Integrand size = 20, antiderivative size = 250 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=-\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a b d^3 \sin (e+f x)}{f^4}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2} \]
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Time = 0.17 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3398, 3377, 2717, 3392, 32, 3391} \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a b d^3 \sin (e+f x)}{f^4}+\frac {3 b^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {3 b^2 c d^2 x}{4 f^2}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {b^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b^2 (c+d x)^4}{8 d}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac {3 b^2 d^3 x^2}{8 f^2} \]
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Rule 32
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \sin (e+f x)+b^2 (c+d x)^3 \sin ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \sin (e+f x) \, dx+b^2 \int (c+d x)^3 \sin ^2(e+f x) \, dx \\ & = \frac {a^2 (c+d x)^4}{4 d}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}+\frac {1}{2} b^2 \int (c+d x)^3 \, dx-\frac {\left (3 b^2 d^2\right ) \int (c+d x) \sin ^2(e+f x) \, dx}{2 f^2}+\frac {(6 a b d) \int (c+d x)^2 \cos (e+f x) \, dx}{f} \\ & = \frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {\left (12 a b d^2\right ) \int (c+d x) \sin (e+f x) \, dx}{f^2}-\frac {\left (3 b^2 d^2\right ) \int (c+d x) \, dx}{4 f^2} \\ & = -\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {\left (12 a b d^3\right ) \int \cos (e+f x) \, dx}{f^3} \\ & = -\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a b d^3 \sin (e+f x)}{f^4}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.93 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\frac {2 \left (2 a^2+b^2\right ) f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-32 a b f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (-6+f^2 x^2\right )\right ) \cos (e+f x)-3 b^2 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (-1+2 f^2 x^2\right )\right ) \cos (2 (e+f x))+96 a b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (-2+f^2 x^2\right )\right ) \sin (e+f x)-2 b^2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (-3+2 f^2 x^2\right )\right ) \sin (2 (e+f x))}{16 f^4} \]
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Time = 0.43 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {-4 \left (d x +c \right ) f \,b^{2} \left (\left (d x +c \right )^{2} f^{2}-\frac {3 d^{2}}{2}\right ) \sin \left (2 f x +2 e \right )-6 \left (\left (d x +c \right )^{2} f^{2}-\frac {d^{2}}{2}\right ) d \,b^{2} \cos \left (2 f x +2 e \right )-32 \left (\left (d x +c \right )^{2} f^{2}-6 d^{2}\right ) \left (d x +c \right ) f a b \cos \left (f x +e \right )+96 \left (\left (d x +c \right )^{2} f^{2}-2 d^{2}\right ) a d b \sin \left (f x +e \right )+16 \left (\frac {d x}{2}+c \right ) x \left (\frac {1}{2} d^{2} x^{2}+c d x +c^{2}\right ) \left (a^{2}+\frac {b^{2}}{2}\right ) f^{4}-32 a b \,c^{3} f^{3}+6 b^{2} c^{2} d \,f^{2}+192 a b c \,d^{2} f -3 b^{2} d^{3}}{16 f^{4}}\) | \(213\) |
risch | \(\frac {a^{2} d^{3} x^{4}}{4}+\frac {d^{3} b^{2} x^{4}}{8}+a^{2} c \,d^{2} x^{3}+\frac {d^{2} b^{2} c \,x^{3}}{2}+\frac {3 a^{2} d \,c^{2} x^{2}}{2}+\frac {3 d \,b^{2} c^{2} x^{2}}{4}+a^{2} c^{3} x +\frac {b^{2} c^{3} x}{2}+\frac {a^{2} c^{4}}{4 d}+\frac {b^{2} c^{4}}{8 d}-\frac {2 a b \left (d^{3} f^{2} x^{3}+3 c \,d^{2} f^{2} x^{2}+3 c^{2} d \,f^{2} x +c^{3} f^{2}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (f x +e \right )}{f^{3}}+\frac {6 a b d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2}\right ) \sin \left (f x +e \right )}{f^{4}}-\frac {3 b^{2} d \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}-d^{2}\right ) \cos \left (2 f x +2 e \right )}{16 f^{4}}-\frac {b^{2} \left (2 d^{3} f^{2} x^{3}+6 c \,d^{2} f^{2} x^{2}+6 c^{2} d \,f^{2} x +2 c^{3} f^{2}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (2 f x +2 e \right )}{8 f^{3}}\) | \(342\) |
