Optimal. Leaf size=56 \[ a d \cos (c) \text {Ci}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+\frac {b \sin (c+d x)}{d^2}-\frac {b x \cos (c+d x)}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3339, 3297, 3303, 3299, 3302, 3296, 2637} \[ a d \cos (c) \text {CosIntegral}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+\frac {b \sin (c+d x)}{d^2}-\frac {b x \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3339
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right ) \sin (c+d x)}{x^2} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^2}+b x \sin (c+d x)\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^2} \, dx+b \int x \sin (c+d x) \, dx\\ &=-\frac {b x \cos (c+d x)}{d}-\frac {a \sin (c+d x)}{x}+\frac {b \int \cos (c+d x) \, dx}{d}+(a d) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {b x \cos (c+d x)}{d}+\frac {b \sin (c+d x)}{d^2}-\frac {a \sin (c+d x)}{x}+(a d \cos (c)) \int \frac {\cos (d x)}{x} \, dx-(a d \sin (c)) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {b x \cos (c+d x)}{d}+a d \cos (c) \text {Ci}(d x)+\frac {b \sin (c+d x)}{d^2}-\frac {a \sin (c+d x)}{x}-a d \sin (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.14, size = 56, normalized size = 1.00 \[ a d \cos (c) \text {Ci}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+\frac {b \sin (c+d x)}{d^2}-\frac {b x \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 79, normalized size = 1.41 \[ -\frac {2 \, a d^{3} x \sin \relax (c) \operatorname {Si}\left (d x\right ) + 2 \, b d x^{2} \cos \left (d x + c\right ) - {\left (a d^{3} x \operatorname {Ci}\left (d x\right ) + a d^{3} x \operatorname {Ci}\left (-d x\right )\right )} \cos \relax (c) + 2 \, {\left (a d^{2} - b x\right )} \sin \left (d x + c\right )}{2 \, d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.56, size = 489, normalized size = 8.73 \[ -\frac {a d^{3} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{3} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{3} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d^{3} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - a d^{3} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{3} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{3} x \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{3} x \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{3} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right ) - a d^{3} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) - a d^{3} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) - 2 \, b d x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} - 8 \, b d x^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - 4 \, a d^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, b d x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 4 \, a d^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, b x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, b x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b d x^{2} + 4 \, a d^{2} \tan \left (\frac {1}{2} \, d x\right ) + 4 \, a d^{2} \tan \left (\frac {1}{2} \, c\right ) - 4 \, b x \tan \left (\frac {1}{2} \, d x\right ) - 4 \, b x \tan \left (\frac {1}{2} \, c\right )}{2 \, {\left (d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} + d^{2} x \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 79, normalized size = 1.41 \[ d \left (\frac {\left (1+2 c \right ) b \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}+\frac {3 c b \cos \left (d x +c \right )}{d^{3}}+a \left (-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.27, size = 69, normalized size = 1.23 \[ \frac {{\left (a {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \cos \relax (c) + a {\left (-i \, \Gamma \left (-1, i \, d x\right ) + i \, \Gamma \left (-1, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{3} - 2 \, b d x \cos \left (d x + c\right ) + 2 \, b \sin \left (d x + c\right )}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b x^{3}\right ) \sin {\left (c + d x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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