Optimal. Leaf size=73 \[ \frac {\text {Ci}(d x) \sin (c)}{a}-\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a}+\frac {\cos (c) \text {Si}(d x)}{a}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a} \]
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Rubi [A]
time = 0.17, 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 = {6874, 3384,
3380, 3383} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 6874
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 = 0.09, size = 63, normalized size = 0.86 \begin {gather*} \frac {\text {Ci}(d x) \sin (c)-\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+\cos (c) \text {Si}(d x)-\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 99, normalized size = 1.36
method | result | size |
derivativedivides | \(\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a}-\frac {b \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a}\) | \(99\) |
default | \(\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a}-\frac {b \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a}\) | \(99\) |
risch | \(-\frac {i {\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a}+\frac {i {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a}+\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a}-\frac {{\mathrm e}^{-i c} \pi \,\mathrm {csgn}\left (d x \right )}{2 a}+\frac {{\mathrm e}^{-i c} \sinIntegral \left (d x \right )}{a}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, -i d x \right )}{2 a}\) | \(162\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 99, normalized size = 1.36 \begin {gather*} \frac {{\left (\operatorname {Ci}\left (d x\right ) + \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right ) + {\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 \left (c\right ) \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} \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*} \int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 3.45, size = 838, normalized size = 11.48 \begin {gather*} -\frac {\Im \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} - \Im \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} - \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \Re \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - \Im \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \Im \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, \Im \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right ) - 4 \, \Im \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right ) + 8 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {1}{2} \, c\right ) \tan \left (\frac {a d}{2 \, b}\right ) - \Im \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} + \Im \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} + \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \operatorname {Si}\left (d x\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} - 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) \tan \left (\frac {a d}{2 \, b}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) \tan \left (\frac {a d}{2 \, b}\right ) + \Im \left ( \operatorname {Ci}\left (d x + \frac {a d}{b}\right ) \right ) - \Im \left ( \operatorname {Ci}\left (d x\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-d x - \frac {a d}{b}\right ) \right ) + \Im \left ( \operatorname {Ci}\left (-d x\right ) \right ) - 2 \, \operatorname {Si}\left (d x\right ) + 2 \, \operatorname {Si}\left (\frac {b d x + a d}{b}\right )}{2 \, {\left (a \tan \left (\frac {1}{2} \, c\right )^{2} \tan \left (\frac {a d}{2 \, b}\right )^{2} + a \tan \left (\frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {a d}{2 \, b}\right )^{2} + a\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 = -0.01 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{x\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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