Integrand size = 19, antiderivative size = 55 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\frac {a x}{2}-\frac {b \cos (e+f x)}{f}+\frac {b \cos ^3(e+f x)}{3 f}-\frac {a \cos (e+f x) \sin (e+f x)}{2 f} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2827, 2715, 8, 2713} \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=-\frac {a \sin (e+f x) \cos (e+f x)}{2 f}+\frac {a x}{2}+\frac {b \cos ^3(e+f x)}{3 f}-\frac {b \cos (e+f x)}{f} \]
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 2827
Rubi steps \begin{align*} \text {integral}& = a \int \sin ^2(e+f x) \, dx+b \int \sin ^3(e+f x) \, dx \\ & = -\frac {a \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} a \int 1 \, dx-\frac {b \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {a x}{2}-\frac {b \cos (e+f x)}{f}+\frac {b \cos ^3(e+f x)}{3 f}-\frac {a \cos (e+f x) \sin (e+f x)}{2 f} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\frac {a (e+f x)}{2 f}-\frac {3 b \cos (e+f x)}{4 f}+\frac {b \cos (3 (e+f x))}{12 f}-\frac {a \sin (2 (e+f x))}{4 f} \]
[In]
[Out]
Time = 1.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {6 a f x -9 b \cos \left (f x +e \right )+b \cos \left (3 f x +3 e \right )-3 a \sin \left (2 f x +2 e \right )-8 b}{12 f}\) | \(47\) |
risch | \(\frac {a x}{2}-\frac {3 b \cos \left (f x +e \right )}{4 f}+\frac {b \cos \left (3 f x +3 e \right )}{12 f}-\frac {a \sin \left (2 f x +2 e \right )}{4 f}\) | \(48\) |
derivativedivides | \(\frac {-\frac {b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(49\) |
default | \(\frac {-\frac {b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\) | \(49\) |
parts | \(\frac {a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}\) | \(51\) |
norman | \(\frac {\frac {a \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a x}{2}-\frac {4 b}{3 f}-\frac {a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {3 a x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {3 a x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {a x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {4 b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\) | \(121\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\frac {2 \, b \cos \left (f x + e\right )^{3} + 3 \, a f x - 3 \, a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \, b \cos \left (f x + e\right )}{6 \, f} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.67 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\begin {cases} \frac {a x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {b \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 b \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\left (e \right )}\right ) \sin ^{2}{\left (e \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\frac {3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a + 4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b}{12 \, f} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\frac {1}{2} \, a x + \frac {b \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {3 \, b \cos \left (f x + e\right )}{4 \, f} - \frac {a \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
[In]
[Out]
Time = 8.73 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x)) \, dx=\frac {a\,x}{2}-\frac {-a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+4\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {4\,b}{3}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^3} \]
[In]
[Out]