Optimal. Leaf size=172 \[ -\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )}-\frac {b \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b \csc ^{-1}(c x)}{2 d^2 e} \]
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Rubi [A] time = 0.29, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5226, 1568, 1475, 1651, 844, 216, 725, 206} \[ -\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )}-\frac {b \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b \csc ^{-1}(c x)}{2 d^2 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 216
Rule 725
Rule 844
Rule 1475
Rule 1568
Rule 1651
Rule 5226
Rubi steps
\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^2} \, dx}{2 c e}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} \left (e+\frac {d}{x}\right )^2 x^4} \, dx}{2 c e}\\ &=-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \operatorname {Subst}\left (\int \frac {x^2}{(e+d x)^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}+\frac {(b c) \operatorname {Subst}\left (\int \frac {e-\left (d-\frac {e^2}{c^2 d}\right ) x}{(e+d x) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 e \left (c^2 d^2-e^2\right )}\\ &=\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2 e}+\frac {\left (b c \left (2-\frac {e^2}{c^2 d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(e+d x) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {b \csc ^{-1}(c x)}{2 d^2 e}-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}-\frac {\left (b c \left (2-\frac {e^2}{c^2 d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{d^2-\frac {e^2}{c^2}-x^2} \, dx,x,\frac {d+\frac {e}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {b \csc ^{-1}(c x)}{2 d^2 e}-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 247, normalized size = 1.44 \[ \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b c e x \sqrt {1-\frac {1}{c^2 x^2}}}{d \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b \left (2 c^2 d^2-e^2\right ) \log \left (c x \left (c d-\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}\right )+e\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2-2 c^2 d^2\right ) \log (d+e x)}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}-\frac {b \sin ^{-1}\left (\frac {1}{c x}\right )}{d^2 e}-\frac {b \sec ^{-1}(c x)}{e (d+e x)^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 1117, normalized size = 6.49 \[ \left [-\frac {a c^{4} d^{6} - b c^{3} d^{5} e - 2 \, a c^{2} d^{4} e^{2} + b c d^{3} e^{3} + a d^{2} e^{4} - {\left (b c^{3} d^{3} e^{3} - b c d e^{5}\right )} x^{2} - {\left (2 \, b c^{2} d^{4} e - b d^{2} e^{3} + {\left (2 \, b c^{2} d^{2} e^{3} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} - b d e^{4}\right )} x\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 (b c^{3} d^{4} e^{2} - b c d^{2} e^{4}\right )} x + {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d^{4} e^{2} - b d^{2} e^{4} + {\left (b c^{2} d^{3} e^{3} - b d e^{5}\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 {a c^{4} d^{6} - b c^{3} d^{5} e - 2 \, a c^{2} d^{4} e^{2} + b c d^{3} e^{3} + a d^{2} e^{4} - {\left (b c^{3} d^{3} e^{3} - b c d e^{5}\right )} x^{2} - 2 \, {\left (2 \, b c^{2} d^{4} e - b d^{2} e^{3} + {\left (2 \, b c^{2} d^{2} e^{3} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} - b d e^{4}\right )} x\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 (b c^{3} d^{4} e^{2} - b c d^{2} e^{4}\right )} x + {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4}\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, {\left (b c^{4} d^{6} - 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} - 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e - 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d^{4} e^{2} - b d^{2} e^{4} + {\left (b c^{2} d^{3} e^{3} - b d e^{5}\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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1005, normalized size = 5.84 \[ -\frac {c^{2} a}{2 \left (c e x +d c \right )^{2} e}-\frac {c^{2} b \,\mathrm {arcsec}\left (c x \right )}{2 \left (c e x +d c \right )^{2} e}-\frac {c^{2} b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {c^{2} b \sqrt {c^{2} x^{2}-1}\, d \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {c^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {c^{2} b \sqrt {c^{2} x^{2}-1}\, d \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {c^{2} b e x}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {b e}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}+\frac {b e \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \right )}-\frac {b e \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 c^{2} d x -2 e}{c e x +d c}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +d c \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 \[ \frac {{\left ({\left (c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e\right )} \int \frac {x e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} - 2 \, d e^{2} x + {\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} - 2 \, d e^{2} x - d^{2} e + {\left (c^{2} d^{2} e - e^{3}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - d^{2} e + {\left (c^{2} d^{2} e - e^{3}\right )} x^{2}}\,{d x} - \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {a}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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