Optimal. Leaf size=73 \[ -\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\cos (a+b x)}{d (c+d x)} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3297, 3303, 3299, 3302} \[ -\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\cos (a+b x)}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rubi steps
\begin {align*} \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx &=-\frac {\cos (a+b x)}{d (c+d x)}-\frac {b \int \frac {\sin (a+b x)}{c+d x} \, dx}{d}\\ &=-\frac {\cos (a+b x)}{d (c+d x)}-\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}-\frac {\left (b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}\\ &=-\frac {\cos (a+b x)}{d (c+d x)}-\frac {b \text {Ci}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 65, normalized size = 0.89 \[ -\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (b \left (\frac {c}{d}+x\right )\right )+b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+\frac {d \cos (a+b x)}{c+d x}}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 123, normalized size = 1.68 \[ -\frac {2 \, {\left (b d x + b c\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) + 2 \, d \cos \left (b x + a\right ) + {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {b d x + b c}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 523, normalized size = 7.16 \[ -\frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + b^{3} c \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - a b^{2} d \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) - b^{3} c \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + a b^{2} d \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + b^{2} d \cos \left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )\right )} d^{2}}{{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 109, normalized size = 1.49 \[ b \left (-\frac {\cos \left (b x +a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}-\frac {\frac {\Si \left (b x +a +\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}-\frac {\Ci \left (b x +a +\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}}{d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.92, size = 166, normalized size = 2.27 \[ -\frac {8 \, b^{2} {\left (E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - b^{2} {\left (8 i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - 8 i \, E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{16 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]
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 \[ \int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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