Integrand size = 18, antiderivative size = 116 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\frac {1}{2} b^2 c x+\frac {1}{4} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {2 a b (c+d x) \cos (e+f x)}{f}+\frac {2 a b d \sin (e+f x)}{f^2}-\frac {b^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}+\frac {b^2 d \sin ^2(e+f x)}{4 f^2} \]
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Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3398, 3377, 2717, 3391} \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\frac {a^2 (c+d x)^2}{2 d}-\frac {2 a b (c+d x) \cos (e+f x)}{f}+\frac {2 a b d \sin (e+f x)}{f^2}-\frac {b^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} b^2 c x+\frac {b^2 d \sin ^2(e+f x)}{4 f^2}+\frac {1}{4} b^2 d x^2 \]
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Rule 2717
Rule 3377
Rule 3391
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (c+d x)+2 a b (c+d x) \sin (e+f x)+b^2 (c+d x) \sin ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 (c+d x)^2}{2 d}+(2 a b) \int (c+d x) \sin (e+f x) \, dx+b^2 \int (c+d x) \sin ^2(e+f x) \, dx \\ & = \frac {a^2 (c+d x)^2}{2 d}-\frac {2 a b (c+d x) \cos (e+f x)}{f}-\frac {b^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}+\frac {b^2 d \sin ^2(e+f x)}{4 f^2}+\frac {1}{2} b^2 \int (c+d x) \, dx+\frac {(2 a b d) \int \cos (e+f x) \, dx}{f} \\ & = \frac {1}{2} b^2 c x+\frac {1}{4} b^2 d x^2+\frac {a^2 (c+d x)^2}{2 d}-\frac {2 a b (c+d x) \cos (e+f x)}{f}+\frac {2 a b d \sin (e+f x)}{f^2}-\frac {b^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}+\frac {b^2 d \sin ^2(e+f x)}{4 f^2} \\ \end{align*}
Time = 4.77 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=-\frac {2 \left (2 a^2+b^2\right ) (e+f x) (-2 c f+d (e-f x))+16 a b f (c+d x) \cos (e+f x)+b^2 d \cos (2 (e+f x))-16 a b d \sin (e+f x)+2 b^2 f (c+d x) \sin (2 (e+f x))}{8 f^2} \]
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Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {a^{2} d \,x^{2}}{2}+a^{2} c x +\frac {b^{2} d \,x^{2}}{4}+\frac {b^{2} c x}{2}-\frac {2 a b \left (d x +c \right ) \cos \left (f x +e \right )}{f}+\frac {2 a b d \sin \left (f x +e \right )}{f^{2}}-\frac {b^{2} d \cos \left (2 f x +2 e \right )}{8 f^{2}}-\frac {b^{2} \left (d x +c \right ) \sin \left (2 f x +2 e \right )}{4 f}\) | \(105\) |
parallelrisch | \(\frac {-2 b^{2} f \left (d x +c \right ) \sin \left (2 f x +2 e \right )-b^{2} d \cos \left (2 f x +2 e \right )-16 a b f \left (d x +c \right ) \cos \left (f x +e \right )+16 a b d \sin \left (f x +e \right )+\left (\left (2 d \,x^{2}+4 c x \right ) f^{2}+d \right ) b^{2}-16 a b c f +8 \left (\frac {d x}{2}+c \right ) x \,f^{2} a^{2}}{8 f^{2}}\) | \(111\) |
parts | \(a^{2} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {b^{2} \left (\frac {d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\right )}{f}+\frac {2 a b \left (\frac {d \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f}-c \cos \left (f x +e \right )+\frac {d e \cos \left (f x +e \right )}{f}\right )}{f}\) | \(184\) |
derivativedivides | \(\frac {a^{2} c \left (f x +e \right )-\frac {a^{2} d e \left (f x +e \right )}{f}+\frac {a^{2} d \left (f x +e \right )^{2}}{2 f}-2 a b c \cos \left (f x +e \right )+\frac {2 a b d e \cos \left (f x +e \right )}{f}+\frac {2 a b d \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f}+b^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {b^{2} d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {b^{2} d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}}{f}\) | \(216\) |
default | \(\frac {a^{2} c \left (f x +e \right )-\frac {a^{2} d e \left (f x +e \right )}{f}+\frac {a^{2} d \left (f x +e \right )^{2}}{2 f}-2 a b c \cos \left (f x +e \right )+\frac {2 a b d e \cos \left (f x +e \right )}{f}+\frac {2 a b d \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f}+b^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {b^{2} d e \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {b^{2} d \left (\left (f x +e \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}+\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}}{f}\) | \(216\) |
