Optimal. Leaf size=158 \[ \frac {b x \sqrt {d-c^2 d x^2}}{6 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {a+b \cosh ^{-1}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.12, antiderivative size = 170, normalized size of antiderivative = 1.08, number of steps
used = 4, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {272, 45, 5922,
12, 393, 212} \begin {gather*} -\frac {a+b \cosh ^{-1}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^3 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 212
Rule 272
Rule 393
Rule 5922
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2-3 c^2 x^2}{3 c^4 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2-3 c^2 x^2}{\left (1-c^2 x^2\right )^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (5 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {a+b \cosh ^{-1}(c x)}{c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {(1-c x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 (1+c x) \sqrt {d-c^2 d x^2}}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 122, normalized size = 0.77 \begin {gather*} \frac {4 a-6 a c^2 x^2-b c x \sqrt {-1+c x} \sqrt {1+c x}+b \left (4-6 c^2 x^2\right ) \cosh ^{-1}(c x)-5 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)}{6 c^4 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \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. \(313\) vs.
\(2(138)=276\).
time = 4.22, size = 314, normalized size = 1.99
method | result | size |
default | \(a \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2}}{d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{3}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{4}}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 175, normalized size = 1.11 \begin {gather*} \frac {1}{12} \, b c {\left (\frac {2 \, \sqrt {-d} x}{c^{6} d^{3} x^{2} - c^{4} d^{3}} + \frac {5 \, \sqrt {-d} \log \left (c x + 1\right )}{c^{5} d^{3}} - \frac {5 \, \sqrt {-d} \log \left (c x - 1\right )}{c^{5} d^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 469, normalized size = 2.97 \begin {gather*} \left [\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + 8 \, {\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 5 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 8 \, {\left (3 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}, \frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + 5 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 4 \, {\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, {\left (3 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {x^3\,\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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