parts | \(\text {Expression too large to display}\) | \(916\) |
norman | \(\frac {\left (\frac {1}{4} a^{2} d^{3}+\frac {1}{8} b^{2} d^{3}\right ) x^{4}+\left (\frac {1}{2} a^{2} d^{3}+\frac {1}{4} b^{2} d^{3}\right ) x^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {1}{4} a^{2} d^{3}+\frac {1}{8} b^{2} d^{3}\right ) x^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {b^{2} d^{3} x^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+d^{2} c \left (2 a^{2}+b^{2}\right ) x^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {8 a b \,c^{3} f^{3}+6 b^{2} c^{2} d \,f^{2}-48 a b c \,d^{2} f -3 b^{2} d^{3}}{4 f^{4}}+\frac {\left (4 a^{2} c^{3} f^{3}+2 b^{2} c^{3} f^{3}-24 a b \,c^{2} d \,f^{2}-3 b^{2} c \,d^{2} f +48 a b \,d^{3}\right ) x}{4 f^{3}}+\frac {\left (8 a b \,c^{3} f^{3}-6 b^{2} c^{2} d \,f^{2}-48 a b c \,d^{2} f +3 b^{2} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f^{4}}+\frac {b \left (-2 b \,c^{3} f^{3}+24 a \,c^{2} d \,f^{2}+3 b c \,d^{2} f -48 a \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f^{4}}+\frac {b \left (2 b \,c^{3} f^{3}+24 a \,c^{2} d \,f^{2}-3 b c \,d^{2} f -48 a \,d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{4}}+\frac {3 d \left (4 a^{2} c^{2} f^{2}+2 b^{2} c^{2} f^{2}-16 a b c d f -b^{2} d^{2}\right ) x^{2}}{8 f^{2}}+\frac {d^{2} \left (2 a^{2} c f +b^{2} c f -4 a b d \right ) x^{3}}{2 f}+\frac {\left (4 a^{2} c^{3} f^{3}+2 b^{2} c^{3} f^{3}+24 a b \,c^{2} d \,f^{2}-3 b^{2} c \,d^{2} f -48 a b \,d^{3}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f^{3}}-\frac {b^{2} d^{3} x^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {c \left (4 a^{2} c^{2} f^{2}+2 b^{2} c^{2} f^{2}+9 b^{2} d^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{2}}+\frac {3 d \left (4 a^{2} c^{2} f^{2}+2 b^{2} c^{2} f^{2}+3 b^{2} d^{2}\right ) x^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f^{2}}+\frac {3 d \left (4 a^{2} c^{2} f^{2}+2 b^{2} c^{2} f^{2}+16 a b c d f -b^{2} d^{2}\right ) x^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f^{2}}+\frac {d^{2} \left (2 a^{2} c f +b^{2} c f +4 a b d \right ) x^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {3 b d \left (2 b \,c^{2} f^{2}+16 a c d f -b \,d^{2}\right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{3}}+\frac {3 d b \left (-2 b \,c^{2} f^{2}+16 a c d f +b \,d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f^{3}}+\frac {3 d^{2} b \left (-b c f +4 d a \right ) x^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}+\frac {3 d^{2} b \left (b c f +4 d a \right ) x^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{2}}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\) | \(922\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1125\) |
default | \(\text {Expression too large to display}\) | \(1125\) |
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Time = 0.29 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.53 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} d^{3} f^{4} x^{4} + 4 \, {\left (2 \, a^{2} + b^{2}\right )} c d^{2} f^{4} x^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} + b^{2}\right )} c^{2} d f^{4} + b^{2} d^{3} f^{2}\right )} x^{2} - 3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} - b^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} f^{4} + 3 \, b^{2} c d^{2} f^{2}\right )} x - 16 \, {\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + a b c^{3} f^{3} - 6 \, a b c d^{2} f + 3 \, {\left (a b c^{2} d f^{3} - 2 \, a b d^{3} f\right )} x\right )} \cos \left (f x + e\right ) + 2 \, {\left (24 \, a b d^{3} f^{2} x^{2} + 48 \, a b c d^{2} f^{2} x + 24 \, a b c^{2} d f^{2} - 48 \, a b d^{3} - {\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 2 \, b^{2} c^{3} f^{3} - 3 \, b^{2} c d^{2} f + 3 \, {\left (2 \, b^{2} c^{2} d f^{3} - b^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (255) = 510\).
