Optimal. Leaf size=131 \[ -\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 \pi ^{3/2} b c^5}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {3 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi ^2 c^4}-\frac {b x^2}{4 \pi ^{3/2} c^3}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^5} \]
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Rubi [A] time = 0.26, antiderivative size = 181, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5751, 5758, 5675, 30, 266, 43} \[ -\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {3 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi ^2 c^4}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 \pi ^{3/2} b c^5}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 \pi c^3 \sqrt {\pi c^2 x^2+\pi }}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 \pi c^5 \sqrt {\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 30
Rule 43
Rule 266
Rule 5675
Rule 5751
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^2 \pi }+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 \pi ^2}-\frac {3 \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{2 c^4 \pi }-\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 c^3 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{2 c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {3 b x^2 \sqrt {1+c^2 x^2}}{4 c^3 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 \pi ^2}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^5 \pi ^{3/2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{2 c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^3 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {3 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^4 \pi ^2}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^5 \pi ^{3/2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^5 \pi \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 147, normalized size = 1.12 \[ \frac {\sinh ^{-1}(c x) \left (-12 a \sqrt {c^2 x^2+1}+9 b c x+b \sinh \left (3 \sinh ^{-1}(c x)\right )\right )+4 a c^3 x^3+12 a c x-4 b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )-6 b \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2-b \sqrt {c^2 x^2+1} \cosh \left (2 \sinh ^{-1}(c x)\right )}{8 \pi ^{3/2} c^5 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b x^{4} \operatorname {arsinh}\left (c x\right ) + a x^{4}\right )}}{\pi ^{2} c^{4} x^{4} + 2 \, \pi ^{2} c^{2} x^{2} + \pi ^{2}}, x\right ) \]
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 [B] time = 0.33, size = 269, normalized size = 2.05 \[ \frac {a \,x^{3}}{2 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {3 a x}{2 c^{4} \pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {3 a \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{4} \pi \sqrt {\pi \,c^{2}}}-\frac {3 b \arcsinh \left (c x \right )^{2}}{4 c^{5} \pi ^{\frac {3}{2}}}+\frac {b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{2 \pi ^{\frac {3}{2}} c^{4}}-\frac {b \,x^{2}}{4 c^{3} \pi ^{\frac {3}{2}}}+\frac {2 b \arcsinh \left (c x \right )}{c^{5} \pi ^{\frac {3}{2}}}-\frac {b}{8 \pi ^{\frac {3}{2}} c^{5}}-\frac {b \arcsinh \left (c x \right ) x^{2}}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) x}{\pi ^{\frac {3}{2}} c^{4} \sqrt {c^{2} x^{2}+1}}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} c^{5} \left (c^{2} x^{2}+1\right )}-\frac {b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{5} \pi ^{\frac {3}{2}}} \]
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 \[ \frac {1}{2} \, a {\left (\frac {x^{3}}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{2}} + \frac {3 \, x}{\pi \sqrt {\pi + \pi c^{2} x^{2}} c^{4}} - \frac {3 \, \operatorname {arsinh}\left (c x\right )}{\pi ^{\frac {3}{2}} c^{5}}\right )} + b \int \frac {x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {a x^{4}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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