Optimal. Leaf size=172 \[ -\frac{1}{2} b d^2 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+\frac{1}{4} d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{16} b c d^2 x \left (c^2 x^2+1\right )^{3/2}-\frac{11}{32} b c d^2 x \sqrt{c^2 x^2+1}-\frac{11}{32} b d^2 \sinh ^{-1}(c x) \]
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Rubi [A] time = 0.202739, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5726, 5659, 3716, 2190, 2279, 2391, 195, 215} \[ \frac{1}{2} b d^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{4} d^2 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{16} b c d^2 x \left (c^2 x^2+1\right )^{3/2}-\frac{11}{32} b c d^2 x \sqrt{c^2 x^2+1}-\frac{11}{32} b d^2 \sinh ^{-1}(c x) \]
Warning: Unable to verify antiderivative.
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Rule 5726
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+d \int \frac{\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac{1}{4} \left (b c d^2\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx\\ &=-\frac{1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}+\frac{1}{2} d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+d^2 \int \frac{a+b \sinh ^{-1}(c x)}{x} \, dx-\frac{1}{16} \left (3 b c d^2\right ) \int \sqrt{1+c^2 x^2} \, dx-\frac{1}{2} \left (b c d^2\right ) \int \sqrt{1+c^2 x^2} \, dx\\ &=-\frac{11}{32} b c d^2 x \sqrt{1+c^2 x^2}-\frac{1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}+\frac{1}{2} d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+d^2 \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac{1}{32} \left (3 b c d^2\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{4} \left (b c d^2\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{11}{32} b c d^2 x \sqrt{1+c^2 x^2}-\frac{1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac{11}{32} b d^2 \sinh ^{-1}(c x)+\frac{1}{2} d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}-\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{11}{32} b c d^2 x \sqrt{1+c^2 x^2}-\frac{1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac{11}{32} b d^2 \sinh ^{-1}(c x)+\frac{1}{2} d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\left (b d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{11}{32} b c d^2 x \sqrt{1+c^2 x^2}-\frac{1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac{11}{32} b d^2 \sinh ^{-1}(c x)+\frac{1}{2} d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=-\frac{11}{32} b c d^2 x \sqrt{1+c^2 x^2}-\frac{1}{16} b c d^2 x \left (1+c^2 x^2\right )^{3/2}-\frac{11}{32} b d^2 \sinh ^{-1}(c x)+\frac{1}{2} d^2 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^2 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac{1}{2} b d^2 \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.200318, size = 173, normalized size = 1.01 \[ \frac{d^2 \left (16 b^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-16 a^2+8 a b c^4 x^4+32 a b c^2 x^2+b \sinh ^{-1}(c x) \left (-32 a+b \left (8 c^4 x^4+32 c^2 x^2+13\right )+32 b \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )+32 a b \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+24 a b-2 b^2 c^3 x^3 \sqrt{c^2 x^2+1}-13 b^2 c x \sqrt{c^2 x^2+1}-16 b^2 \sinh ^{-1}(c x)^2\right )}{32 b} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.116, size = 231, normalized size = 1.3 \begin{align*}{\frac{{d}^{2}a{c}^{4}{x}^{4}}{4}}+{d}^{2}a{c}^{2}{x}^{2}+{d}^{2}a\ln \left ( cx \right ) +{\frac{{d}^{2}b{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{4}}{4}}+{d}^{2}b{\it Arcsinh} \left ( cx \right ){c}^{2}{x}^{2}+{\frac{13\,b{d}^{2}{\it Arcsinh} \left ( cx \right ) }{32}}-{\frac{{d}^{2}b{c}^{3}{x}^{3}}{16}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{13\,{d}^{2}bcx}{32}\sqrt{{c}^{2}{x}^{2}+1}}+{d}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +{d}^{2}b{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) -{\frac{{d}^{2}b \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{2}}+{d}^{2}b{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +{d}^{2}b{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) \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 c^{4} d^{2} x^{4} + a c^{2} d^{2} x^{2} + a d^{2} \log \left (x\right ) + \int b c^{4} d^{2} x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \, b c^{2} d^{2} x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \frac{b d^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x}\,{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{a c^{4} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} + 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arsinh}\left (c x\right )}{x}, 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*} d^{2} \left (\int \frac{a}{x}\, dx + \int 2 a c^{2} x\, dx + \int a c^{4} x^{3}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{x}\, dx + \int 2 b c^{2} x \operatorname{asinh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \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 )}^{2}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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