Optimal. Leaf size=72 \[ \frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\sin (c+d x)}{b (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 72, 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} \[ \frac {d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\sin (c+d x)}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx &=-\frac {\sin (c+d x)}{b (a+b x)}+\frac {d \int \frac {\cos (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac {\sin (c+d x)}{b (a+b x)}+\frac {\left (d \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}-\frac {\left (d \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^2}-\frac {\sin (c+d x)}{b (a+b x)}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 66, normalized size = 0.92 \[ \frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )-d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )-\frac {b \sin (c+d x)}{a+b x}}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 123, normalized size = 1.71 \[ \frac {2 \, {\left (b d x + a d\right )} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + {\left ({\left (b d x + a d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b d x + a d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - 2 \, b \sin \left (d x + c\right )}{2 \, {\left (b^{3} x + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 518, normalized size = 7.19 \[ \frac {{\left ({\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - b c d^{2} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a d^{3} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Ci}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + {\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) - b c d^{2} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + a d^{3} \sin \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b c + a d}{b}\right ) + b d^{2} \sin \left (-\frac {{\left (b x + a\right )} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}}{b}\right )\right )} b^{2}}{{\left ({\left (b x + a\right )} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} - b^{5} c + a b^{4} d\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 107, normalized size = 1.49 \[ d \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.08, size = 164, normalized size = 2.28 \[ \frac {d^{2} {\left (-i \, E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + d^{2} {\left (E_{2}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left ({\left (d x + c\right )} b^{2} - b^{2} c + a b d\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sin {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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