Optimal. Leaf size=179 \[ \frac {a d^2 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {a d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^4}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {a d \cos (c+d x)}{2 b^3 (a+b x)}-\frac {\sin (c+d x)}{b^2 (a+b x)}+\frac {a \sin (c+d x)}{2 b^2 (a+b x)^2} \]
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Rubi [A] time = 0.35, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3299, 3302} \[ \frac {a d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 b^4}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^4}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}-\frac {\sin (c+d x)}{b^2 (a+b x)}+\frac {a \sin (c+d x)}{2 b^2 (a+b x)^2}+\frac {a d \cos (c+d x)}{2 b^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \sin (c+d x)}{(a+b x)^3} \, dx &=\int \left (-\frac {a \sin (c+d x)}{b (a+b x)^3}+\frac {\sin (c+d x)}{b (a+b x)^2}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{b}-\frac {a \int \frac {\sin (c+d x)}{(a+b x)^3} \, dx}{b}\\ &=\frac {a \sin (c+d x)}{2 b^2 (a+b x)^2}-\frac {\sin (c+d x)}{b^2 (a+b x)}+\frac {d \int \frac {\cos (c+d x)}{a+b x} \, dx}{b^2}-\frac {(a d) \int \frac {\cos (c+d x)}{(a+b x)^2} \, dx}{2 b^2}\\ &=\frac {a d \cos (c+d x)}{2 b^3 (a+b x)}+\frac {a \sin (c+d x)}{2 b^2 (a+b x)^2}-\frac {\sin (c+d x)}{b^2 (a+b x)}+\frac {\left (a d^2\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{2 b^3}+\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^2}-\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^2}\\ &=\frac {a d \cos (c+d x)}{2 b^3 (a+b x)}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a \sin (c+d x)}{2 b^2 (a+b x)^2}-\frac {\sin (c+d x)}{b^2 (a+b x)}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {\left (a d^2 \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^3}+\frac {\left (a d^2 \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{2 b^3}\\ &=\frac {a d \cos (c+d x)}{2 b^3 (a+b x)}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^3}+\frac {a d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^4}+\frac {a \sin (c+d x)}{2 b^2 (a+b x)^2}-\frac {\sin (c+d x)}{b^2 (a+b x)}+\frac {a d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^4}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 157, normalized size = 0.88 \[ \frac {d (a+b x)^2 \left (\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \sin \left (c-\frac {a d}{b}\right )+2 b \cos \left (c-\frac {a d}{b}\right )\right )+\text {Si}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac {a d}{b}\right )-2 b \sin \left (c-\frac {a d}{b}\right )\right )\right )+b \cos (d x) (a d \cos (c) (a+b x)-b \sin (c) (a+2 b x))-b \sin (d x) (a d \sin (c) (a+b x)+b \cos (c) (a+2 b x))}{2 b^4 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 346, normalized size = 1.93 \[ \frac {2 \, {\left (a b^{2} d x + a^{2} b d\right )} \cos \left (d x + c\right ) + 2 \, {\left ({\left (b^{3} d x^{2} + 2 \, a b^{2} d x + a^{2} b d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{3} d x^{2} + 2 \, a b^{2} d x + a^{2} b d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + {\left (a b^{2} d^{2} x^{2} + 2 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (2 \, b^{3} x + a b^{2}\right )} \sin \left (d x + c\right ) - {\left ({\left (a b^{2} d^{2} x^{2} + 2 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b^{2} d^{2} x^{2} + 2 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 4 \, {\left (b^{3} d x^{2} + 2 \, a b^{2} d x + a^{2} b d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 419, normalized size = 2.34 \[ \frac {-\frac {d^{3} \left (d a -c b \right ) \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{b}+\frac {d^{3} \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{b}-d^{3} c \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {x \sin {\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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