Optimal. Leaf size=128 \[ -\frac {d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+c^2 d \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{2} b c^2 d \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-\frac {b c d \sqrt {c^2 x^2+1}}{2 x}+\frac {1}{2} b c^2 d \sinh ^{-1}(c x) \]
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Rubi [A] time = 0.13, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5728, 277, 215, 5659, 3716, 2190, 2279, 2391} \[ \frac {1}{2} b c^2 d \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-\frac {d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+c^2 d \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d \sqrt {c^2 x^2+1}}{2 x}+\frac {1}{2} b c^2 d \sinh ^{-1}(c x) \]
Warning: Unable to verify antiderivative.
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Rule 215
Rule 277
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5728
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} (b c d) \int \frac {\sqrt {1+c^2 x^2}}{x^2} \, dx+\left (c^2 d\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{2 x}-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\left (c^2 d\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{2} \left (b c^3 d\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{2 x}+\frac {1}{2} b c^2 d \sinh ^{-1}(c x)-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}-\left (2 c^2 d\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{2 x}+\frac {1}{2} b c^2 d \sinh ^{-1}(c x)-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+c^2 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\left (b c^2 d\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{2 x}+\frac {1}{2} b c^2 d \sinh ^{-1}(c x)-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+c^2 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \left (b c^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=-\frac {b c d \sqrt {1+c^2 x^2}}{2 x}+\frac {1}{2} b c^2 d \sinh ^{-1}(c x)-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+c^2 d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac {1}{2} b c^2 d \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 111, normalized size = 0.87 \[ a c^2 d \log (x)-\frac {a d}{2 x^2}+\frac {1}{2} b c^2 d \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac {b c d \sqrt {c^2 x^2+1}}{2 x}-\frac {1}{2} b c^2 d \sinh ^{-1}(c x)^2+b c^2 d \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac {b d \sinh ^{-1}(c x)}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{2} d x^{2} + a d + {\left (b c^{2} d x^{2} + b d\right )} \operatorname {arsinh}\left (c x\right )}{x^{3}}, x\right ) \]
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 [A] time = 0.24, size = 175, normalized size = 1.37 \[ c^{2} d a \ln \left (c x \right )-\frac {d a}{2 x^{2}}-\frac {c^{2} d b \arcsinh \left (c x \right )^{2}}{2}-\frac {b c d \sqrt {c^{2} x^{2}+1}}{2 x}+\frac {c^{2} d b}{2}-\frac {d b \arcsinh \left (c x \right )}{2 x^{2}}+c^{2} d b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+c^{2} d b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+c^{2} d b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+c^{2} d b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) \]
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 \[ b c^{2} d \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x}\,{d x} + a c^{2} d \log \relax (x) - \frac {1}{2} \, b d {\left (\frac {\sqrt {c^{2} x^{2} + 1} c}{x} + \frac {\operatorname {arsinh}\left (c x\right )}{x^{2}}\right )} - \frac {a d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x^3} \,d x \]
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 \[ d \left (\int \frac {a}{x^{3}}\, dx + \int \frac {a c^{2}}{x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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