Optimal. Leaf size=62 \[ a^2 \sin (c) \text {Ci}(d x)+a^2 \cos (c) \text {Si}(d x)-\frac {2 a b \cos (c+d x)}{d}+\frac {b^2 \sin (c+d x)}{d^2}-\frac {b^2 x \cos (c+d x)}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 62, 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 = {6742, 2638, 3303, 3299, 3302, 3296, 2637} \[ a^2 \sin (c) \text {CosIntegral}(d x)+a^2 \cos (c) \text {Si}(d x)-\frac {2 a b \cos (c+d x)}{d}+\frac {b^2 \sin (c+d x)}{d^2}-\frac {b^2 x \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \sin (c+d x)}{x} \, dx &=\int \left (2 a b \sin (c+d x)+\frac {a^2 \sin (c+d x)}{x}+b^2 x \sin (c+d x)\right ) \, dx\\ &=a^2 \int \frac {\sin (c+d x)}{x} \, dx+(2 a b) \int \sin (c+d x) \, dx+b^2 \int x \sin (c+d x) \, dx\\ &=-\frac {2 a b \cos (c+d x)}{d}-\frac {b^2 x \cos (c+d x)}{d}+\frac {b^2 \int \cos (c+d x) \, dx}{d}+\left (a^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx+\left (a^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {2 a b \cos (c+d x)}{d}-\frac {b^2 x \cos (c+d x)}{d}+a^2 \text {Ci}(d x) \sin (c)+\frac {b^2 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.31, size = 51, normalized size = 0.82 \[ a^2 \sin (c) \text {Ci}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {b (b \sin (c+d x)-d (2 a+b x) \cos (c+d x))}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 78, normalized size = 1.26 \[ \frac {2 \, a^{2} d^{2} \cos \relax (c) \operatorname {Si}\left (d x\right ) + 2 \, b^{2} \sin \left (d x + c\right ) - 2 \, {\left (b^{2} d x + 2 \, a b d\right )} \cos \left (d x + c\right ) + {\left (a^{2} d^{2} \operatorname {Ci}\left (d x\right ) + a^{2} d^{2} \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c)}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.74, size = 551, normalized size = 8.89 \[ -\frac {a^{2} d^{2} \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a^{2} d^{2} \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a^{2} d^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a^{2} d^{2} \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a^{2} d^{2} \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, b^{2} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a^{2} d^{2} \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2} d^{2} \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, a^{2} d^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2} d^{2} \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - a^{2} d^{2} \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a^{2} d^{2} \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 4 \, a b d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, b^{2} d x \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, a^{2} d^{2} \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a^{2} d^{2} \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, b^{2} d x \tan \left (\frac {1}{2} \, c\right )^{2} - a^{2} d^{2} \Im \left (\operatorname {Ci}\left (d x\right ) \right ) + a^{2} d^{2} \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) - 2 \, a^{2} d^{2} \operatorname {Si}\left (d x\right ) - 4 \, a b d \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a b d \tan \left (\frac {1}{2} \, c\right )^{2} - 4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b^{2} d x + 4 \, a b d - 4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{2 \, {\left (d^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + d^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 79, normalized size = 1.27 \[ \frac {\left (1+c \right ) b^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}-\frac {2 a b \cos \left (d x +c \right )}{d}+\frac {2 c \,b^{2} \cos \left (d x +c \right )}{d^{2}}+a^{2} \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.74, size = 80, normalized size = 1.29 \[ \frac {{\left (a^{2} {\left (-i \, {\rm Ei}\left (i \, d x\right ) + i \, {\rm Ei}\left (-i \, d x\right )\right )} \cos \relax (c) + a^{2} {\left ({\rm Ei}\left (i \, d x\right ) + {\rm Ei}\left (-i \, d x\right )\right )} \sin \relax (c)\right )} d^{2} + 2 \, b^{2} \sin \left (d x + c\right ) - 2 \, {\left (b^{2} d x + 2 \, a b d\right )} \cos \left (d x + c\right )}{2 \, d^{2}} \]
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 \[ b^2\,\cos \relax (c)\,\left (\frac {\sin \left (d\,x\right )}{d^2}-\frac {x\,\cos \left (d\,x\right )}{d}\right )+b^2\,\sin \relax (c)\,\left (\frac {\cos \left (d\,x\right )}{d^2}+\frac {x\,\sin \left (d\,x\right )}{d}\right )+a^2\,\mathrm {cosint}\left (d\,x\right )\,\sin \relax (c)+a^2\,\mathrm {sinint}\left (d\,x\right )\,\cos \relax (c)-\frac {2\,a\,b\,\cos \left (d\,x\right )\,\cos \relax (c)}{d}+\frac {2\,a\,b\,\sin \left (d\,x\right )\,\sin \relax (c)}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.91, size = 90, normalized size = 1.45 \[ a^{2} \sin {\relax (c )} \operatorname {Ci}{\left (d x \right )} + a^{2} \cos {\relax (c )} \operatorname {Si}{\left (d x \right )} + 2 a b \left (\begin {cases} - \cos {\relax (c )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) + b^{2} x \left (\begin {cases} - \cos {\relax (c )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - b^{2} \left (\begin {cases} - x \cos {\relax (c )} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\relax (c )} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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