Optimal. Leaf size=210 \[ \frac{1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{8} \pi x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\pi ^{3/2} b \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac{3}{8} \pi ^{3/2} b c x^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac{\pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c}+\frac{1}{32} \pi ^{3/2} b^2 x \left (c^2 x^2+1\right )^{3/2}+\frac{15}{64} \pi ^{3/2} b^2 x \sqrt{c^2 x^2+1}-\frac{9 \pi ^{3/2} b^2 \sinh ^{-1}(c x)}{64 c} \]
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Rubi [A] time = 0.227046, antiderivative size = 294, normalized size of antiderivative = 1.4, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac{\pi \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{c^2 x^2+1}}+\frac{1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{8} \pi x \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\pi b \left (c^2 x^2+1\right )^{3/2} \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac{3 \pi b c x^2 \sqrt{\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{c^2 x^2+1}}+\frac{15}{64} \pi b^2 x \sqrt{\pi c^2 x^2+\pi }+\frac{1}{32} \pi b^2 x \left (c^2 x^2+1\right ) \sqrt{\pi c^2 x^2+\pi }-\frac{9 \pi b^2 \sqrt{\pi c^2 x^2+\pi } \sinh ^{-1}(c x)}{64 c \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5661
Rule 321
Rule 215
Rule 5717
Rule 195
Rubi steps
\begin{align*} \int \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} (3 \pi ) \int \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{\left (b c \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} \pi x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (3 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \sqrt{1+c^2 x^2}}+\frac{\left (b^2 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \sqrt{1+c^2 x^2}}\\ &=\frac{1}{32} b^2 \pi x \left (1+c^2 x^2\right ) \sqrt{\pi +c^2 \pi x^2}-\frac{3 b c \pi x^2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}-\frac{b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} \pi x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{32 \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 c^2 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \sqrt{1+c^2 x^2}}\\ &=\frac{15}{64} b^2 \pi x \sqrt{\pi +c^2 \pi x^2}+\frac{1}{32} b^2 \pi x \left (1+c^2 x^2\right ) \sqrt{\pi +c^2 \pi x^2}-\frac{3 b c \pi x^2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}-\frac{b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} \pi x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{1+c^2 x^2}}+\frac{\left (3 b^2 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{64 \sqrt{1+c^2 x^2}}-\frac{\left (3 b^2 \pi \sqrt{\pi +c^2 \pi x^2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}\\ &=\frac{15}{64} b^2 \pi x \sqrt{\pi +c^2 \pi x^2}+\frac{1}{32} b^2 \pi x \left (1+c^2 x^2\right ) \sqrt{\pi +c^2 \pi x^2}-\frac{9 b^2 \pi \sqrt{\pi +c^2 \pi x^2} \sinh ^{-1}(c x)}{64 c \sqrt{1+c^2 x^2}}-\frac{3 b c \pi x^2 \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \sqrt{1+c^2 x^2}}-\frac{b \pi \left (1+c^2 x^2\right )^{3/2} \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{3}{8} \pi x \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\pi \sqrt{\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.621436, size = 202, normalized size = 0.96 \[ \frac{\pi ^{3/2} \left (64 a^2 c^3 x^3 \sqrt{c^2 x^2+1}+160 a^2 c x \sqrt{c^2 x^2+1}+4 \sinh ^{-1}(c x) \left (4 a \left (6 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )-16 b^2 \cosh \left (2 \sinh ^{-1}(c x)\right )-b^2 \cosh \left (4 \sinh ^{-1}(c x)\right )\right )+8 b \sinh ^{-1}(c x)^2 \left (12 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )-64 a b \cosh \left (2 \sinh ^{-1}(c x)\right )-4 a b \cosh \left (4 \sinh ^{-1}(c