Optimal. Leaf size=177 \[ \frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b d^2 (c x-1)^{7/2} (c x+1)^{7/2}}{49 c^3}-\frac {b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{175 c^3}+\frac {4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{315 c^3}-\frac {8 b d^2 \sqrt {c x-1} \sqrt {c x+1}}{105 c^3} \]
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Rubi [A] time = 0.25, antiderivative size = 223, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {270, 5731, 12, 520, 1251, 771} \[ \frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b d^2 \left (1-c^2 x^2\right )^4}{49 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{175 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{315 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 \left (1-c^2 x^2\right )}{105 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 520
Rule 771
Rule 1251
Rule 5731
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )}{105 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{105} \left (b c d^2\right ) \int \frac {x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x \left (35-42 c^2 x+15 c^4 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{c^2 \sqrt {-1+c^2 x}}-\frac {4 \sqrt {-1+c^2 x}}{c^2}+\frac {3 \left (-1+c^2 x\right )^{3/2}}{c^2}+\frac {15 \left (-1+c^2 x\right )^{5/2}}{c^2}\right ) \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b d^2 \left (1-c^2 x^2\right )}{105 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{175 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d^2 \left (1-c^2 x^2\right )^4}{49 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 116, normalized size = 0.66 \[ \frac {d^2 \left (105 a c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )-b \sqrt {c x-1} \sqrt {c x+1} \left (225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right )+105 b c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right ) \cosh ^{-1}(c x)\right )}{11025 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 153, normalized size = 0.86 \[ \frac {1575 \, a c^{7} d^{2} x^{7} - 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} d^{2} x^{7} - 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{6} d^{2} x^{6} - 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} + 818 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{11025 \, c^{3}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 120, normalized size = 0.68 \[ \frac {d^{2} a \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \,\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \mathrm {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} x^{6}-612 c^{4} x^{4}+409 c^{2} x^{2}+818\right )}{11025}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 261, normalized size = 1.47 \[ \frac {1}{7} \, a c^{4} d^{2} x^{7} - \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} \]
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 \[ \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.13, size = 209, normalized size = 1.18 \[ \begin {cases} \frac {a c^{4} d^{2} x^{7}}{7} - \frac {2 a c^{2} d^{2} x^{5}}{5} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{4} d^{2} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {b c^{3} d^{2} x^{6} \sqrt {c^{2} x^{2} - 1}}{49} - \frac {2 b c^{2} d^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {68 b c d^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{1225} + \frac {b d^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {409 b d^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{11025 c} - \frac {818 b d^{2} \sqrt {c^{2} x^{2} - 1}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{3} \left (a + \frac {i \pi b}{2}\right )}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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