \left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{\sqrt{c^2 x^2+1}}-\frac{4 a^2 \sqrt{c^2 d x^2+d}}{x^2}-4 a^2 c^2 \sqrt{d} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+4 a^2 c^2 \sqrt{d} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.318, size = 870, normalized size = 2.4 \begin{align*} \text{result too large to display} \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^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{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{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x^{3}}\, 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{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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