Optimal. Leaf size=73 \[ -\frac {\sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\sin (c) \text {Ci}(d x)}{a}+\frac {\cos (c) \text {Si}(d x)}{a} \]
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Rubi [A] time = 0.26, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6742, 3303, 3299, 3302} \[ -\frac {\sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a}+\frac {\sin (c) \text {CosIntegral}(d x)}{a}+\frac {\cos (c) \text {Si}(d x)}{a} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x (a+b x)} \, dx &=\int \left (\frac {\sin (c+d x)}{a x}-\frac {b \sin (c+d x)}{a (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x} \, dx}{a}-\frac {b \int \frac {\sin (c+d x)}{a+b x} \, dx}{a}\\ &=\frac {\cos (c) \int \frac {\sin (d x)}{x} \, dx}{a}-\frac {\left (b \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a}+\frac {\sin (c) \int \frac {\cos (d x)}{x} \, dx}{a}-\frac {\left (b \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a}\\ &=\frac {\text {Ci}(d x) \sin (c)}{a}-\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a}+\frac {\cos (c) \text {Si}(d x)}{a}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 63, normalized size = 0.86 \[ \frac {-\sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )-\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+\sin (c) \text {Ci}(d x)+\cos (c) \text {Si}(d x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 99, normalized size = 1.36 \[ \frac {{\left (\operatorname {Ci}\left (d x\right ) + \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c) + {\left (\operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right ) + 2 \, \cos \relax (c) \operatorname {Si}\left (d x\right ) - 2 \, \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 2.06, size = 838, normalized size = 11.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 99, normalized size = 1.36 \[ -\frac {b \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a}+\frac {\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )} x}\,{d x} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )}{x\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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