Optimal. Leaf size=76 \[ \frac {a+b \cosh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5798, 5718, 207} \[ \frac {a+b \cosh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 5718
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \cosh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \cosh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 90, normalized size = 1.18 \[ -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d^2 \left (c^2 x^2-1\right )}-\frac {b \sqrt {-d \left (c^2 x^2-1\right )} \tanh ^{-1}(c x)}{c^2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 327, normalized size = 4.30 \[ \left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 4 \, \sqrt {-c^{2} d x^{2} + d} a}{4 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}}, -\frac {{\left (b c^{2} x^{2} - b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 2 \, \sqrt {-c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-c^{2} d x^{2} + d} a}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}}\right ] \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 198, normalized size = 2.61 \[ \frac {a}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\frac {\frac {{\left (c \sqrt {d} x + \sqrt {c x + 1} \sqrt {c x - 1} \sqrt {d}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c x + 1}} + \frac {\sqrt {c x + 1} \sqrt {c x - 1} \sqrt {d}}{\sqrt {-c x + 1}}}{\sqrt {c x + 1} c^{3} d^{2} x + {\left (c x + 1\right )} \sqrt {c x - 1} c^{2} d^{2}} - \int \frac {c^{2} x^{3} + c x^{2} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )} - x}{\sqrt {-c x + 1} {\left ({\left (c^{2} d^{\frac {3}{2}} x^{2} - d^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, \log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} + 2 \, {\left (c^{3} d^{\frac {3}{2}} x^{3} - c d^{\frac {3}{2}} x\right )} e^{\left (\log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )} + {\left (c^{4} d^{\frac {3}{2}} x^{4} - c^{2} d^{\frac {3}{2}} x^{2}\right )} \sqrt {c x + 1}\right )}}\,{d x}\right )} + \frac {a}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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