Optimal. Leaf size=155 \[ \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x}-\frac {2 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{3 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5932, 5917, 29,
30} \begin {gather*} -\frac {2 c^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^3}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 x^2 \sqrt {d-c^2 d x^2}}-\frac {2 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log (x)}{3 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 30
Rule 5917
Rule 5932
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^3} \, dx}{3 \sqrt {d-c^2 d x^2}}+\frac {\left (2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x} \, dx}{3 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2 \sqrt {d-c^2 d x^2}}-\frac {(1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{3 x \sqrt {d-c^2 d x^2}}-\frac {2 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \log (x)}{3 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 174, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {d-c^2 d x^2} \left (b c x+6 b c^3 x^3+2 a \sqrt {-1+c x} \sqrt {1+c x}+4 a c^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}+2 b \sqrt {-1+c x} \sqrt {1+c x} \left (1+2 c^2 x^2\right ) \cosh ^{-1}(c x)-4 b c^3 x^3 \log (-1+c x)-4 b c^3 x^3 \log \left (1+\frac {1}{-1+c x}\right )\right )}{6 d x^3 \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. \(854\) vs.
\(2(131)=262\).
time = 4.98, size = 855, normalized size = 5.52
method | result | size |
default | \(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (c x -1\right ) \left (c x +1\right ) c^{6}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}{d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \mathrm {arccosh}\left (c x \right ) c^{6}}{d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (c x -1\right ) \left (c x +1\right ) c^{4}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,\mathrm {arccosh}\left (c x \right ) c^{4}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3}}{2 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right )}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) c^{2}}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) x}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, c}{6 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) x^{3}}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}\) | \(855\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 134, normalized size = 0.86 \begin {gather*} \frac {1}{6} \, {\left (\frac {4 \, c^{2} \sqrt {-d} \log \left (x\right )}{d} - \frac {\sqrt {-d}}{d x^{2}}\right )} b c - \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 479, normalized size = 3.09 \begin {gather*} \left [-\frac {2 \, {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} - 1} + 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}, \frac {4 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 2 \, {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} - 1} - 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{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 {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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