x)\right )+32 b^2 \sinh ^{-1}(c x)^3+32 b^2 \sinh \left (2 \sinh ^{-1}(c x)\right )+b^2 \sinh \left (4 \sinh ^{-1}(c x)\right )\right )}{256 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.081, size = 350, normalized size = 1.7 \begin{align*}{\frac{{a}^{2}x}{4} \left ( \pi \,{c}^{2}{x}^{2}+\pi \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}\pi \,x}{8}\sqrt{\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{3\,{a}^{2}{\pi }^{2}}{8}\ln \left ({\pi \,{c}^{2}x{\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+\sqrt{\pi \,{c}^{2}{x}^{2}+\pi } \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{{b}^{2}{\pi }^{{\frac{3}{2}}}{c}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{x}^{3}}{4}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{b}^{2}{\pi }^{{\frac{3}{2}}}{c}^{3}{\it Arcsinh} \left ( cx \right ){x}^{4}}{8}}+{\frac{{b}^{2}{\pi }^{{\frac{3}{2}}}{c}^{2}{x}^{3}}{32}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{5\,{b}^{2}{\pi }^{3/2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}x}{8}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{5\,{b}^{2}{\pi }^{3/2}c{\it Arcsinh} \left ( cx \right ){x}^{2}}{8}}+{\frac{{b}^{2}{\pi }^{{\frac{3}{2}}} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{8\,c}}+{\frac{17\,{b}^{2}{\pi }^{3/2}x}{64}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{17\,{b}^{2}{\pi }^{3/2}{\it Arcsinh} \left ( cx \right ) }{64\,c}}+{\frac{ab{\pi }^{{\frac{3}{2}}}{c}^{2}{\it Arcsinh} \left ( cx \right ){x}^{3}}{2}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{ab{\pi }^{{\frac{3}{2}}}{c}^{3}{x}^{4}}{8}}+{\frac{5\,ab{\pi }^{3/2}{\it Arcsinh} \left ( cx \right ) x}{4}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{5\,ab{\pi }^{3/2}c{x}^{2}}{8}}+{\frac{3\,ab{\pi }^{3/2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{8\,c}}-{\frac{ab{\pi }^{{\frac{3}{2}}}}{2\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\pi + \pi c^{2} x^{2}}{\left (\pi a^{2} c^{2} x^{2} + \pi a^{2} +{\left (\pi b^{2} c^{2} x^{2} + \pi b^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (\pi a b c^{2} x^{2} + \pi a b\right )} \operatorname{arsinh}\left (c x\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 151.738, size = 405, normalized size = 1.93 \begin{align*} \begin{cases} \frac{\pi ^{\frac{3}{2}} a^{2} c^{2} x^{3} \sqrt{c^{2} x^{2} + 1}}{4} + \frac{5 \pi ^{\frac{3}{2}} a^{2} x \sqrt{c^{2} x^{2} + 1}}{8} + \frac{3 \pi ^{\frac{3}{2}} a^{2} \operatorname{asinh}{\left (c x \right )}}{8 c} - \frac{\pi ^{\frac{3}{2}} a b c^{3} x^{4}}{8} + \frac{\pi ^{\frac{3}{2}} a b c^{2} x^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{2} - \frac{5 \pi ^{\frac{3}{2}} a b c x^{2}}{8} + \frac{5 \pi ^{\frac{3}{2}} a b x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}{\left (c x \right )}}{4} + \frac{3 \pi ^{\frac{3}{2}} a b \operatorname{asinh}^{2}{\left (c x \right )}}{8 c} - \frac{\pi ^{\frac{3}{2}} b^{2} c^{3} x^{4} \operatorname{asinh}{\left (c x \right )}}{8} + \frac{\pi ^{\frac{3}{2}} b^{2} c^{2} x^{3} \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (c x \right )}}{4} + \frac{\pi ^{\frac{3}{2}} b^{2} c^{2} x^{3} \sqrt{c^{2} x^{2} + 1}}{32} - \frac{5 \pi ^{\frac{3}{2}} b^{2} c x^{2} \operatorname{asinh}{\left (c x \right )}}{8} + \frac{5 \pi ^{\frac{3}{2}} b^{2} x \sqrt{c^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (c x \right )}}{8} + \frac{17 \pi ^{\frac{3}{2}} b^{2} x \sqrt{c^{2} x^{2} + 1}}{64} + \frac{\pi ^{\frac{3}{2}} b^{2} \operatorname{asinh}^{3}{\left (c x \right )}}{8 c} - \frac{17 \pi ^{\frac{3}{2}} b^{2} \operatorname{asinh}{\left (c x \right )}}{64 c} & \text{for}\: c \neq 0 \\\pi ^{\frac{3}{2}} a^{2} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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