Optimal. Leaf size=179 \[ \frac{3 b \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{16 d^2}-\frac{3 b \cos \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b c}{d}+6 b x\right )}{16 d^2}-\frac{3 b \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{16 d^2}+\frac{3 b \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{16 d^2}-\frac{3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac{\sin (6 a+6 b x)}{32 d (c+d x)} \]
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Rubi [A] time = 0.297098, antiderivative size = 179, normalized size of antiderivative = 1., 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 4406
Rule 3297
Rule 3303
Rule 3299
Rule 3302
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*}
Mathematica [A] time = 0.9808, size = 189, normalized size = 1.06 \[ \frac{6 b (c+d x) \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )-6 b (c+d x) \cos \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b (c+d x)}{d}\right )-6 b (c+d x) \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+6 b (c+d x) \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b (c+d x)}{d}\right )-3 d \sin (2 a) \cos (2 b x)+d \sin (6 a) \cos (6 b x)-3 d \cos (2 a) \sin (2 b x)+d \cos (6 a) \sin (6 b x)}{32 d^2 (c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 256, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{{b}^{2}}{192} \left ( -6\,{\frac{\sin \left ( 6\,bx+6\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+6\,{\frac{1}{d} \left ( 6\,{\frac{1}{d}{\it Si} \left ( 6\,bx+6\,a+6\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 6\,{\frac{-ad+bc}{d}} \right ) }+6\,{\frac{1}{d}{\it Ci} \left ( 6\,bx+6\,a+6\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 6\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) }+{\frac{3\,{b}^{2}}{64} \left ( -2\,{\frac{\sin \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 2\,{\frac{-ad+bc}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 2\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.76057, size = 406, normalized size = 2.27 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.596735, size = 632, normalized size = 3.53 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{3} \sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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