Optimal. Leaf size=65 \[ \frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4406, 12, 3303, 3299, 3302} \[ \frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rubi steps
\begin {align*} \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx &=\int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx\\ &=\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx\\ &=\frac {1}{2} \cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\frac {1}{2} \sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx\\ &=\frac {\text {Ci}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 60, normalized size = 0.92 \[ \frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )+\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 80, normalized size = 1.23 \[ \frac {{\left (\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + \operatorname {Ci}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.85, size = 569, normalized size = 8.75 \[ \frac {\Im \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a)^{2} \tan \left (\frac {b c}{d}\right )^{2} - \Im \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a)^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \relax (a)^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Re \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a)^{2} \tan \left (\frac {b c}{d}\right ) + 2 \, \Re \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a)^{2} \tan \left (\frac {b c}{d}\right ) - 2 \, \Re \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a) \tan \left (\frac {b c}{d}\right )^{2} - 2 \, \Re \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a) \tan \left (\frac {b c}{d}\right )^{2} - \Im \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a)^{2} + \Im \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a)^{2} - 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \relax (a)^{2} + 4 \, \Im \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a) \tan \left (\frac {b c}{d}\right ) - 4 \, \Im \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a) \tan \left (\frac {b c}{d}\right ) + 8 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \relax (a) \tan \left (\frac {b c}{d}\right ) - \Im \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} + \Im \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} - 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Re \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a) + 2 \, \Re \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \relax (a) - 2 \, \Re \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) - 2 \, \Re \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) + \Im \left (\operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) - \Im \left (\operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) + 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{4 \, {\left (d \tan \relax (a)^{2} \tan \left (\frac {b c}{d}\right )^{2} + d \tan \relax (a)^{2} + d \tan \left (\frac {b c}{d}\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 1.29 \[ \frac {\Si \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{2 d}-\frac {\Ci \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 141, normalized size = 2.17 \[ -\frac {b {\left (i \, E_{1}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (E_{1}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{1}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )}{c+d\,x} \,d x \]
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 \[ \int \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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