Optimal. Leaf size=184 \[ \frac{9 b^2 \sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b^2 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{8 d^3}-\frac{3 b^2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d^3}+\frac{9 b^2 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b \sin ^2(a+b x) \cos (a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.353643, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3314, 3303, 3299, 3302, 3312} \[ \frac{9 b^2 \sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b^2 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{8 d^3}-\frac{3 b^2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d^3}+\frac{9 b^2 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}-\frac{3 b \sin ^2(a+b x) \cos (a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 3314
Rule 3303
Rule 3299
Rule 3302
Rule 3312
Rubi steps
\begin{align*} \int \frac{\sin ^3(a+b x)}{(c+d x)^3} \, dx &=-\frac{3 b \cos (a+b x) \sin ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2}+\frac{\left (3 b^2\right ) \int \frac{\sin (a+b x)}{c+d x} \, dx}{d^2}-\frac{\left (9 b^2\right ) \int \frac{\sin ^3(a+b x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac{3 b \cos (a+b x) \sin ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2}-\frac{\left (9 b^2\right ) \int \left (\frac{3 \sin (a+b x)}{4 (c+d x)}-\frac{\sin (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{2 d^2}+\frac{\left (3 b^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d^2}+\frac{\left (3 b^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d^2}\\ &=\frac{3 b^2 \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d^3}-\frac{3 b \cos (a+b x) \sin ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2}+\frac{3 b^2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^3}+\frac{\left (9 b^2\right ) \int \frac{\sin (3 a+3 b x)}{c+d x} \, dx}{8 d^2}-\frac{\left (27 b^2\right ) \int \frac{\sin (a+b x)}{c+d x} \, dx}{8 d^2}\\ &=\frac{3 b^2 \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d^3}-\frac{3 b \cos (a+b x) \sin ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2}+\frac{3 b^2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^3}+\frac{\left (9 b^2 \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx}{8 d^2}-\frac{\left (27 b^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{8 d^2}+\frac{\left (9 b^2 \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx}{8 d^2}-\frac{\left (27 b^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{8 d^2}\\ &=\frac{9 b^2 \text{Ci}\left (\frac{3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac{3 b c}{d}\right )}{8 d^3}-\frac{3 b^2 \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{8 d^3}-\frac{3 b \cos (a+b x) \sin ^2(a+b x)}{2 d^2 (c+d x)}-\frac{\sin ^3(a+b x)}{2 d (c+d x)^2}-\frac{3 b^2 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{8 d^3}+\frac{9 b^2 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b c}{d}+3 b x\right )}{8 d^3}\\ \end{align*}
Mathematica [A] time = 0.79574, size = 221, normalized size = 1.2 \[ \frac{6 b^2 (c+d x)^2 \left (3 \sin \left (3 a-\frac{3 b c}{d}\right ) \text{CosIntegral}\left (\frac{3 b (c+d x)}{d}\right )-\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (b \left (\frac{c}{d}+x\right )\right )-\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (b \left (\frac{c}{d}+x\right )\right )+3 \cos \left (3 a-\frac{3 b c}{d}\right ) \text{Si}\left (\frac{3 b (c+d x)}{d}\right )\right )-6 d \cos (b x) (b \cos (a) (c+d x)+d \sin (a))+2 d \cos (3 b x) (3 b \cos (3 a) (c+d x)+d \sin (3 a))+6 d \sin (b x) (b \sin (a) (c+d x)-d \cos (a))+2 d \sin (3 b x) (d \cos (3 a)-3 b \sin (3 a) (c+d x))}{16 d^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 313, normalized size = 1.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{{b}^{3}}{12} \left ( -{\frac{3\,\sin \left ( 3\,bx+3\,a \right ) }{2\, \left ( \left ( bx+a \right ) d-da+cb \right ) ^{2}d}}+{\frac{3}{2\,d} \left ( -3\,{\frac{\cos \left ( 3\,bx+3\,a \right ) }{ \left ( \left ( bx+a \right ) d-da+cb \right ) d}}-3\,{\frac{1}{d} \left ( 3\,{\frac{1}{d}{\it Si} \left ( 3\,bx+3\,a+3\,{\frac{-da+cb}{d}} \right ) \cos \left ( 3\,{\frac{-da+cb}{d}} \right ) }-3\,{\frac{1}{d}{\it Ci} \left ( 3\,bx+3\,a+3\,{\frac{-da+cb}{d}} \right ) \sin \left ( 3\,{\frac{-da+cb}{d}} \right ) } \right ) } \right ) } \right ) }+{\frac{3\,{b}^{3}}{4} \left ( -{\frac{\sin \left ( bx+a \right ) }{2\, \left ( \left ( bx+a \right ) d-da+cb \right ) ^{2}d}}+{\frac{1}{2\,d} \left ( -{\frac{\cos \left ( bx+a \right ) }{ \left ( \left ( bx+a \right ) d-da+cb \right ) d}}-{\frac{1}{d} \left ({\frac{1}{d}{\it Si} \left ( bx+a+{\frac{-da+cb}{d}} \right ) \cos \left ({\frac{-da+cb}{d}} \right ) }-{\frac{1}{d}{\it Ci} \left ( bx+a+{\frac{-da+cb}{d}} \right ) \sin \left ({\frac{-da+cb}{d}} \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.97265, size = 454, normalized size = 2.47 \begin{align*} \frac{b^{3}{\left (-3 i \, E_{3}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + 3 i \, E_{3}\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^{3}{\left (i \, E_{3}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) - i \, E_{3}\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 ) - 3 \, b^{3}{\left (E_{3}\left (\frac{i \, b c + i \,{\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{3}\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^{3}{\left (E_{3}\left (\frac{3 i \, b c + 3 i \,{\left (b x + a\right )} d - 3 i \, a d}{d}\right ) + E_{3}\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 )}{8 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} +{\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \,{\left (b c d^{2} - a d^{3}\right )}{\left (b x + a\right )}\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97958, size = 919, normalized size = 4.99 \begin{align*} \frac{24 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} + 18 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \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 ) - 6 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{Si}\left (\frac{b d x + b c}{d}\right ) - 24 \,{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + 8 \,{\left (d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sin \left (b x + a\right ) - 3 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (\frac{b d x + b c}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (-\frac{b d x + b c}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right ) + 9 \,{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname{Ci}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\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 )}{16 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\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(-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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