norman | \(\frac {\left (\frac {1}{2} a^{2} d +\frac {1}{4} b^{2} d \right ) x^{2}+\left (a^{2} d +\frac {1}{2} b^{2} d \right ) x^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {1}{2} a^{2} d +\frac {1}{4} b^{2} d \right ) x^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {b \left (-b c f +4 d a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}+\frac {b \left (b c f +4 d a \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{2}}+c \left (2 a^{2}+b^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {b^{2} d x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {4 a b c}{f}+\frac {\left (2 a^{2} c f +b^{2} c f -4 a b d \right ) x}{2 f}-\frac {\left (4 a b c f -b^{2} d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{2}}+\frac {\left (2 a^{2} c f +b^{2} c f +4 a b d \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {b^{2} d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\) | \(299\) |
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Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} d f^{2} x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c f^{2} x - b^{2} d \cos \left (f x + e\right )^{2} - 8 \, {\left (a b d f x + a b c f\right )} \cos \left (f x + e\right ) + 2 \, {\left (4 \, a b d - {\left (b^{2} d f x + b^{2} c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.89 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\begin {cases} a^{2} c x + \frac {a^{2} d x^{2}}{2} - \frac {2 a b c \cos {\left (e + f x \right )}}{f} - \frac {2 a b d x \cos {\left (e + f x \right )}}{f} + \frac {2 a b d \sin {\left (e + f x \right )}}{f^{2}} + \frac {b^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {b^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac {b^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {b^{2} d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {b^{2} d \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} & \text {for}\: f \neq 0 \\\left (a + b \sin {\left (e \right )}\right )^{2} \left (c x + \frac {d x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.74 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\frac {8 \, {\left (f x + e\right )} a^{2} c + 2 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} c + \frac {4 \, {\left (f x + e\right )}^{2} a^{2} d}{f} - \frac {8 \, {\left (f x + e\right )} a^{2} d e}{f} - \frac {2 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} d e}{f} - 16 \, a b c \cos \left (f x + e\right ) + \frac {16 \, a b d e \cos \left (f x + e\right )}{f} - \frac {16 \, {\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} a b d}{f} + \frac {{\left (2 \, {\left (f x + e\right )}^{2} - 2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right )\right )} b^{2} d}{f}}{8 \, f} \]
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Time = 0.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\frac {1}{2} \, a^{2} d x^{2} + \frac {1}{4} \, b^{2} d x^{2} + a^{2} c x + \frac {1}{2} \, b^{2} c x - \frac {b^{2} d \cos \left (2 \, f x + 2 \, e\right )}{8 \, f^{2}} + \frac {2 \, a b d \sin \left (f x + e\right )}{f^{2}} - \frac {2 \, {\left (a b d f x + a b c f\right )} \cos \left (f x + e\right )}{f^{2}} - \frac {{\left (b^{2} d f x + b^{2} c f\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f^{2}} \]
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Time = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.23 \[ \int (c+d x) (a+b \sin (e+f x))^2 \, dx=\frac {a^2\,d\,x^2}{2}+\frac {b^2\,d\,x^2}{4}+a^2\,c\,x+\frac {b^2\,c\,x}{2}-\frac {b^2\,c\,\sin \left (2\,e+2\,f\,x\right )}{4\,f}+\frac {b^2\,d\,{\sin \left (e+f\,x\right )}^2}{4\,f^2}+\frac {4\,a\,b\,c\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{f}-\frac {b^2\,d\,x\,\sin \left (2\,e+2\,f\,x\right )}{4\,f}+\frac {2\,a\,b\,d\,\sin \left (e+f\,x\right )}{f^2}+\frac {2\,a\,b\,d\,x\,\left (2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}{f} \]
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