Optimal. Leaf size=94 \[ \frac {a+b \sinh ^{-1}(c x)}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \text {ArcTan}(c x)}{\pi ^{3/2}}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}-\frac {b \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {b \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {5811, 5816,
4267, 2317, 2438, 209} \begin {gather*} \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 {ArcTan}(c x)}{\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 {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2317
Rule 2438
Rule 4267
Rule 5811
Rule 5816
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 {\text {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 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\pi ^{3/2}}+\frac {b \text {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 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}}+\frac {b \text {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 = 0.23, size = 143, normalized size = 1.52 \begin {gather*} \frac {\frac {a}{\sqrt {1+c^2 x^2}}+\frac {b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-2 b \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-b \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-b \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )}{\pi ^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.54, size = 157, normalized size = 1.67
method | result | size |
default | \(a \left (\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}\right )+\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {2 b \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}-\frac {b \dilog \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}-\frac {b \dilog \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}-\frac {b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {3}{2}}}\) | \(157\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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