Optimal. Leaf size=398 \[ -\frac{b^2 c d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{3 b c^3 d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3}{2} c^2 d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{c d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{c^2 x^2+1}}+\frac{c d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+b c d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{2 b c d \sqrt{c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 c^2 d x \sqrt{c^2 d x^2+d}-\frac{5 b^2 c d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{4 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.421631, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {5739, 5682, 5675, 5661, 321, 215, 5726, 5659, 3716, 2190, 2279, 2391, 195} \[ \frac{b^2 c d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{c^2 x^2+1}}-\frac{3 b c^3 d x^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3}{2} c^2 d x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{c d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{c^2 x^2+1}}-\frac{c d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 x^2+1}}+b c d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{2 b c d \sqrt{c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 c^2 d x \sqrt{c^2 d x^2+d}-\frac{5 b^2 c d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{4 \sqrt{c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Rule 5739
Rule 5682
Rule 5675
Rule 5661
Rule 321
Rule 215
Rule 5726
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 195
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (3 c^2 d\right ) \int \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{\left (2 b c d \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{\sqrt{1+c^2 x^2}}\\ &=b c d \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{2} c^2 d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{\left (2 b c d \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (3 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{\sqrt{1+c^2 x^2}}-\frac{\left (3 b c^3 d \sqrt{d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{1}{2} b^2 c^2 d x \sqrt{d+c^2 d x^2}-\frac{3 b c^3 d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+b c d \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{2} c^2 d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{1+c^2 x^2}}+\frac{\left (2 b c d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 c^4 d \sqrt{d+c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d+c^2 d x^2}-\frac{b^2 c d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{2 \sqrt{1+c^2 x^2}}-\frac{3 b c^3 d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+b c d \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{2} c^2 d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{1+c^2 x^2}}-\frac{\left (4 b c d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (3 b^2 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d+c^2 d x^2}-\frac{5 b^2 c d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt{1+c^2 x^2}}-\frac{3 b c^3 d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+b c d \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{2} c^2 d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{1+c^2 x^2}}+\frac{2 b c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (2 b^2 c d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d+c^2 d x^2}-\frac{5 b^2 c d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt{1+c^2 x^2}}-\frac{3 b c^3 d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+b c d \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{2} c^2 d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{1+c^2 x^2}}+\frac{2 b c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (b^2 c d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ &=\frac{1}{4} b^2 c^2 d x \sqrt{d+c^2 d x^2}-\frac{5 b^2 c d \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt{1+c^2 x^2}}-\frac{3 b c^3 d x^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+b c d \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3}{2} c^2 d x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}}-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt{1+c^2 x^2}}+\frac{2 b c d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}+\frac{b^2 c d \sqrt{d+c^2 d x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 3.03117, size = 369, normalized size = 0.93 \[ \frac{-8 b^2 d \sqrt{c^2 d x^2+d} \left (3 c x \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x) \left (3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x \left (\sinh ^{-1}(c x)+3\right ) \sinh ^{-1}(c x)-6 c x \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )\right )+36 a^2 c d^{3/2} x \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+12 a^2 d \left (c^2 x^2-2\right ) \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+24 a b d \sqrt{c^2 d x^2+d} \left (-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right )-6 a b c d x \sqrt{c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )+b^2 c d x \sqrt{c^2 d x^2+d} \left (4 \sinh ^{-1}(c x)^3+\left (6 \sinh ^{-1}(c x)^2+3\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )-6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 x \sqrt{c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.259, size = 954, 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{{\left (a^{2} c^{2} d x^{2} + a^{2} d +{\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} + a b d\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{x^{2}}, 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{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, 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{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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