Optimal. Leaf size=305 \[ \frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\cos \left (4 e-\frac {4 c f}{d}\right ) \text {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {i \text {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}-\frac {\sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d} \]
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Rubi [A]
time = 0.55, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3809, 3384,
3380, 3383, 3393} \begin {gather*} -\frac {i \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \text {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\text {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 3809
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) (a+i a \tan (e+f x))^2} \, dx &=\int \left (\frac {1}{4 a^2 (c+d x)}+\frac {\cos (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {\cos ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {i \sin (2 e+2 f x)}{2 a^2 (c+d x)}-\frac {\sin ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {i \sin (4 e+4 f x)}{4 a^2 (c+d x)}\right ) \, dx\\ &=\frac {\log (c+d x)}{4 a^2 d}-\frac {i \int \frac {\sin (4 e+4 f x)}{c+d x} \, dx}{4 a^2}-\frac {i \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{2 a^2}+\frac {\int \frac {\cos ^2(2 e+2 f x)}{c+d x} \, dx}{4 a^2}-\frac {\int \frac {\sin ^2(2 e+2 f x)}{c+d x} \, dx}{4 a^2}+\frac {\int \frac {\cos (2 e+2 f x)}{c+d x} \, dx}{2 a^2}\\ &=\frac {\log (c+d x)}{4 a^2 d}-\frac {\int \left (\frac {1}{2 (c+d x)}-\frac {\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{4 a^2}+\frac {\int \left (\frac {1}{2 (c+d x)}+\frac {\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{4 a^2}-\frac {\left (i \cos \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sin \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^2}-\frac {\left (i \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}{2 a^2}+\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cos \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a^2}-\frac {\left (i \sin \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cos \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^2}-\frac {\left (i \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}{2 a^2}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a^2}\\ &=\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {i \text {Ci}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \frac {\int \frac {\cos (4 e+4 f x)}{c+d x} \, dx}{8 a^2}\\ &=\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {i \text {Ci}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \left (\frac {\cos \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\cos \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^2}-\frac {\sin \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\sin \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^2}\right )\\ &=\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {i \text {Ci}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \left (\frac {\cos \left (4 e-\frac {4 c f}{d}\right ) \text {Ci}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^2 d}-\frac {\sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^2 d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 211, normalized size = 0.69 \begin {gather*} \frac {\left (\cos \left (2 e-\frac {2 c f}{d}\right )-i \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \left (2 \text {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right )+\cos \left (2 e-\frac {2 c f}{d}\right ) \log (f (c+d x))+\text {CosIntegral}\left (\frac {4 f (c+d x)}{d}\right ) \left (\cos \left (2 e-\frac {2 c f}{d}\right )-i \sin \left (2 e-\frac {2 c f}{d}\right )\right )+i \log (f (c+d x)) \sin \left (2 e-\frac {2 c f}{d}\right )-2 i \text {Si}\left (\frac {2 f (c+d x)}{d}\right )-i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {4 f (c+d x)}{d}\right )-\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 377, normalized size = 1.24
method | result | size |
risch | \(\frac {\ln \left (d x +c \right )}{4 a^{2} d}-\frac {{\mathrm e}^{\frac {4 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 4 i f x +4 i e +\frac {4 i \left (c f -d e \right )}{d}\right )}{4 a^{2} d}-\frac {{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{2 a^{2} d}\) | \(114\) |
default | \(\frac {-\frac {i f \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{4}-\frac {i f \left (\frac {4 \sinIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \cos \left (\frac {4 c f -4 d e}{d}\right )}{d}-\frac {4 \cosineIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \sin \left (\frac {4 c f -4 d e}{d}\right )}{d}\right )}{16}+\frac {f \left (\frac {4 \sinIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \sin \left (\frac {4 c f -4 d e}{d}\right )}{d}+\frac {4 \cosineIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \cos \left (\frac {4 c f -4 d e}{d}\right )}{d}\right )}{16}+\frac {f \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{4}+\frac {f \ln \left (c f -d e +d \left (f x +e \right )\right )}{4 d}}{a^{2} f}\) | \(377\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 208, normalized size = 0.68 \begin {gather*} -\frac {2 \, f \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) E_{1}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + f \cos \left (\frac {4 \, {\left (c f - d e\right )}}{d}\right ) E_{1}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + i \, f E_{1}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) \sin \left (\frac {4 \, {\left (c f - d e\right )}}{d}\right ) + 2 i \, f E_{1}\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 \log \left ({\left (f x + e\right )} d + c f - d e\right )}{4 \, a^{2} d 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 = 0.28 \begin {gather*} \frac {2 \, {\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 )} + {\rm Ei}\left (-\frac {4 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (-i \, c f + i \, d e\right )}}{d}\right )} + \log \left (\frac {d x + c}{d}\right )}{4 \, a^{2} d} \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 {\int \frac {1}{c \tan ^{2}{\left (e + f x \right )} - 2 i c \tan {\left (e + f x \right )} - c + d x \tan ^{2}{\left (e + f x \right )} - 2 i d x \tan {\left (e + f x \right )} - d x}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 420, normalized size = 1.38 \begin {gather*} \frac {2 \, \cos \left (\frac {2 \, c f}{d}\right ) \cos \left (2 \, e\right ) \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + \cos \left (2 \, e\right )^{2} \log \left (d x + c\right ) + 2 i \, \cos \left (2 \, e\right ) \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {2 \, c f}{d}\right ) + 2 i \, \cos \left (\frac {2 \, c f}{d}\right ) \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (2 \, e\right ) + 2 i \, \cos \left (2 \, e\right ) \log \left (d x + c\right ) \sin \left (2 \, e\right ) - 2 \, \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right ) - \log \left (d x + c\right ) \sin \left (2 \, e\right )^{2} - 2 i \, \cos \left (\frac {2 \, c f}{d}\right ) \cos \left (2 \, e\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 \, \cos \left (2 \, e\right ) \sin \left (\frac {2 \, c f}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 \, \cos \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 2 i \, \sin \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + \cos \left (\frac {4 \, c f}{d}\right ) \operatorname {Ci}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + i \, \operatorname {Ci}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {4 \, c f}{d}\right ) - i \, \cos \left (\frac {4 \, c f}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + \sin \left (\frac {4 \, c f}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (d f x + c f\right )}}{d}\right )}{4 \, {\left (a^{2} d \cos \left (2 \, e\right )^{2} + 2 i \, a^{2} d \cos \left (2 \, e\right ) \sin \left (2 \, e\right ) - a^{2} d \sin \left (2 \, e\right )^{2}\right )}} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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