Optimal. Leaf size=84 \[ \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5713, 5688, 260} \[ \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5688
Rule 5713
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 0.86 \[ \frac {2 a c x-b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )+2 b c x \cosh ^{-1}(c x)}{2 c d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 180, normalized size = 2.14 \[ \frac {a x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 70, normalized size = 0.83 \[ -\frac {b c \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, d} + \frac {b x \operatorname {arcosh}\left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a x}{\sqrt {-c^{2} d x^{2} + d} d} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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