Time = 0.45 (sec) , antiderivative size = 779, normalized size of antiderivative = 3.12 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\begin {cases} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac {a^{2} d^{3} x^{4}}{4} - \frac {2 a b c^{3} \cos {\left (e + f x \right )}}{f} - \frac {6 a b c^{2} d x \cos {\left (e + f x \right )}}{f} + \frac {6 a b c^{2} d \sin {\left (e + f x \right )}}{f^{2}} - \frac {6 a b c d^{2} x^{2} \cos {\left (e + f x \right )}}{f} + \frac {12 a b c d^{2} x \sin {\left (e + f x \right )}}{f^{2}} + \frac {12 a b c d^{2} \cos {\left (e + f x \right )}}{f^{3}} - \frac {2 a b d^{3} x^{3} \cos {\left (e + f x \right )}}{f} + \frac {6 a b d^{3} x^{2} \sin {\left (e + f x \right )}}{f^{2}} + \frac {12 a b d^{3} x \cos {\left (e + f x \right )}}{f^{3}} - \frac {12 a b d^{3} \sin {\left (e + f x \right )}}{f^{4}} + \frac {b^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} c^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac {3 b^{2} c^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 b^{2} c^{2} d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} c^{2} d \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {b^{2} c d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 b^{2} c d^{2} x^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {3 b^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {3 b^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} + \frac {b^{2} d^{3} x^{4} \sin ^{2}{\left (e + f x \right )}}{8} + \frac {b^{2} d^{3} x^{4} \cos ^{2}{\left (e + f x \right )}}{8} - \frac {b^{2} d^{3} x^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} d^{3} x^{2} \sin ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {3 b^{2} d^{3} x^{2} \cos ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {3 b^{2} d^{3} x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} - \frac {3 b^{2} d^{3} \sin ^{2}{\left (e + f x \right )}}{8 f^{4}} & \text {for}\: f \neq 0 \\\left (a + b \sin {\left (e \right )}\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 959 vs. \(2 (234) = 468\).
Time = 0.23 (sec) , antiderivative size = 959, normalized size of antiderivative = 3.84 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.47 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{8} \, b^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {1}{2} \, b^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {3}{4} \, b^{2} c^{2} d x^{2} + a^{2} c^{3} x + \frac {1}{2} \, b^{2} c^{3} x - \frac {3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} - b^{2} d^{3}\right )} \cos \left (2 \, f x + 2 \, e\right )}{16 \, f^{4}} - \frac {2 \, {\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x + a b c^{3} f^{3} - 6 \, a b d^{3} f x - 6 \, a b c d^{2} f\right )} \cos \left (f x + e\right )}{f^{4}} - \frac {{\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 6 \, b^{2} c^{2} d f^{3} x + 2 \, b^{2} c^{3} f^{3} - 3 \, b^{2} d^{3} f x - 3 \, b^{2} c d^{2} f\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{4}} + \frac {6 \, {\left (a b d^{3} f^{2} x^{2} + 2 \, a b c d^{2} f^{2} x + a b c^{2} d f^{2} - 2 \, a b d^{3}\right )} \sin \left (f x + e\right )}{f^{4}} \]
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Time = 2.45 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.99 \[ \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx=\frac {\frac {3\,b^2\,d^3\,\cos \left (2\,e+2\,f\,x\right )}{2}+8\,a^2\,c^3\,f^4\,x+4\,b^2\,c^3\,f^4\,x-96\,a\,b\,d^3\,\sin \left (e+f\,x\right )-2\,b^2\,c^3\,f^3\,\sin \left (2\,e+2\,f\,x\right )+2\,a^2\,d^3\,f^4\,x^4+b^2\,d^3\,f^4\,x^4-16\,a\,b\,c^3\,f^3\,\cos \left (e+f\,x\right )-3\,b^2\,d^3\,f^2\,x^2\,\cos \left (2\,e+2\,f\,x\right )-2\,b^2\,d^3\,f^3\,x^3\,\sin \left (2\,e+2\,f\,x\right )+3\,b^2\,c\,d^2\,f\,\sin \left (2\,e+2\,f\,x\right )+3\,b^2\,d^3\,f\,x\,\sin \left (2\,e+2\,f\,x\right )-3\,b^2\,c^2\,d\,f^2\,\cos \left (2\,e+2\,f\,x\right )+12\,a^2\,c^2\,d\,f^4\,x^2+8\,a^2\,c\,d^2\,f^4\,x^3+6\,b^2\,c^2\,d\,f^4\,x^2+4\,b^2\,c\,d^2\,f^4\,x^3+96\,a\,b\,c\,d^2\,f\,\cos \left (e+f\,x\right )+96\,a\,b\,d^3\,f\,x\,\cos \left (e+f\,x\right )-6\,b^2\,c\,d^2\,f^2\,x\,\cos \left (2\,e+2\,f\,x\right )-6\,b^2\,c^2\,d\,f^3\,x\,\sin \left (2\,e+2\,f\,x\right )+48\,a\,b\,c^2\,d\,f^2\,\sin \left (e+f\,x\right )-6\,b^2\,c\,d^2\,f^3\,x^2\,\sin \left (2\,e+2\,f\,x\right )-16\,a\,b\,d^3\,f^3\,x^3\,\cos \left (e+f\,x\right )+48\,a\,b\,d^3\,f^2\,x^2\,\sin \left (e+f\,x\right )-48\,a\,b\,c\,d^2\,f^3\,x^2\,\cos \left (e+f\,x\right )-48\,a\,b\,c^2\,d\,f^3\,x\,\cos \left (e+f\,x\right )+96\,a\,b\,c\,d^2\,f^2\,x\,\sin \left (e+f\,x\right )}{8\,f^4} \]
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