Optimal. Leaf size=94 \[ \frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2}}-\frac {b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac {b \tan ^{-1}(c x)}{\pi ^{3/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 119, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5755, 5760, 4182, 2279, 2391, 203} \[ -\frac {b \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {b \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2}}-\frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{\pi \sqrt {\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 203
Rule 2279
Rule 2391
Rule 4182
Rule 5755
Rule 5760
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=\frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx}{\pi }-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{\pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}+\frac {b \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}\\ &=\frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac {b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 143, normalized size = 1.52 \[ \frac {\frac {a}{\sqrt {c^2 x^2+1}}-a \log \left (\pi \left (\sqrt {c^2 x^2+1}+1\right )\right )+a \log (x)+\frac {b \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+b \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-b \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-b \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-2 b \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{\pi ^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\pi ^{2} c^{4} x^{5} + 2 \, \pi ^{2} c^{2} x^{3} + \pi ^{2} x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 156, normalized size = 1.66 \[ \frac {a}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {a \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}+\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {2 b \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}-\frac {b \dilog \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}-\frac {b \dilog \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}} \]
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 \[ -a {\left (\frac {\operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right )}{\pi ^{\frac {3}{2}}} - \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}}}\right )} + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {a}{c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} + x \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} + x \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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