Optimal. Leaf size=206 \[ \frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b d^2 (c x-1)^{9/2} (c x+1)^{9/2}}{81 c^5}-\frac {10 b d^2 (c x-1)^{7/2} (c x+1)^{7/2}}{441 c^5}-\frac {b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{525 c^5}+\frac {4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{945 c^5}-\frac {8 b d^2 \sqrt {c x-1} \sqrt {c x+1}}{315 c^5} \]
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Rubi [A] time = 0.29, antiderivative size = 264, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {270, 5731, 12, 520, 1251, 897, 1153} \[ \frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b d^2 \left (1-c^2 x^2\right )^5}{81 c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {10 b d^2 \left (1-c^2 x^2\right )^4}{441 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{525 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{945 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 \left (1-c^2 x^2\right )}{315 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 520
Rule 897
Rule 1153
Rule 1251
Rule 5731
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{315} \left (b c d^2\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \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^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2 \left (8-20 x^2+35 x^4\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{315 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{c^4}-\frac {4 x^2}{c^4}+\frac {3 x^4}{c^4}+\frac {50 x^6}{c^4}+\frac {35 x^8}{c^4}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{315 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b d^2 \left (1-c^2 x^2\right )}{315 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d^2 \left (1-c^2 x^2\right )^2}{945 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3}{525 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b d^2 \left (1-c^2 x^2\right )^4}{441 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^5}{81 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 124, normalized size = 0.60 \[ \frac {d^2 \left (315 a c^5 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )+315 b c^5 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right ) \cosh ^{-1}(c x)-b \sqrt {c x-1} \sqrt {c x+1} \left (1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right )\right )}{99225 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 165, normalized size = 0.80 \[ \frac {11025 \, a c^{9} d^{2} x^{9} - 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} d^{2} x^{9} - 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} d^{2} x^{8} - 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} + 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{5}} \]
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 = 0.02, size = 128, normalized size = 0.62 \[ \frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \,\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-2650 c^{6} x^{6}+789 c^{4} x^{4}+1052 c^{2} x^{2}+2104\right )}{99225}\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 319, normalized size = 1.55 \[ \frac {1}{9} \, a c^{4} d^{2} x^{9} - \frac {2}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{5} \, a d^{2} x^{5} - \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.46, size = 236, normalized size = 1.15 \[ \begin {cases} \frac {a c^{4} d^{2} x^{9}}{9} - \frac {2 a c^{2} d^{2} x^{7}}{7} + \frac {a d^{2} x^{5}}{5} + \frac {b c^{4} d^{2} x^{9} \operatorname {acosh}{\left (c x \right )}}{9} - \frac {b c^{3} d^{2} x^{8} \sqrt {c^{2} x^{2} - 1}}{81} - \frac {2 b c^{2} d^{2} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} + \frac {106 b c d^{2} x^{6} \sqrt {c^{2} x^{2} - 1}}{3969} + \frac {b d^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {263 b d^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{33075 c} - \frac {1052 b d^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{99225 c^{3}} - \frac {2104 b d^{2} \sqrt {c^{2} x^{2} - 1}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{5} \left (a + \frac {i \pi b}{2}\right )}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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