Optimal. Leaf size=202 \[ -\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5802, 103, 151, 12, 93, 208} \[ -\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac {b c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)} \]
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 {a+b \cosh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3} \, dx}{3 e}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {(b 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 {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b 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 {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b 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 {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b 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 {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b 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 {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b 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 {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b 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 {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b 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 {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b 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 = 0.98, size = 259, normalized size = 1.28 \[ -\frac {\frac {2 a+\frac {b c e \sqrt {c x-1} \sqrt {c x+1} (d+e x) \left (c^2 d (4 d+3 e x)-e^2\right )}{\left (e^2-c^2 d^2\right )^2}}{(d+e x)^3}+\frac {i b c^3 \left (2 c^2 d^2+e^2\right ) \log \left (\frac {12 e^2 (e-c d)^2 (c d+e)^2 \left (\sqrt {c x-1} \sqrt {c x+1} \sqrt {e^2-c^2 d^2}-i c^2 d x-i e\right )}{b c^3 \sqrt {e^2-c^2 d^2} \left (2 c^2 d^2+e^2\right ) (d+e x)}\right )}{(e-c d)^2 (c d+e)^2 \sqrt {e^2-c^2 d^2}}+\frac {2 b \cosh ^{-1}(c x)}{(d+e x)^3}}{6 e} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.55, size = 1963, normalized size = 9.72 \[ \text {result too large to display} \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 1137, normalized size = 5.63 \[ -\frac {c^{3} a}{3 \left (c x e +c d \right )^{3} e}-\frac {c^{3} b \,\mathrm {arccosh}\left (c x \right )}{3 \left (c x e +c d \right )^{3} e}-\frac {c^{7} b \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} b \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^{7} b \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 {c^{5} b 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 {2 c^{5} b \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} b \,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^{5} b 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} b \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} b \,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] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, {\left (6 \, c \int \frac {1}{3 \, {\left (c^{3} e^{4} x^{6} + 3 \, c^{3} d e^{3} x^{5} - 3 \, c d^{2} e^{2} x^{2} - c d^{3} e x + {\left (3 \, c^{3} d^{2} e^{2} - c e^{4}\right )} x^{4} + {\left (c^{3} d^{3} e - 3 \, c d e^{3}\right )} x^{3} + {\left (c^{2} e^{4} x^{5} + 3 \, c^{2} d e^{3} x^{4} - 3 \, d^{2} e^{2} x - d^{3} e + {\left (3 \, c^{2} d^{2} e^{2} - e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 3 \, d e^{3}\right )} x^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x} + \frac {2 \, {\left (c^{6} d^{3} + 3 \, c^{4} d e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} e - 3 \, c^{4} d^{4} e^{3} + 3 \, c^{2} d^{2} e^{5} - e^{7}} - \frac {3 \, c^{6} d^{6} - 2 \, c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4} + 2 \, {\left (c^{6} d^{4} e^{2} - c^{2} e^{6}\right )} x^{2} + {\left (5 \, c^{6} d^{5} e - 2 \, c^{4} d^{3} e^{3} - 3 \, c^{2} d e^{5}\right )} x - 2 \, {\left (c^{6} d^{6} - 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} - e^{6}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (c^{6} d^{6} + 3 \, c^{5} d^{5} e + 3 \, c^{4} d^{4} e^{2} + c^{3} d^{3} e^{3} + {\left (c^{6} d^{3} e^{3} + 3 \, c^{5} d^{2} e^{4} + 3 \, c^{4} d e^{5} + c^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{6} d^{4} e^{2} + 3 \, c^{5} d^{3} e^{3} + 3 \, c^{4} d^{2} e^{4} + c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (c^{6} d^{5} e + 3 \, c^{5} d^{4} e^{2} + 3 \, c^{4} d^{3} e^{3} + c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c x + 1\right ) + {\left (c^{6} d^{6} - 3 \, c^{5} d^{5} e + 3 \, c^{4} d^{4} e^{2} - c^{3} d^{3} e^{3} + {\left (c^{6} d^{3} e^{3} - 3 \, c^{5} d^{2} e^{4} + 3 \, c^{4} d e^{5} - c^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{6} d^{4} e^{2} - 3 \, c^{5} d^{3} e^{3} + 3 \, c^{4} d^{2} e^{4} - c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (c^{6} d^{5} e - 3 \, c^{5} d^{4} e^{2} + 3 \, c^{4} d^{3} e^{3} - c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c x - 1\right )}{c^{6} d^{9} e - 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} - d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} - 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} - e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} - 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} - d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} - 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} - d^{2} e^{8}\right )} x}\right )} b - \frac {a}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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