Optimal. Leaf size=155 \[ -\frac {b c \pi ^{3/2}}{2 x}-b c^3 \pi ^{3/2} x+\frac {3}{2} c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-3 c^2 \pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac {3}{2} b c^2 \pi ^{3/2} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac {3}{2} b c^2 \pi ^{3/2} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5807, 5806,
5816, 4267, 2317, 2438, 8, 14} \begin {gather*} \frac {3}{2} \pi c^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-3 \pi ^{3/2} c^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\pi ^{3/2} (-b) c^3 x-\frac {3}{2} \pi ^{3/2} b c^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac {3}{2} \pi ^{3/2} b c^2 \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )-\frac {\pi ^{3/2} b c}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 2317
Rule 2438
Rule 4267
Rule 5806
Rule 5807
Rule 5816
Rubi steps
\begin {align*} \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} \left (3 c^2 \pi \right ) \int \frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1+c^2 x^2}{x^2} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {3}{2} c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (c^2+\frac {1}{x^2}\right ) \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int 1 \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \pi \sqrt {\pi +c^2 \pi x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {\left (3 c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \pi \sqrt {\pi +c^2 \pi x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 b c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \pi \sqrt {\pi +c^2 \pi x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 b c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c \pi \sqrt {\pi +c^2 \pi x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {b c^3 \pi x \sqrt {\pi +c^2 \pi x^2}}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b c^2 \pi \sqrt {\pi +c^2 \pi x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {3 b c^2 \pi \sqrt {\pi +c^2 \pi x^2} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.08, size = 292, normalized size = 1.88 \begin {gather*} \frac {\pi ^{3/2} \left (-8 b c^3 x^3-4 a \sqrt {1+c^2 x^2}+8 a c^2 x^2 \sqrt {1+c^2 x^2}+8 b c^2 x^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-b c^3 x^3 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-b c^2 x^2 \sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )+12 b c^2 x^2 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-12 b c^2 x^2 \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+12 a c^2 x^2 \log (x)-12 a c^2 x^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+12 b c^2 x^2 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-12 b c^2 x^2 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+4 b c x \sinh ^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-4 b \sinh ^{-1}(c x) \sinh ^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 10.32, size = 291, normalized size = 1.88
method | result | size |
default | \(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{2 \pi \,x^{2}}+\frac {3 c^{2} \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )}{2}\right )+b \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) \pi ^{\frac {3}{2}} c^{2}-b \,c^{3} \pi ^{\frac {3}{2}} x -\frac {b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {b c \,\pi ^{\frac {3}{2}}}{2 x}-\frac {b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right )}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}-\frac {3 b \,c^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {3 b \,c^{2} \pi ^{\frac {3}{2}} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{2}+\frac {3 b \,c^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{2}+\frac {3 b \,c^{2} \pi ^{\frac {3}{2}} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{2}\) | \(291\) |
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*} \pi ^{\frac {3}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{3}}\, dx + \int \frac {a c^{2} \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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