Optimal. Leaf size=195 \[ -\frac {c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.24, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5802, 103, 151, 12, 93, 208} \[ -\frac {c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac {c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 151
Rule 208
Rule 5802
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3} \, dx}{3 e}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {c \int \frac {-2 c^2 d+c^2 e x}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{6 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c \int \frac {c^2 \left (2 c^2 d^2+e^2\right )}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (c^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (c^3 \left (2 c^2 d^2+e^2\right )\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 )}{3 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {\cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{3 (c d-e)^{5/2} e (c d+e)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.56, size = 244, normalized size = 1.25 \[ \frac {1}{6} \left (\frac {c \sqrt {c x-1} \sqrt {c x+1} \left (e^2-c^2 d (4 d+3 e x)\right )}{\left (e^2-c^2 d^2\right )^2 (d+e x)^2}-\frac {i c^3 \left (2 c^2 d^2+e^2\right ) \log \left (\frac {12 e^2 (e-c d)^2 (c d+e)^2 \left (\sqrt {c x-1} \sqrt {c x+1} \sqrt {e^2-c^2 d^2}-i c^2 d x-i e\right )}{c^3 \sqrt {e^2-c^2 d^2} \left (2 c^2 d^2+e^2\right ) (d+e x)}\right )}{e (e-c d)^2 (c d+e)^2 \sqrt {e^2-c^2 d^2}}-\frac {2 \cosh ^{-1}(c x)}{e (d+e x)^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.49, size = 1799, normalized size = 9.23 \[ \text {result too large to display} \]
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 = 1108, normalized size = 5.68 \[ -\frac {c^{3} \mathrm {arccosh}\left (c x \right )}{3 \left (c x e +c d \right )^{3} e}-\frac {c^{7} \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^{2} d^{2}}{3 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {2 c^{7} \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^{3}}{3 e \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{5} e \sqrt {c x +1}\, \sqrt {c x -1}\, x d}{2 \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}-\frac {c^{5} e^{2} \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^{2}}{6 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{7} \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^{4}}{3 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {2 c^{5} \sqrt {c x +1}\, \sqrt {c x -1}\, d^{2}}{3 \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}-\frac {c^{5} e \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}{3 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{5} \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}}{6 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {c^{3} e^{2} \sqrt {c x +1}\, \sqrt {c x -1}}{6 \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}} \]
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 \[ \int \frac {\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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