Optimal. Leaf size=265 \[ -\frac {4144 c^2 \sqrt {1+a^2 x^2}}{1125 a}-\frac {272 c^2 \left (1+a^2 x^2\right )^{3/2}}{3375 a}-\frac {6 c^2 \left (1+a^2 x^2\right )^{5/2}}{625 a}+\frac {298}{75} c^2 x \sinh ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac {8 c^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.30, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {5786, 5772,
5798, 267, 5784, 455, 45, 200, 12, 1261, 712} \begin {gather*} \frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)-\frac {6 c^2 \left (a^2 x^2+1\right )^{5/2}}{625 a}-\frac {272 c^2 \left (a^2 x^2+1\right )^{3/2}}{3375 a}-\frac {4144 c^2 \sqrt {a^2 x^2+1}}{1125 a}+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}-\frac {4 c^2 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {8 c^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2}{5 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {298}{75} c^2 x \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 200
Rule 267
Rule 455
Rule 712
Rule 1261
Rule 5772
Rule 5784
Rule 5786
Rule 5798
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^3 \, dx-\frac {1}{5} \left (3 a c^2\right ) \int x \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2 \, dx\\ &=-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{25} \left (6 c^2\right ) \int \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x) \, dx+\frac {1}{15} \left (8 c^2\right ) \int \sinh ^{-1}(a x)^3 \, dx-\frac {1}{5} \left (4 a c^2\right ) \int x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \, dx\\ &=\frac {6}{25} c^2 x \sinh ^{-1}(a x)+\frac {4}{25} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{15} \left (8 c^2\right ) \int \left (1+a^2 x^2\right ) \sinh ^{-1}(a x) \, dx-\frac {1}{25} \left (6 a c^2\right ) \int \frac {x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt {1+a^2 x^2}} \, dx-\frac {1}{5} \left (8 a c^2\right ) \int \frac {x \sinh ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {58}{75} c^2 x \sinh ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac {8 c^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac {1}{5} \left (16 c^2\right ) \int \sinh ^{-1}(a x) \, dx-\frac {1}{125} \left (2 a c^2\right ) \int \frac {x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {1+a^2 x^2}} \, dx-\frac {1}{15} \left (8 a c^2\right ) \int \frac {x \left (1+\frac {a^2 x^2}{3}\right )}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {298}{75} c^2 x \sinh ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac {8 c^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3-\frac {1}{125} \left (a c^2\right ) \text {Subst}\left (\int \frac {15+10 a^2 x+3 a^4 x^2}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {1}{15} \left (4 a c^2\right ) \text {Subst}\left (\int \frac {1+\frac {a^2 x}{3}}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {1}{5} \left (16 a c^2\right ) \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {16 c^2 \sqrt {1+a^2 x^2}}{5 a}+\frac {298}{75} c^2 x \sinh ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac {8 c^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3-\frac {1}{125} \left (a c^2\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1+a^2 x}}+4 \sqrt {1+a^2 x}+3 \left (1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )-\frac {1}{15} \left (4 a c^2\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+a^2 x}}+\frac {1}{3} \sqrt {1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {4144 c^2 \sqrt {1+a^2 x^2}}{1125 a}-\frac {272 c^2 \left (1+a^2 x^2\right )^{3/2}}{3375 a}-\frac {6 c^2 \left (1+a^2 x^2\right )^{5/2}}{625 a}+\frac {298}{75} c^2 x \sinh ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)-\frac {8 c^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2}{5 a}-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 137, normalized size = 0.52 \begin {gather*} \frac {c^2 \left (-2 \sqrt {1+a^2 x^2} \left (31841+842 a^2 x^2+81 a^4 x^4\right )+30 a x \left (2235+190 a^2 x^2+27 a^4 x^4\right ) \sinh ^{-1}(a x)-225 \sqrt {1+a^2 x^2} \left (149+38 a^2 x^2+9 a^4 x^4\right ) \sinh ^{-1}(a x)^2+1125 a x \left (15+10 a^2 x^2+3 a^4 x^4\right ) \sinh ^{-1}(a x)^3\right )}{16875 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{2} \arcsinh \left (a x \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 210, normalized size = 0.79 \begin {gather*} -\frac {1}{75} \, {\left (9 \, \sqrt {a^{2} x^{2} + 1} a^{2} c^{2} x^{4} + 38 \, \sqrt {a^{2} x^{2} + 1} c^{2} x^{2} + \frac {149 \, \sqrt {a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname {arsinh}\left (a x\right )^{3} - \frac {2}{16875} \, {\left (81 \, \sqrt {a^{2} x^{2} + 1} a^{2} c^{2} x^{4} + 842 \, \sqrt {a^{2} x^{2} + 1} c^{2} x^{2} - \frac {15 \, {\left (27 \, a^{4} c^{2} x^{5} + 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname {arsinh}\left (a x\right )}{a} + \frac {31841 \, \sqrt {a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 204, normalized size = 0.77 \begin {gather*} \frac {1125 \, {\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 225 \, {\left (9 \, a^{4} c^{2} x^{4} + 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (27 \, a^{5} c^{2} x^{5} + 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, {\left (81 \, a^{4} c^{2} x^{4} + 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt {a^{2} x^{2} + 1}}{16875 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 262, normalized size = 0.99 \begin {gather*} \begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {asinh}^{3}{\left (a x \right )}}{5} + \frac {6 a^{4} c^{2} x^{5} \operatorname {asinh}{\left (a x \right )}}{125} - \frac {3 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{25} - \frac {6 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} + 1}}{625} + \frac {2 a^{2} c^{2} x^{3} \operatorname {asinh}^{3}{\left (a x \right )}}{3} + \frac {76 a^{2} c^{2} x^{3} \operatorname {asinh}{\left (a x \right )}}{225} - \frac {38 a c^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{75} - \frac {1684 a c^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{16875} + c^{2} x \operatorname {asinh}^{3}{\left (a x \right )} + \frac {298 c^{2} x \operatorname {asinh}{\left (a x \right )}}{75} - \frac {149 c^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{75 a} - \frac {63682 c^{2} \sqrt {a^{2} x^{2} + 1}}{16875 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\mathrm {asinh}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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