Optimal. Leaf size=179 \[ \frac {3 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}-\frac {3 b \cos \left (6 a-\frac {6 b c}{d}\right ) \text {Ci}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}+\frac {3 b \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)} \]
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Rubi [A] time = 0.30, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4406, 3297, 3303, 3299, 3302} \[ \frac {3 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}-\frac {3 b \cos \left (6 a-\frac {6 b c}{d}\right ) \text {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}+\frac {3 b \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rubi steps
\begin {align*} \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx &=\int \left (\frac {3 \sin (2 a+2 b x)}{32 (c+d x)^2}-\frac {\sin (6 a+6 b x)}{32 (c+d x)^2}\right ) \, dx\\ &=-\left (\frac {1}{32} \int \frac {\sin (6 a+6 b x)}{(c+d x)^2} \, dx\right )+\frac {3}{32} \int \frac {\sin (2 a+2 b x)}{(c+d x)^2} \, dx\\ &=-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)}+\frac {(3 b) \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx}{16 d}-\frac {(3 b) \int \frac {\cos (6 a+6 b x)}{c+d x} \, dx}{16 d}\\ &=-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)}-\frac {\left (3 b \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{c+d x} \, dx}{16 d}+\frac {\left (3 b \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{16 d}+\frac {\left (3 b \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{c+d x} \, dx}{16 d}-\frac {\left (3 b \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{16 d}\\ &=\frac {3 b \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}-\frac {3 b \cos \left (6 a-\frac {6 b c}{d}\right ) \text {Ci}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)}-\frac {3 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}+\frac {3 b \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}\\ \end {align*}
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Mathematica [A] time = 0.96, size = 189, normalized size = 1.06 \[ \frac {6 b (c+d x) \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Ci}\left (\frac {2 b (c+d x)}{d}\right )-6 b (c+d x) \cos \left (6 a-\frac {6 b c}{d}\right ) \text {Ci}\left (\frac {6 b (c+d x)}{d}\right )-6 b (c+d x) \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+6 b (c+d x) \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b (c+d x)}{d}\right )-3 d \sin (2 a) \cos (2 b x)+d \sin (6 a) \cos (6 b x)-3 d \cos (2 a) \sin (2 b x)+d \cos (6 a) \sin (6 b x)}{32 d^2 (c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 248, normalized size = 1.39 \[ \frac {6 \, {\left (b d x + b c\right )} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) - 6 \, {\left (b d x + b c\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + 3 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {6 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + 32 \, {\left (d \cos \left (b x + a\right )^{5} - d \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )}{32 \, {\left (d^{3} x + c d^{2}\right )}} \]
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 = 0.02, size = 256, normalized size = 1.43 \[ \frac {-\frac {b^{2} \left (-\frac {6 \sin \left (6 b x +6 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}+\frac {\frac {36 \Si \left (6 b x +6 a +\frac {-6 d a +6 c b}{d}\right ) \sin \left (\frac {-6 d a +6 c b}{d}\right )}{d}+\frac {36 \Ci \left (6 b x +6 a +\frac {-6 d a +6 c b}{d}\right ) \cos \left (\frac {-6 d a +6 c b}{d}\right )}{d}}{d}\right )}{192}+\frac {3 b^{2} \left (-\frac {2 \sin \left (2 b x +2 a \right )}{\left (\left (b x +a \right ) d -d a +c b \right ) d}+\frac {\frac {4 \Si \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}+\frac {4 \Ci \left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}}{d}\right )}{64}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.48, size = 301, normalized size = 1.68 \[ \frac {b^{2} {\left (-3 i \, E_{2}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + 3 i \, E_{2}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b^{2} {\left (i \, E_{2}\left (\frac {6 i \, b c + 6 i \, {\left (b x + a\right )} d - 6 i \, a d}{d}\right ) - i \, E_{2}\left (-\frac {6 i \, b c + 6 i \, {\left (b x + a\right )} d - 6 i \, a d}{d}\right )\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, b^{2} {\left (E_{2}\left (\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{2}\left (-\frac {2 i \, b c + 2 i \, {\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b^{2} {\left (E_{2}\left (\frac {6 i \, b c + 6 i \, {\left (b x + a\right )} d - 6 i \, a d}{d}\right ) + E_{2}\left (-\frac {6 i \, b c + 6 i \, {\left (b x + a\right )} d - 6 i \, a d}{d}\right )\right )} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )}{64 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]
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 \[ \int \frac {{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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