Optimal. Leaf size=151 \[ -\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {152 b d \sqrt {c x-1} \sqrt {c x+1}}{3675 c^5}-\frac {76 b d x^2 \sqrt {c x-1} \sqrt {c x+1}}{3675 c^3}+\frac {1}{49} b c d x^6 \sqrt {c x-1} \sqrt {c x+1}-\frac {19 b d x^4 \sqrt {c x-1} \sqrt {c x+1}}{1225 c} \]
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Rubi [A] time = 0.15, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {14, 5731, 12, 460, 100, 74} \[ -\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {76 b d x^2 \sqrt {c x-1} \sqrt {c x+1}}{3675 c^3}-\frac {152 b d \sqrt {c x-1} \sqrt {c x+1}}{3675 c^5}+\frac {1}{49} b c d x^6 \sqrt {c x-1} \sqrt {c x+1}-\frac {19 b d x^4 \sqrt {c x-1} \sqrt {c x+1}}{1225 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 74
Rule 100
Rule 460
Rule 5731
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{35} (b c d) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{245} (19 b c d) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {19 b d x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c}+\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(19 b d) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1225 c}\\ &=-\frac {19 b d x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c}+\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(76 b d) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1225 c}\\ &=-\frac {76 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^3}-\frac {19 b d x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c}+\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(76 b d) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3675 c^3}\\ &=-\frac {76 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^3}-\frac {19 b d x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c}+\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(152 b d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3675 c^3}\\ &=-\frac {152 b d \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^5}-\frac {76 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3675 c^3}-\frac {19 b d x^4 \sqrt {-1+c x} \sqrt {1+c x}}{1225 c}+\frac {1}{49} b c d x^6 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 91, normalized size = 0.60 \[ \frac {d \left (-105 a x^5 \left (5 c^2 x^2-7\right )-105 b x^5 \left (5 c^2 x^2-7\right ) \cosh ^{-1}(c x)+\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right )}{c^5}\right )}{3675} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 113, normalized size = 0.75 \[ -\frac {525 \, a c^{7} d x^{7} - 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} - 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} d x^{6} - 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} - 152 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{5}} \]
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 = 0.03, size = 98, normalized size = 0.65 \[ \frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-57 c^{4} x^{4}-76 c^{2} x^{2}-152\right )}{3675}\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 184, normalized size = 1.22 \[ -\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d \]
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 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.02, size = 158, normalized size = 1.05 \[ \begin {cases} - \frac {a c^{2} d x^{7}}{7} + \frac {a d x^{5}}{5} - \frac {b c^{2} d x^{7} \operatorname {acosh}{\left (c x \right )}}{7} + \frac {b c d x^{6} \sqrt {c^{2} x^{2} - 1}}{49} + \frac {b d x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {19 b d x^{4} \sqrt {c^{2} x^{2} - 1}}{1225 c} - \frac {76 b d x^{2} \sqrt {c^{2} x^{2} - 1}}{3675 c^{3}} - \frac {152 b d \sqrt {c^{2} x^{2} - 1}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {d x^{5} \left (a + \frac {i \pi b}{2}\right )}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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