Optimal. Leaf size=72 \[ \frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\sin (a+b x)}{d (c+d x)}-\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2} \]
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Rubi [A]
time = 0.09, 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 = {3378, 3384,
3380, 3383} \begin {gather*} \frac {b \cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\sin (a+b x)}{d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rubi steps
\begin {align*} \int \frac {\sin (a+b x)}{(c+d x)^2} \, dx &=-\frac {\sin (a+b x)}{d (c+d x)}+\frac {b \int \frac {\cos (a+b x)}{c+d x} \, dx}{d}\\ &=-\frac {\sin (a+b x)}{d (c+d x)}+\frac {\left (b \cos \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 {\left (b \sin \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 {b \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\sin (a+b x)}{d (c+d x)}-\frac {b \sin \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 = 0.15, size = 66, normalized size = 0.92 \begin {gather*} \frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (b \left (\frac {c}{d}+x\right )\right )-\frac {d \sin (a+b x)}{c+d x}-b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 112, normalized size = 1.56
method | result | size |
derivativedivides | \(b \left (-\frac {\sin \left (b x +a \right )}{\left (-d a +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}+\frac {\cosineIntegral \left (b x +a +\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}}{d}\right )\) | \(112\) |
default | \(b \left (-\frac {\sin \left (b x +a \right )}{\left (-d a +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +c b}{d}\right ) \sin \left (\frac {-d a +c b}{d}\right )}{d}+\frac {\cosineIntegral \left (b x +a +\frac {-d a +c b}{d}\right ) \cos \left (\frac {-d a +c b}{d}\right )}{d}}{d}\right )\) | \(112\) |
risch | \(-\frac {b \,{\mathrm e}^{\frac {i \left (d a -c b \right )}{d}} \expIntegral \left (1, -i b x -i a -\frac {-i a d +i b c}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{d}} \expIntegral \left (1, i b x +i a -\frac {i \left (d a -c b \right )}{d}\right )}{2 d^{2}}-\frac {\left (-2 d x b -2 c b \right ) \sin \left (b x +a \right )}{2 d \left (d x +c \right ) \left (-d x b -c b \right )}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.36, size = 164, normalized size = 2.28 \begin {gather*} -\frac {b^{2} {\left (i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, 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 (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 )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 124, normalized size = 1.72 \begin {gather*} -\frac {2 \, {\left (b d x + b c\right )} \sin \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\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 )} \cos \left (-\frac {b c - a d}{d}\right ) + 2 \, d \sin \left (b x + a\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \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 (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs.
\(2 (72) = 144\).
time = 4.26, size = 521, normalized size = 7.24 \begin {gather*} \frac {{\left ({\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 {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 ) + b^{3} c \cos \left (-\frac {b c - a d}{d}\right ) \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 ) - a b^{2} d \cos \left (-\frac {b c - a d}{d}\right ) \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 ) + {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \sin \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 \sin \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 \sin \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 \sin \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} \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 (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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