Optimal. Leaf size=168 \[ -\frac {f \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {i f \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))} \]
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Rubi [A] time = 0.24, 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 = {3724, 3303, 3299, 3302} \[ -\frac {f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {i f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^2}+\frac {i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {1}{d (c+d x) (a+i a \tan (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3724
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 = 0.85, size = 224, normalized size = 1.33 \[ \frac {\sec (e+f x) \left (\cos \left (\frac {c f}{d}\right )+i \sin \left (\frac {c f}{d}\right )\right ) \left (-2 f (c+d x) \text {Ci}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )-i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+2 f (c+d x) \text {Si}\left (\frac {2 f (c+d x)}{d}\right ) \left (\sin \left (e-\frac {f (c+d x)}{d}\right )+i \cos \left (e-\frac {f (c+d x)}{d}\right )\right )+d \left (-\sin \left (f \left (x-\frac {c}{d}\right )+e\right )+\sin \left (f \left (\frac {c}{d}+x\right )+e\right )+i \cos \left (f \left (x-\frac {c}{d}\right )+e\right )+i \cos \left (f \left (\frac {c}{d}+x\right )+e\right )\right )\right )}{2 a d^2 (c+d x) (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 84, normalized size = 0.50 \[ \frac {{\left ({\left ({\left (-2 i \, d f x - 2 i \, c f\right )} {\rm Ei}\left (\frac {-2 i \, d f x - 2 i \, c f}{d}\right ) e^{\left (\frac {-2 i \, d e + 2 i \, c f}{d}\right )} - d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - d\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, {\left (a d^{3} x + a c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 74.99, size = 1099, normalized size = 6.54 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 285, normalized size = 1.70 \[ \frac {-\frac {i f^{2} \left (-\frac {2 \sin \left (2 f x +2 e \right )}{\left (\left (f x +e \right ) d +c f -d e \right ) d}+\frac {\frac {4 \Si \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {4 \Ci \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}}{d}\right )}{4}+\frac {f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (\left (f x +e \right ) d +c f -d e \right ) d}-\frac {2 \left (\frac {2 \Si \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \Ci \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {f^{2}}{2 \left (\left (f x +e \right ) d +c f -d e \right ) d}}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 117, normalized size = 0.70 \[ -\frac {f^{2} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) + i \, f^{2} E_{2}\left (\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + f^{2}}{2 \, {\left ({\left (f x + e\right )} a d^{2} - a d^{2} e + a c d f\right )} f} \]
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 \[ \int \frac {1}{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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 \[ - \frac {i \int \frac {1}{c^{2} \tan {\left (e + f x \right )} - i c^{2} + 2 c d x \tan {\left (e + f x \right )} - 2 i c d x + d^{2} x^{2} \tan {\left (e + f x \right )} - i d^{2} x^{2}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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