Optimal. Leaf size=299 \[ -\frac {3 b \sin (c) \text {Ci}(d x)}{a^4}+\frac {3 b \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a^4}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}+\frac {3 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^4}+\frac {2 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 b \sin (c+d x)}{a^3 (a+b x)}+\frac {d \cos (c) \text {Ci}(d x)}{a^3}-\frac {d \sin (c) \text {Si}(d x)}{a^3}-\frac {\sin (c+d x)}{a^3 x}-\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{2 a^2 b}-\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 a^2 b}-\frac {b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac {d \cos (c+d x)}{2 a^2 (a+b x)} \]
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Rubi [A] time = 0.67, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3299, 3302} \[ -\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 a^2 b}-\frac {3 b \sin (c) \text {CosIntegral}(d x)}{a^4}+\frac {3 b \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^4}+\frac {2 d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 a^2 b}-\frac {2 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}+\frac {3 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^4}-\frac {2 b \sin (c+d x)}{a^3 (a+b x)}-\frac {b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac {d \cos (c+d x)}{2 a^2 (a+b x)}+\frac {d \cos (c) \text {CosIntegral}(d x)}{a^3}-\frac {d \sin (c) \text {Si}(d x)}{a^3}-\frac {\sin (c+d x)}{a^3 x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x^2 (a+b x)^3} \, dx &=\int \left (\frac {\sin (c+d x)}{a^3 x^2}-\frac {3 b \sin (c+d x)}{a^4 x}+\frac {b^2 \sin (c+d x)}{a^2 (a+b x)^3}+\frac {2 b^2 \sin (c+d x)}{a^3 (a+b x)^2}+\frac {3 b^2 \sin (c+d x)}{a^4 (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^2} \, dx}{a^3}-\frac {(3 b) \int \frac {\sin (c+d x)}{x} \, dx}{a^4}+\frac {\left (3 b^2\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{a^4}+\frac {\left (2 b^2\right ) \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{a^3}+\frac {b^2 \int \frac {\sin (c+d x)}{(a+b x)^3} \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{a^3 x}-\frac {b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac {2 b \sin (c+d x)}{a^3 (a+b x)}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{a^3}+\frac {(2 b d) \int \frac {\cos (c+d x)}{a+b x} \, dx}{a^3}+\frac {(b d) \int \frac {\cos (c+d x)}{(a+b x)^2} \, dx}{2 a^2}-\frac {(3 b \cos (c)) \int \frac {\sin (d x)}{x} \, dx}{a^4}+\frac {\left (3 b^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}{a^4}-\frac {(3 b \sin (c)) \int \frac {\cos (d x)}{x} \, dx}{a^4}+\frac {\left (3 b^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}{a^4}\\ &=-\frac {d \cos (c+d x)}{2 a^2 (a+b x)}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}+\frac {3 b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^4}-\frac {\sin (c+d x)}{a^3 x}-\frac {b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac {2 b \sin (c+d x)}{a^3 (a+b x)}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}+\frac {3 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^4}-\frac {d^2 \int \frac {\sin (c+d x)}{a+b x} \, dx}{2 a^2}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{a^3}+\frac {\left (2 b 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}{a^3}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{a^3}-\frac {\left (2 b 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}{a^3}\\ &=-\frac {d \cos (c+d x)}{2 a^2 (a+b x)}+\frac {d \cos (c) \text {Ci}(d x)}{a^3}+\frac {2 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}+\frac {3 b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^4}-\frac {\sin (c+d x)}{a^3 x}-\frac {b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac {2 b \sin (c+d x)}{a^3 (a+b x)}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d \sin (c) \text {Si}(d x)}{a^3}+\frac {3 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^4}-\frac {2 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {\left (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 a^2}-\frac {\left (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 a^2}\\ &=-\frac {d \cos (c+d x)}{2 a^2 (a+b x)}+\frac {d \cos (c) \text {Ci}(d x)}{a^3}+\frac {2 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {3 b \text {Ci}(d x) \sin (c)}{a^4}+\frac {3 b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^4}-\frac {d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 a^2 b}-\frac {\sin (c+d x)}{a^3 x}-\frac {b \sin (c+d x)}{2 a^2 (a+b x)^2}-\frac {2 b \sin (c+d x)}{a^3 (a+b x)}-\frac {3 b \cos (c) \text {Si}(d x)}{a^4}-\frac {d \sin (c) \text {Si}(d x)}{a^3}+\frac {3 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^4}-\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 a^2 b}-\frac {2 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 2.07, size = 540, normalized size = 1.81 \[ -\frac {a^4 d^2 x \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d^2 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d x \sin (c) \text {Si}(d x)+4 a^3 b d x \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b \sin (c+d x)+a^3 b d x \cos (c+d x)+x (a+b x)^2 \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (a^2 d^2-6 b^2\right ) \sin \left (c-\frac {a d}{b}\right )-4 a b d \cos \left (c-\frac {a d}{b}\right )\right )+a^2 b^2 d^2 x^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+4 a^2 b^2 d x^2 \sin (c) \text {Si}(d x)+8 a^2 b^2 d x^2 \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+6 a^2 b^2 x \cos (c) \text {Si}(d x)-6 a^2 b^2 x \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a^2 b^2 d x^2 \cos (c+d x)+9 a^2 b^2 x \sin (c+d x)-6 b^4 x^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a b^3 d x^3 \sin (c) \text {Si}(d x)+4 a b^3 d x^3 \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+12 a b^3 x^2 \cos (c) \text {Si}(d x)-12 a b^3 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+6 a b^3 x^2 \sin (c+d x)+2 b x (a+b x)^2 \text {Ci}(d x) (3 b \sin (c)-a d \cos (c))+6 b^4 x^3 \cos (c) \text {Si}(d x)}{2 a^4 b x (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 689, normalized size = 2.30 \[ -\frac {2 \, {\left (a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a b^{3} d x^{3} + 2 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \operatorname {Ci}\left (d x\right ) + {\left (a b^{3} d x^{3} + 2 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \operatorname {Ci}\left (-d x\right ) - 6 \, {\left (b^{4} x^{3} + 2 \, a b^{3} x^{2} + a^{2} b^{2} x\right )} \operatorname {Si}\left (d x\right )\right )} \cos \relax (c) - 2 \, {\left (2 \, {\left (a b^{3} d x^{3} + 2 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (a b^{3} d x^{3} + 2 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - {\left ({\left (a^{2} b^{2} d^{2} - 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} - 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} - 6 \, a^{2} b^{2}\right )} x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (6 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x + 2 \, a^{3} b\right )} \sin \left (d x + c\right ) + 2 \, {\left (3 \, {\left (b^{4} x^{3} + 2 \, a b^{3} x^{2} + a^{2} b^{2} x\right )} \operatorname {Ci}\left (d x\right ) + 3 \, {\left (b^{4} x^{3} + 2 \, a b^{3} x^{2} + a^{2} b^{2} x\right )} \operatorname {Ci}\left (-d x\right ) + 2 \, {\left (a b^{3} d x^{3} + 2 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \operatorname {Si}\left (d x\right )\right )} \sin \relax (c) - {\left ({\left ({\left (a^{2} b^{2} d^{2} - 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} - 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} - 6 \, a^{2} b^{2}\right )} x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left ({\left (a^{2} b^{2} d^{2} - 6 \, b^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{2} - 6 \, a b^{3}\right )} x^{2} + {\left (a^{4} d^{2} - 6 \, a^{2} b^{2}\right )} x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + 8 \, {\left (a b^{3} d x^{3} + 2 \, a^{2} b^{2} d x^{2} + a^{3} b d x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\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 [A] time = 0.03, size = 405, normalized size = 1.35 \[ d \left (\frac {-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )}{a^{3}}+\frac {d \,b^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{a^{2}}+\frac {3 b^{2} \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d \,a^{4}}-\frac {3 b \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{d \,a^{4}}+\frac {2 b^{2} \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{3}}\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {\sin {\left (c + d x \right )}}{x^{2} \left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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