Optimal. Leaf size=185 \[ -\frac{\sin \left (5 a-\frac{5 b c}{d}\right ) \text{CosIntegral}\left (\frac{5 b c}{d}+5 b x\right )}{16 d}+\frac{\sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}-\frac{\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b c}{d}+5 b x\right )}{16 d} \]
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Rubi [A] time = 0.338659, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4406, 3303, 3299, 3302} \[ -\frac{\sin \left (5 a-\frac{5 b c}{d}\right ) \text{CosIntegral}\left (\frac{5 b c}{d}+5 b x\right )}{16 d}+\frac{\sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}-\frac{\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b c}{d}+5 b x\right )}{16 d} \]
Antiderivative was successfully verified.
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Rule 4406
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac{\sin (a+b x)}{8 (c+d x)}+\frac{\sin (3 a+3 b x)}{16 (c+d x)}-\frac{\sin (5 a+5 b x)}{16 (c+d x)}\right ) \, dx\\ &=\frac{1}{16} \int \frac{\sin (3 a+3 b x)}{c+d x} \, dx-\frac{1}{16} \int \frac{\sin (5 a+5 b x)}{c+d x} \, dx+\frac{1}{8} \int \frac{\sin (a+b x)}{c+d x} \, dx\\ &=-\left (\frac{1}{16} \cos \left (5 a-\frac{5 b c}{d}\right ) \int \frac{\sin \left (\frac{5 b c}{d}+5 b x\right )}{c+d x} \, dx\right )+\frac{1}{16} \cos \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\sin \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{8} \cos \left (a-\frac{b c}{d}\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx-\frac{1}{16} \sin \left (5 a-\frac{5 b c}{d}\right ) \int \frac{\cos \left (\frac{5 b c}{d}+5 b x\right )}{c+d x} \, dx+\frac{1}{16} \sin \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\cos \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{8} \sin \left (a-\frac{b c}{d}\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=-\frac{\text{Ci}\left (\frac{5 b c}{d}+5 b x\right ) \sin \left (5 a-\frac{5 b c}{d}\right )}{16 d}+\frac{\text{Ci}\left (\frac{3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac{3 b c}{d}\right )}{16 d}+\frac{\text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{8 d}+\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d}+\frac{\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{16 d}-\frac{\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b c}{d}+5 b x\right )}{16 d}\\ \end{align*}
Mathematica [A] time = 0.523339, size = 154, normalized size = 0.83 \[ \frac{\sin \left (5 a-\frac{5 b c}{d}\right ) \left (-\text{CosIntegral}\left (\frac{5 b (c+d x)}{d}\right )\right )+\sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b (c+d x)}{d}\right )+2 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (b \left (\frac{c}{d}+x\right )\right )+2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (b \left (\frac{c}{d}+x\right )\right )+\cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b (c+d x)}{d}\right )-\cos \left (5 a-\frac{5 b c}{d}\right ) \text{Si}\left (\frac{5 b (c+d x)}{d}\right )}{16 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 253, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{b}{80} \left ( 5\,{\frac{1}{d}{\it Si} \left ( 5\,bx+5\,a+5\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 5\,{\frac{-ad+bc}{d}} \right ) }-5\,{\frac{1}{d}{\it Ci} \left ( 5\,bx+5\,a+5\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 5\,{\frac{-ad+bc}{d}} \right ) } \right ) }+{\frac{b}{8} \left ({\frac{1}{d}{\it Si} \left ( bx+a+{\frac{-ad+bc}{d}} \right ) \cos \left ({\frac{-ad+bc}{d}} \right ) }-{\frac{1}{d}{\it Ci} \left ( bx+a+{\frac{-ad+bc}{d}} \right ) \sin \left ({\frac{-ad+bc}{d}} \right ) } \right ) }+{\frac{b}{48} \left ( 3\,{\frac{1}{d}{\it Si} \left ( 3\,bx+3\,a+3\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 3\,{\frac{-ad+bc}{d}} \right ) }-3\,{\frac{1}{d}{\it Ci} \left ( 3\,bx+3\,a+3\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 3\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.72738, size = 549, normalized size = 2.97 \begin{align*} \frac{b{\left (-2 i \, E_{1}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + 2 i \, E_{1}\left (-\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac{b c - a d}{d}\right ) + b{\left (-i \, E_{1}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + i \, E_{1}\left (-\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right )\right )} \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + b{\left (i \, E_{1}\left (\frac{5 i \, b c + 5 i \,{\left (b x + a\right )} d - 5 i \, a d}{d}\right ) - i \, E_{1}\left (-\frac{5 i \, b c + 5 i \,{\left (b x + a\right )} d - 5 i \, a d}{d}\right )\right )} \cos \left (-\frac{5 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, b{\left (E_{1}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\left (-\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right ) - b{\left (E_{1}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + E_{1}\left (-\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right )\right )} \sin \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + b{\left (E_{1}\left (\frac{5 i \, b c + 5 i \,{\left (b x + a\right )} d - 5 i \, a d}{d}\right ) + E_{1}\left (-\frac{5 i \, b c + 5 i \,{\left (b x + a\right )} d - 5 i \, a d}{d}\right )\right )} \sin \left (-\frac{5 \,{\left (b c - a d\right )}}{d}\right )}{32 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.50882, size = 612, normalized size = 3.31 \begin{align*} \frac{2 \,{\left (\operatorname{Ci}\left (\frac{b d x + b c}{d}\right ) + \operatorname{Ci}\left (-\frac{b d x + b c}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right ) +{\left (\operatorname{Ci}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) -{\left (\operatorname{Ci}\left (\frac{5 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{5 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac{5 \,{\left (b c - a d\right )}}{d}\right ) - 2 \, \cos \left (-\frac{5 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{5 \,{\left (b d x + b c\right )}}{d}\right ) + 2 \, \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{Si}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) + 4 \, \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{Si}\left (\frac{b d x + b c}{d}\right )}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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