Optimal. Leaf size=338 \[ -\frac {2 b \sqrt {c^2 d x^2+d} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {2 b \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {2 a b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+\frac {2 b^2 \sqrt {c^2 d x^2+d} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 b^2 \sqrt {c^2 d x^2+d} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+2 b^2 \sqrt {c^2 d x^2+d}-\frac {2 b^2 c x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.34, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5742, 5760, 4182, 2531, 2282, 6589, 5653, 261} \[ -\frac {2 b \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {2 b \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {2 b^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 b^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 a b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+2 b^2 \sqrt {c^2 d x^2+d}-\frac {2 b^2 c x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 2282
Rule 2531
Rule 4182
Rule 5653
Rule 5742
Rule 5760
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} \, dx &=\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\sqrt {d+c^2 d x^2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b c \sqrt {d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {2 a b c x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\sqrt {d+c^2 d x^2} \operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c \sqrt {d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {2 a b c x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {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 (2 b \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 (2 b \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 (2 b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=2 b^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {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 {2 b \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 {2 b \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 (2 b^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 (2 b^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}}\\ &=2 b^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {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 {2 b \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 {2 b \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 (2 b^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 (2 b^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}}\\ &=2 b^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {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 {2 b \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 {2 b \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 {2 b^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 {2 b^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*}
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Mathematica [A] time = 1.17, size = 352, normalized size = 1.04 \[ a^2 \sqrt {c^2 d x^2+d}-a^2 \sqrt {d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+d\right )+a^2 \sqrt {d} \log (c x)+\frac {2 a b \sqrt {c^2 d x^2+d} \left (\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {b^2 \sqrt {c^2 d x^2+d} \left (2 \sqrt {c^2 x^2+1}+\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-2 \sinh ^{-1}(c x) \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+2 \text {Li}_3\left (-e^{-\sinh ^{-1}(c x)}\right )-2 \text {Li}_3\left (e^{-\sinh ^{-1}(c x)}\right )-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x)^2 \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 823, normalized size = 2.43 \[ -\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right ) a^{2}+a^{2} \sqrt {c^{2} d \,x^{2}+d}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} x^{2}}{c^{2} x^{2}+1}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{c^{2} x^{2}+1}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{c^{2} x^{2}+1}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c x}{\sqrt {c^{2} x^{2}+1}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{2} x^{2}+1}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (\sqrt {d} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - \sqrt {c^{2} d x^{2} + d}\right )} a^{2} + \int \frac {\sqrt {c^{2} d x^{2} + d} b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x} + \frac {2 \, \sqrt {c^{2} d x^{2} + d} a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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