Optimal. Leaf size=168 \[ -\frac {i f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))} \]
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Rubi [A]
time = 0.17, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3805, 3384,
3380, 3383} \begin {gather*} -\frac {f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {i f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 3805
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))} \, dx &=-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}-\frac {(i f) \int \frac {\cos (2 e+2 f x)}{c+d x} \, dx}{a d}-\frac {f \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{a d}\\ &=-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}-\frac {\left (i f \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}-\frac {\left (f \cos \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}+\frac {\left (i f \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}-\frac {\left (f \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}\\ &=-\frac {i f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {f \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.88, size = 224, normalized size = 1.33 \begin {gather*} \frac {\sec (e+f x) \left (\cos \left (\frac {c f}{d}\right )+i \sin \left (\frac {c f}{d}\right )\right ) \left (d \left (i \cos \left (e+f \left (-\frac {c}{d}+x\right )\right )+i \cos \left (e+f \left (\frac {c}{d}+x\right )\right )-\sin \left (e+f \left (-\frac {c}{d}+x\right )\right )+\sin \left (e+f \left (\frac {c}{d}+x\right )\right )\right )-2 f (c+d x) \text {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )-i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+2 f (c+d x) \left (i \cos \left (e-\frac {f (c+d x)}{d}\right )+\sin \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 a d^2 (c+d x) (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 285, normalized size = 1.70
method | result | size |
risch | \(-\frac {1}{2 d a \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-2 i \left (f x +e \right )}}{2 a \left (d x f +c f \right ) d}+\frac {i f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{a \,d^{2}}\) | \(96\) |
default | \(\frac {-\frac {i f^{2} \left (-\frac {2 \sin \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {4 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {4 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}}{d}\right )}{4}+\frac {f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {f^{2}}{2 \left (c f -d e +d \left (f x +e \right )\right ) d}}{a f}\) | \(285\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 127, normalized size = 0.76 \begin {gather*} -\frac {f^{2} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + i \, f^{2} E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) + f^{2}}{2 \, {\left ({\left (f x + e\right )} a d^{2} + a c d f - a d^{2} e\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 86, normalized size = 0.51 \begin {gather*} -\frac {{\left ({\left (2 \, {\left (i \, d f x + i \, c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, c f + i \, d e\right )}}{d}\right )} + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + d\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, {\left (a d^{3} x + a c d^{2}\right )}} \end {gather*}
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 \begin {gather*} - \frac {i \int \frac {1}{c^{2} \tan {\left (e + f x \right )} - i c^{2} + 2 c d x \tan {\left (e + f x \right )} - 2 i c d x + d^{2} x^{2} \tan {\left (e + f x \right )} - i d^{2} x^{2}}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1099 vs. \(2 (166) = 332\).
time = 2.77, size = 1099, normalized size = 6.54 \begin {gather*} -\frac {{\left (2 i \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Ci}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) - 2 i \, c f^{3} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Ci}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) + 2 i \, d f^{2} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Ci}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e - 2 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} \operatorname {Ci}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) + 2 \, c f^{3} \operatorname {Ci}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) - 2 \, d f^{2} \operatorname {Ci}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) + 2 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) - 2 \, c f^{3} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) + 2 \, d f^{2} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) e \operatorname {Si}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) + 2 i \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) - 2 i \, c f^{3} \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) + 2 i \, d f^{2} e \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) - d f^{2} \cos \left (\frac {2 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right ) - i \, d f^{2} \sin \left (\frac {2 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right ) - d f^{2}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} a d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - a c d^{4} f + a d^{5} e\right )} f} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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