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 = 0.23, antiderivative size = 294, normalized size of antiderivative = 1.40, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, 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 195
Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5682
Rule 5684
Rule 5717
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*}
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Mathematica [A] time = 0.57, size = 202, normalized size = 0.96 \[ \frac {\pi ^{3/2} \left (160 a^2 c x \sqrt {c^2 x^2+1}+64 a^2 c^3 x^3 \sqrt {c^2 x^2+1}+4 \sinh ^{-1}(c x) \left (4 a \left (6 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )-16 b^2 \cosh \left (2 \sinh ^{-1}(c x)\right )-b^2 \cosh \left (4 \sinh ^{-1}(c x)\right )\right )+8 b \sinh ^{-1}(c x)^2 \left (12 a+8 b \sinh \left (2 \sinh ^{-1}(c x)\right )+b \sinh \left (4 \sinh ^{-1}(c x)\right )\right )-64 a b \cosh \left (2 \sinh ^{-1}(c x)\right )-4 a b \cosh \left (4 \sinh ^{-1}(c x)\right )+32 b^2 \sinh ^{-1}(c x)^3+32 b^2 \sinh \left (2 \sinh ^{-1}(c x)\right )+b^2 \sinh \left (4 \sinh ^{-1}(c x)\right )\right )}{256 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi a^{2} c^{2} x^{2} + \pi a^{2} + {\left (\pi b^{2} c^{2} x^{2} + \pi b^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (\pi a b c^{2} x^{2} + \pi a b\right )} \operatorname {arsinh}\left (c x\right )\right )}, x\right ) \]
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 = 0.12, size = 350, normalized size = 1.67 \[ \frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {3}{2}} c^{2} \arcsinh \left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3}}{4}-\frac {b^{2} \pi ^{\frac {3}{2}} c^{3} \arcsinh \left (c x \right ) x^{4}}{8}+\frac {b^{2} \pi ^{\frac {3}{2}} c^{2} x^{3} \sqrt {c^{2} x^{2}+1}}{32}+\frac {5 b^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x}{8}-\frac {5 b^{2} \pi ^{\frac {3}{2}} c \arcsinh \left (c x \right ) x^{2}}{8}+\frac {17 b^{2} \pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}}{64}+\frac {b^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{3}}{8 c}-\frac {17 b^{2} \pi ^{\frac {3}{2}} \arcsinh \left (c x \right )}{64 c}+\frac {a b \,\pi ^{\frac {3}{2}} c^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{2}-\frac {a b \,\pi ^{\frac {3}{2}} c^{3} x^{4}}{8}+\frac {5 a b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{4}-\frac {5 a b \,\pi ^{\frac {3}{2}} c \,x^{2}}{8}+\frac {3 a b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2}}{8 c}-\frac {a b \,\pi ^{\frac {3}{2}}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 95.26, size = 405, normalized size = 1.93 \[ \begin {cases} \frac {\pi ^{\frac {3}{2}} a^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a^{2} x \sqrt {c^{2} x^{2} + 1}}{8} + \frac {3 \pi ^{\frac {3}{2}} a^{2} \operatorname {asinh}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} a b c^{3} x^{4}}{8} + \frac {\pi ^{\frac {3}{2}} a b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {5 \pi ^{\frac {3}{2}} a b c x^{2}}{8} + \frac {5 \pi ^{\frac {3}{2}} a b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4} + \frac {3 \pi ^{\frac {3}{2}} a b \operatorname {asinh}^{2}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b^{2} c^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{32} - \frac {5 \pi ^{\frac {3}{2}} b^{2} c x^{2} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {5 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {17 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1}}{64} + \frac {\pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}^{3}{\left (c x \right )}}{8 c} - \frac {17 \pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}{\left (c x \right )}}{64 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} a^{2} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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