Optimal. Leaf size=194 \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{b^2 c^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac{b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{b^2 c^2 \log (x)}{d} \]
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Rubi [A] time = 0.388449, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5747, 5720, 5461, 4182, 2531, 2282, 6589, 5723, 29} \[ \frac{b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac{b^2 c^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac{b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}-\frac{b c \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac{2 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{b^2 c^2 \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5720
Rule 5461
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5723
Rule 29
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}-c^2 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx+\frac{(b c) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \sqrt{1+c^2 x^2}} \, dx}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}-\frac{c^2 \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{\left (b^2 c^2\right ) \int \frac{1}{x} \, dx}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{b^2 c^2 \log (x)}{d}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac{b^2 c^2 \log (x)}{d}+\frac{\left (2 b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac{\left (2 b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac{b^2 c^2 \log (x)}{d}+\frac{b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac{b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac{b^2 c^2 \log (x)}{d}+\frac{b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac{b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ &=-\frac{b c \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac{b^2 c^2 \log (x)}{d}+\frac{b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac{b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac{b^2 c^2 \text{Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac{b^2 c^2 \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ \end{align*}
Mathematica [C] time = 1.06281, size = 419, normalized size = 2.16 \[ -\frac{-4 a b c^2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-4 a b c^2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+2 a b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+2 b^2 c^2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+2 b^2 c^2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )-b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )-a^2 c^2 \log \left (c^2 x^2+1\right )+2 a^2 c^2 \log (x)+\frac{a^2}{x^2}+\frac{2 a b c \sqrt{c^2 x^2+1}}{x}-4 a b c^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-4 a b c^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+4 a b c^2 \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac{2 a b \sinh ^{-1}(c x)}{x^2}+\frac{2 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{x}-2 b^2 c^2 \log (c x)-\frac{4}{3} b^2 c^2 \sinh ^{-1}(c x)^3-2 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )+2 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{12} i \pi ^3 b^2 c^2+\frac{b^2 \sinh ^{-1}(c x)^2}{x^2}}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.148, size = 719, normalized size = 3.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (\frac{c^{2} \log \left (c^{2} x^{2} + 1\right )}{d} - \frac{2 \, c^{2} \log \left (x\right )}{d} - \frac{1}{d x^{2}}\right )} a^{2} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{2} d x^{5} + d x^{3}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{2} d x^{5} + d 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^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{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*} \frac{\int \frac{a^{2}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx}{d} \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{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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