Integrand size = 16, antiderivative size = 172 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {b \csc ^{-1}(c x)}{2 d^2 e}-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}} \]
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Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5334, 1582, 1489, 1665, 858, 222, 739, 212} \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=-\frac {a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )}-\frac {b \csc ^{-1}(c x)}{2 d^2 e} \]
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Rule 212
Rule 222
Rule 739
Rule 858
Rule 1489
Rule 1582
Rule 1665
Rule 5334
Rubi steps \begin{align*} \text {integral}& = -\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 \text {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) \text {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 \text {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 ) \text {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 ) \text {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 ) \text {arctanh}\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*}
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.44 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}} x}{d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b \sec ^{-1}(c x)}{e (d+e x)^2}-\frac {b \arcsin \left (\frac {1}{c x}\right )}{d^2 e}+\frac {b \left (-2 c^2 d^2+e^2\right ) \log (d+e x)}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}+\frac {b \left (2 c^2 d^2-e^2\right ) \log \left (e+c \left (c d-\sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(573\) vs. \(2(159)=318\).
Time = 2.07 (sec) , antiderivative size = 574, normalized size of antiderivative = 3.34
method | result | size |
parts | \(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \operatorname {arcsec}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{2} e x -\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{3}+2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{2} e x +2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{3}+\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}+\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e^{3} c x +\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}-\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) e^{3} c x -\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c d \,e^{2}\right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \,d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(574\) |
derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{3}+\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{2} e x -2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e^{3} c x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x \,d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(583\) |
default | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{3}+\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c^{3} d^{2} e x -2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e^{3} c x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}\, e -2 d x \,c^{2}-2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x \,d^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c e x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(583\) |
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Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (156) = 312\).
Time = 0.55 (sec) , antiderivative size = 1117, normalized size of antiderivative = 6.49 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\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 ] \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}} \,d x } \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \]
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