Optimal. Leaf size=179 \[ -\frac {23}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-2 \pi ^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-b \pi ^{5/2} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+b \pi ^{5/2} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5808, 5806,
5816, 4267, 2317, 2438, 8, 200} \begin {gather*} \frac {1}{5} \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi ^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-2 \pi ^{5/2} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{25} \pi ^{5/2} b c^5 x^5-\frac {11}{45} \pi ^{5/2} b c^3 x^3-\pi ^{5/2} b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\pi ^{5/2} b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )-\frac {23}{15} \pi ^{5/2} b c x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 200
Rule 2317
Rule 2438
Rule 4267
Rule 5806
Rule 5808
Rule 5816
Rubi steps
\begin {align*} \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi ^2 \int \frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \sqrt {1+c^2 x^2}}-\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {8 b c \pi ^2 x \sqrt {\pi +c^2 \pi x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {11 b c^3 \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2}}{45 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^5 \sqrt {\pi +c^2 \pi x^2}}{25 \sqrt {1+c^2 x^2}}+\pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (\pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (b c \pi ^2 \sqrt {\pi +c^2 \pi x^2}\right ) \int 1 \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {23 b c \pi ^2 x \sqrt {\pi +c^2 \pi x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {11 b c^3 \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2}}{45 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^5 \sqrt {\pi +c^2 \pi x^2}}{25 \sqrt {1+c^2 x^2}}+\pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (\pi ^2 \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 )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {23 b c \pi ^2 x \sqrt {\pi +c^2 \pi x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {11 b c^3 \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2}}{45 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^5 \sqrt {\pi +c^2 \pi x^2}}{25 \sqrt {1+c^2 x^2}}+\pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \pi ^2 \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 (b \pi ^2 \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 )}{\sqrt {1+c^2 x^2}}+\frac {\left (b \pi ^2 \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 )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {23 b c \pi ^2 x \sqrt {\pi +c^2 \pi x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {11 b c^3 \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2}}{45 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^5 \sqrt {\pi +c^2 \pi x^2}}{25 \sqrt {1+c^2 x^2}}+\pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \pi ^2 \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 (b \pi ^2 \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 )}{\sqrt {1+c^2 x^2}}+\frac {\left (b \pi ^2 \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 )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {23 b c \pi ^2 x \sqrt {\pi +c^2 \pi x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {11 b c^3 \pi ^2 x^3 \sqrt {\pi +c^2 \pi x^2}}{45 \sqrt {1+c^2 x^2}}-\frac {b c^5 \pi ^2 x^5 \sqrt {\pi +c^2 \pi x^2}}{25 \sqrt {1+c^2 x^2}}+\pi ^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \pi ^2 \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 {b \pi ^2 \sqrt {\pi +c^2 \pi x^2} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b \pi ^2 \sqrt {\pi +c^2 \pi x^2} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 257, normalized size = 1.44 \begin {gather*} \frac {1}{225} \pi ^{5/2} \left (-345 b c x-55 b c^3 x^3-9 b c^5 x^5+345 a \sqrt {1+c^2 x^2}+165 a c^2 x^2 \sqrt {1+c^2 x^2}+45 a c^4 x^4 \sqrt {1+c^2 x^2}+345 b \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+165 b c^2 x^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+45 b c^4 x^4 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+225 b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-225 b \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+225 a \log (x)-225 a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+225 b \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-225 b \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.79, size = 284, normalized size = 1.59
method | result | size |
default | \(a \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{5}}{25}-\frac {11 b \,c^{3} \pi ^{\frac {5}{2}} x^{3}}{45}+\frac {23 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}-b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {11 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}}{15}-\frac {23 b c \,\pi ^{\frac {5}{2}} x}{15}+\frac {b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}}{5}+b \,\pi ^{\frac {5}{2}} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {5}{2}} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )\) | \(284\) |
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 {5}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int 2 a c^{2} x \sqrt {c^{2} x^{2} + 1}\, dx + \int a c^{4} x^{3} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 2 b c^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, 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 )}^{5/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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