Optimal. Leaf size=179 \[ \frac {1}{5} \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi ^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-2 \pi ^{5/2} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{25} \pi ^{5/2} b c^5 x^5-\frac {11}{45} \pi ^{5/2} b c^3 x^3-\pi ^{5/2} b \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\pi ^{5/2} b \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )-\frac {23}{15} \pi ^{5/2} b c x \]
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Rubi [A] time = 0.43, antiderivative size = 329, normalized size of antiderivative = 1.84, 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 = {5744, 5742, 5760, 4182, 2279, 2391, 8, 194} \[ -\frac {\pi ^2 b \sqrt {\pi c^2 x^2+\pi } \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {\pi ^2 b \sqrt {\pi c^2 x^2+\pi } \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {1}{5} \left (\pi c^2 x^2+\pi \right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} \pi \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\pi ^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {2 \pi ^2 \sqrt {\pi c^2 x^2+\pi } \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {\pi ^2 b c^5 x^5 \sqrt {\pi c^2 x^2+\pi }}{25 \sqrt {c^2 x^2+1}}-\frac {11 \pi ^2 b c^3 x^3 \sqrt {\pi c^2 x^2+\pi }}{45 \sqrt {c^2 x^2+1}}-\frac {23 \pi ^2 b c x \sqrt {\pi c^2 x^2+\pi }}{15 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 2279
Rule 2391
Rule 4182
Rule 5742
Rule 5744
Rule 5760
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.36, size = 257, normalized size = 1.44 \[ \frac {1}{225} \pi ^{5/2} \left (165 a c^2 x^2 \sqrt {c^2 x^2+1}+345 a \sqrt {c^2 x^2+1}-225 a \log \left (\pi \left (\sqrt {c^2 x^2+1}+1\right )\right )+45 a c^4 x^4 \sqrt {c^2 x^2+1}+225 a \log (x)-9 b c^5 x^5-55 b c^3 x^3+165 b c^2 x^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+345 b \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+45 b c^4 x^4 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+225 b \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-225 b \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-345 b c x+225 b \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-225 b \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi ^{2} a c^{4} x^{4} + 2 \, \pi ^{2} a c^{2} x^{2} + \pi ^{2} a + {\left (\pi ^{2} b c^{4} x^{4} + 2 \, \pi ^{2} b c^{2} x^{2} + \pi ^{2} b\right )} \operatorname {arsinh}\left (c x\right )\right )}}{x}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 284, normalized size = 1.59 \[ \frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} a}{5}+\frac {a \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}-a \,\pi ^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )+a \sqrt {\pi \,c^{2} x^{2}+\pi }\, \pi ^{2}-b \,\pi ^{\frac {5}{2}} \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {5}{2}} \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{5}}{25}-\frac {11 b \,c^{3} \pi ^{\frac {5}{2}} x^{3}}{45}+\frac {23 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}-\frac {23 b c \,\pi ^{\frac {5}{2}} x}{15}+\frac {b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}}{5}+\frac {11 b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}}{15} \]
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 \[ -\frac {1}{15} \, {\left (15 \, \pi ^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - 15 \, \pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}} - 5 \, \pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} - 3 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}\right )} a + b \int \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x} \,d x \]
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 \[ \pi ^{\frac {5}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int 2 a c^{2} x \sqrt {c^{2} x^{2} + 1}\, dx + \int a c^{4} x^{3} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 2 b c^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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