Optimal. Leaf size=132 \[ \frac {c^3 d \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{e (c d-e)^{3/2} (c d+e)^{3/2}}-\frac {c \sqrt {c x-1} \sqrt {c x+1}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5802, 96, 93, 208} \[ -\frac {c \sqrt {c x-1} \sqrt {c x+1}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {c^3 d \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{e (c d-e)^{3/2} (c d+e)^{3/2}}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 93
Rule 96
Rule 208
Rule 5802
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {c \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{2 e}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (c^3 d\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{2 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (c^3 d\right ) \operatorname {Subst}\left (\int \frac {1}{c d-e-(c d+e) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\cosh ^{-1}(c x)}{2 e (d+e x)^2}+\frac {c^3 d \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 190, normalized size = 1.44 \[ \frac {c (d+e x) \left (-e \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d^2-e^2}-c^2 d (d+e x) \log \left (-\sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d^2-e^2}+c^2 d x+e\right )+c^2 d (d+e x) \log (d+e x)\right )-\left (c^2 d^2-e^2\right )^{3/2} \cosh ^{-1}(c x)}{2 e (c d-e) (c d+e) \sqrt {c^2 d^2-e^2} (d+e x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 1044, normalized size = 7.91 \[ \left [-\frac {c^{4} d^{6} - c^{2} d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4}\right )} x^{2} + {\left (c^{3} d^{3} e^{2} x^{2} + 2 \, c^{3} d^{4} e x + c^{3} d^{5}\right )} \sqrt {c^{2} d^{2} - e^{2}} \log \left (\frac {c^{3} d^{2} x + c d e - \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} - \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + 2 \, {\left (c^{4} d^{5} e - c^{2} d^{3} e^{3}\right )} x - {\left ({\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (c^{4} d^{6} - 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (c^{3} d^{5} e - c d^{3} e^{3} + {\left (c^{3} d^{4} e^{2} - c d^{2} e^{4}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (c^{4} d^{8} e - 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} - 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} - 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}}, -\frac {c^{4} d^{6} - c^{2} d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{3} d^{3} e^{2} x^{2} + 2 \, c^{3} d^{4} e x + c^{3} d^{5}\right )} \sqrt {-c^{2} d^{2} + e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + 2 \, {\left (c^{4} d^{5} e - c^{2} d^{3} e^{3}\right )} x - {\left ({\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (c^{4} d^{6} - 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (c^{3} d^{5} e - c d^{3} e^{3} + {\left (c^{3} d^{4} e^{2} - c d^{2} e^{4}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{2 \, {\left (c^{4} d^{8} e - 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} - 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} - 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 338, normalized size = 2.56 \[ -\frac {c^{2} \mathrm {arccosh}\left (c x \right )}{2 \left (c x e +c d \right )^{2} e}-\frac {c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) x d}{2 e \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) d^{2}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{2} \sqrt {c x +1}\, \sqrt {c x -1}}{2 \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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