Optimal. Leaf size=458 \[ \frac {16 b d^2 x \sqrt {d-c^2 d x^2}}{3003 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^2 x^3 \sqrt {d-c^2 d x^2}}{9009 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^5 \sqrt {d-c^2 d x^2}}{5005 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^7 \sqrt {d-c^2 d x^2}}{21021 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {53 b c d^2 x^9 \sqrt {d-c^2 d x^2}}{3861 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {27 b c^3 d^2 x^{11} \sqrt {d-c^2 d x^2}}{1573 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{13} \sqrt {d-c^2 d x^2}}{169 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^8 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^8 d^2}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{13/2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^8 d^4} \]
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Rubi [A]
time = 0.19, antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {272, 45, 5922,
12, 1824} \begin {gather*} \frac {\left (d-c^2 d x^2\right )^{13/2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^8 d^4}-\frac {3 \left (d-c^2 d x^2\right )^{11/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^8 d}-\frac {53 b c d^2 x^9 \sqrt {d-c^2 d x^2}}{3861 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b d^2 x^7 \sqrt {d-c^2 d x^2}}{21021 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b d^2 x \sqrt {d-c^2 d x^2}}{3003 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^{13} \sqrt {d-c^2 d x^2}}{169 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^2 x^3 \sqrt {d-c^2 d x^2}}{9009 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {27 b c^3 d^2 x^{11} \sqrt {d-c^2 d x^2}}{1573 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^2 x^5 \sqrt {d-c^2 d x^2}}{5005 c^3 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 1824
Rule 5922
Rubi steps
\begin {align*} \int x^7 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^7 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {16 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac {8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac {6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac {d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^3 \left (-16-56 c^2 x^2-126 c^4 x^4-231 c^6 x^6\right )}{3003 c^8} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {16 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac {8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac {6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac {d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (-16-56 c^2 x^2-126 c^4 x^4-231 c^6 x^6\right ) \, dx}{3003 c^7 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {16 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac {8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac {6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac {d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-16-8 c^2 x^2-6 c^4 x^4-5 c^6 x^6+371 c^8 x^8-567 c^{10} x^{10}+231 c^{12} x^{12}\right ) \, dx}{3003 c^7 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {16 b d^2 x \sqrt {d-c^2 d x^2}}{3003 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^2 x^3 \sqrt {d-c^2 d x^2}}{9009 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^5 \sqrt {d-c^2 d x^2}}{5005 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^7 \sqrt {d-c^2 d x^2}}{21021 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {53 b c d^2 x^9 \sqrt {d-c^2 d x^2}}{3861 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {27 b c^3 d^2 x^{11} \sqrt {d-c^2 d x^2}}{1573 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{13} \sqrt {d-c^2 d x^2}}{169 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac {8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac {6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac {d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 193, normalized size = 0.42 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (b \left (16 x+\frac {8 c^2 x^3}{3}+\frac {6 c^4 x^5}{5}+\frac {5 c^6 x^7}{7}-\frac {371 c^8 x^9}{9}+\frac {567 c^{10} x^{11}}{11}-\frac {231 c^{12} x^{13}}{13}\right )+231 c^5 x^6 (-1+c x)^{7/2} (1+c x)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {2 (-1+c x)^{7/2} (1+c x)^{7/2} \left (8+28 c^2 x^2+63 c^4 x^4\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c}\right )}{3003 c^7 \sqrt {-1+c x} \sqrt {1+c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2373\) vs.
\(2(386)=772\).
time = 3.95, size = 2374, normalized size = 5.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(2374\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 313, normalized size = 0.68 \begin {gather*} -\frac {1}{3003} \, {\left (\frac {231 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{6}}{c^{2} d} + \frac {126 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{4} d} + \frac {56 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{8} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{3003} \, {\left (\frac {231 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{6}}{c^{2} d} + \frac {126 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{4} d} + \frac {56 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{8} d}\right )} a - \frac {{\left (800415 \, c^{12} \sqrt {-d} d^{2} x^{13} - 2321865 \, c^{10} \sqrt {-d} d^{2} x^{11} + 1856855 \, c^{8} \sqrt {-d} d^{2} x^{9} - 32175 \, c^{6} \sqrt {-d} d^{2} x^{7} - 54054 \, c^{4} \sqrt {-d} d^{2} x^{5} - 120120 \, c^{2} \sqrt {-d} d^{2} x^{3} - 720720 \, \sqrt {-d} d^{2} x\right )} b}{135270135 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 353, normalized size = 0.77 \begin {gather*} \frac {45045 \, {\left (231 \, b c^{14} d^{2} x^{14} - 798 \, b c^{12} d^{2} x^{12} + 938 \, b c^{10} d^{2} x^{10} - 376 \, b c^{8} d^{2} x^{8} - b c^{6} d^{2} x^{6} - 2 \, b c^{4} d^{2} x^{4} - 8 \, b c^{2} d^{2} x^{2} + 16 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (800415 \, b c^{13} d^{2} x^{13} - 2321865 \, b c^{11} d^{2} x^{11} + 1856855 \, b c^{9} d^{2} x^{9} - 32175 \, b c^{7} d^{2} x^{7} - 54054 \, b c^{5} d^{2} x^{5} - 120120 \, b c^{3} d^{2} x^{3} - 720720 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 45045 \, {\left (231 \, a c^{14} d^{2} x^{14} - 798 \, a c^{12} d^{2} x^{12} + 938 \, a c^{10} d^{2} x^{10} - 376 \, a c^{8} d^{2} x^{8} - a c^{6} d^{2} x^{6} - 2 \, a c^{4} d^{2} x^{4} - 8 \, a c^{2} d^{2} x^{2} + 16 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{135270135 \, {\left (c^{10} x^{2} - c^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 x^7\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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