Optimal. Leaf size=98 \[ \frac {x^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac {2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^4}+\frac {2 b x}{3 \sqrt {\pi } c^3}-\frac {b x^3}{9 \sqrt {\pi } c} \]
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Rubi [A] time = 0.16, antiderivative size = 142, normalized size of antiderivative = 1.45, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5758, 5717, 8, 30} \[ \frac {x^2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2}-\frac {2 \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^4}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {\pi c^2 x^2+\pi }}+\frac {2 b x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }-\frac {2 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{3 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi }+\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {2 b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {\pi +c^2 \pi x^2}}-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi }+\frac {x^2 \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi }\\ \end {align*}
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Mathematica [A] time = 0.14, size = 82, normalized size = 0.84 \[ \frac {3 a \sqrt {c^2 x^2+1} \left (c^2 x^2-2\right )+b \left (6 c x-c^3 x^3\right )+3 b \sqrt {c^2 x^2+1} \left (c^2 x^2-2\right ) \sinh ^{-1}(c x)}{9 \sqrt {\pi } c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 132, normalized size = 1.35 \[ \frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 3 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 6 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 6 \, a\right )}}{9 \, {\left (\pi c^{6} x^{2} + \pi c^{4}\right )}} \]
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 = 0.11, size = 133, normalized size = 1.36 \[ a \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )+\frac {b \left (3 \arcsinh \left (c x \right ) c^{4} x^{4}-3 \arcsinh \left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-6 \arcsinh \left (c x \right )+6 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{4} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 117, normalized size = 1.19 \[ \frac {1}{3} \, b {\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{2}} - \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{2}} - \frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{4}}\right )} - \frac {{\left (c^{2} x^{3} - 6 \, x\right )} b}{9 \, \sqrt {\pi } c^{3}} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.94, size = 122, normalized size = 1.24 \[ \frac {a \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{3}}{9 c} + \frac {x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{2}} + \frac {2 x}{3 c^{3}} - \frac {2 \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{4}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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