Optimal. Leaf size=227 \[ \frac {i f^2 \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))} \]
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Rubi [A] time = 0.31, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3725, 3724, 3303, 3299, 3302} \[ \frac {i f^2 \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3724
Rule 3725
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx &=\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}+\frac {(i f) \int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))} \, dx}{d}\\ &=\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}-\frac {\left (i f^2\right ) \int \frac {\sin \left (2 \left (e+\frac {\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{a d^2}-\frac {f^2 \int \frac {\cos \left (2 \left (e+\frac {\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{a d^2}\\ &=\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {\left (i f^2 \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^2}+\frac {\left (f^2 \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^2}+\frac {\left (i f^2 \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^2}-\frac {\left (f^2 \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^2}\\ &=\frac {i f}{2 a d^2 (c+d x)}+\frac {f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f^2 \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}\\ \end {align*}
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Mathematica [A] time = 1.59, size = 283, normalized size = 1.25 \[ \frac {\left (\cos \left (f \left (x-\frac {c}{d}\right )+e\right )+i \sin \left (f \left (x-\frac {c}{d}\right )+e\right )\right ) \left (4 f^2 (c+d x)^2 \text {Ci}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )+i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+i \left (4 f^2 (c+d x)^2 \text {Si}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )+i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+d \left (d \sin \left (f \left (x-\frac {c}{d}\right )+e\right )+d \sin \left (f \left (\frac {c}{d}+x\right )+e\right )+2 i c f \sin \left (f \left (\frac {c}{d}+x\right )+e\right )+2 i d f x \sin \left (f \left (\frac {c}{d}+x\right )+e\right )+i d \cos \left (f \left (x-\frac {c}{d}\right )+e\right )+(2 c f+2 d f x-i d) \cos \left (f \left (\frac {c}{d}+x\right )+e\right )\right )\right )\right )}{4 a d^3 (c+d x)^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.83, size = 118, normalized size = 0.52 \[ \frac {4 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (\frac {2 i \, d f x + 2 i \, c f}{d}\right ) e^{\left (\frac {2 i \, d e - 2 i \, c f}{d}\right )} - d^{2} + {\left (2 i \, d^{2} f x + 2 i \, c d f + d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{4 \, {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 1630, normalized size = 7.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.64, size = 143, normalized size = 0.63 \[ -\frac {1}{4 d \left (d x +c \right )^{2} a}-\frac {f^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{4 a \,d^{3} \left (i f x +\frac {i c f}{d}\right )^{2}}-\frac {f^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{2 a \,d^{3} \left (i f x +\frac {i c f}{d}\right )}-\frac {f^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \Ei \left (1, -2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{a \,d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 160, normalized size = 0.70 \[ \frac {2 \, f^{3} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{3}\left (-\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - 2 i \, f^{3} E_{3}\left (-\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{3}}{4 \, {\left ({\left (f x + e\right )}^{2} a d^{3} + a d^{3} e^{2} - 2 \, a c d^{2} e f + a c^{2} d f^{2} - 2 \, {\left (a d^{3} e - a c d^{2} f\right )} {\left (f x + e\right )}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^3} \,d x \]
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 \[ - \frac {i \int \frac {1}{c^{3} \cot {\left (e + f x \right )} - i c^{3} + 3 c^{2} d x \cot {\left (e + f x \right )} - 3 i c^{2} d x + 3 c d^{2} x^{2} \cot {\left (e + f x \right )} - 3 i c d^{2} x^{2} + d^{3} x^{3} \cot {\left (e + f x \right )} - i d^{3} x^{3}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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