Optimal. Leaf size=110 \[ \frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (c^2 x^2+1\right )}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {b c x}{2 d^2 \sqrt {c^2 x^2+1}}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \]
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Rubi [A] time = 0.18, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5755, 5720, 5461, 4182, 2279, 2391, 191} \[ -\frac {b \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (c^2 x^2+1\right )}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {b c x}{2 d^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2279
Rule 2391
Rule 4182
Rule 5461
Rule 5720
Rule 5755
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx &=\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{x \left (d+c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}+\frac {2 \operatorname {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}+\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \sinh ^{-1}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [B] time = 0.42, size = 234, normalized size = 2.13 \[ -\frac {\frac {a^2}{b}-\frac {a}{c^2 x^2+1}+a \log \left (c^2 x^2+1\right )+2 a \sinh ^{-1}(c x)-2 a \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+2 b \text {Li}_2\left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+2 b \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+\frac {b c x}{\sqrt {c^2 x^2+1}}-\frac {b \sinh ^{-1}(c x)}{c^2 x^2+1}+2 b \sinh ^{-1}(c x) \log \left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}+1\right )+2 b \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-2 b \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (c x\right ) + a}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 283, normalized size = 2.57 \[ \frac {a \ln \left (c x \right )}{d^{2}}+\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {b c x}{2 d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right )}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d^{2}}-\frac {b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d^{2}}+\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d^{2}}+\frac {b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d^{2}} \]
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 \[ \frac {1}{2} \, a {\left (\frac {1}{c^{2} d^{2} x^{2} + d^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{d^{2}} + \frac {2 \, \log \relax (x)}{d^{2}}\right )} + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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