Optimal. Leaf size=56 \[ -\frac {2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi }}-\frac {b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }} \]
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Rubi [A] time = 0.12, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5760, 4182, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi }} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 5760
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {\pi }}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {\pi }}+\frac {b \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {\pi }}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 96, normalized size = 1.71 \[ \frac {-a \log \left (\pi \left (\sqrt {c^2 x^2+1}+1\right )\right )+a \log (x)+b \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-b \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-b \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )}{\sqrt {\pi }} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\pi c^{2} x^{3} + \pi x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {\pi + \pi c^{2} x^{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 72, normalized size = 1.29 \[ -\frac {a \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\sqrt {\pi }}+\frac {b \left (4 \dilog \left (\frac {1}{c x +\sqrt {c^{2} x^{2}+1}}\right )-\dilog \left (\frac {1}{\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}}\right )\right )}{2 \sqrt {\pi }} \]
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 \[ b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{\sqrt {\pi + \pi c^{2} x^{2}} x}\,{d x} - \frac {a \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right )}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]
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 \[ \frac {\int \frac {a}{x \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x \sqrt {c^{2} x^{2} + 1}}\, dx}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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