Integrand size = 21, antiderivative size = 101 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {1}{8} \left (4 a^2+3 b^2\right ) x-\frac {2 a b \cos (e+f x)}{f}+\frac {2 a b \cos ^3(e+f x)}{3 f}-\frac {\left (4 a^2+3 b^2\right ) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {b^2 \cos (e+f x) \sin ^3(e+f x)}{4 f} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2868, 2713, 3093, 2715, 8} \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=-\frac {\left (4 a^2+3 b^2\right ) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} x \left (4 a^2+3 b^2\right )+\frac {2 a b \cos ^3(e+f x)}{3 f}-\frac {2 a b \cos (e+f x)}{f}-\frac {b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f} \]
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 2868
Rule 3093
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \sin ^3(e+f x) \, dx+\int \sin ^2(e+f x) \left (a^2+b^2 \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {b^2 \cos (e+f x) \sin ^3(e+f x)}{4 f}+\frac {1}{4} \left (4 a^2+3 b^2\right ) \int \sin ^2(e+f x) \, dx-\frac {(2 a b) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {2 a b \cos (e+f x)}{f}+\frac {2 a b \cos ^3(e+f x)}{3 f}-\frac {\left (4 a^2+3 b^2\right ) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {b^2 \cos (e+f x) \sin ^3(e+f x)}{4 f}+\frac {1}{8} \left (4 a^2+3 b^2\right ) \int 1 \, dx \\ & = \frac {1}{8} \left (4 a^2+3 b^2\right ) x-\frac {2 a b \cos (e+f x)}{f}+\frac {2 a b \cos ^3(e+f x)}{3 f}-\frac {\left (4 a^2+3 b^2\right ) \cos (e+f x) \sin (e+f x)}{8 f}-\frac {b^2 \cos (e+f x) \sin ^3(e+f x)}{4 f} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.16 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {a^2 (e+f x)}{2 f}+\frac {3 b^2 (e+f x)}{8 f}-\frac {3 a b \cos (e+f x)}{2 f}+\frac {a b \cos (3 (e+f x))}{6 f}-\frac {a^2 \sin (2 (e+f x))}{4 f}-\frac {b^2 \sin (2 (e+f x))}{4 f}+\frac {b^2 \sin (4 (e+f x))}{32 f} \]
[In]
[Out]
Time = 1.60 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+b^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(89\) |
default | \(\frac {a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+b^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(89\) |
parallelrisch | \(\frac {48 a^{2} f x +36 b^{2} f x +3 b^{2} \sin \left (4 f x +4 e \right )+16 a b \cos \left (3 f x +3 e \right )-24 \sin \left (2 f x +2 e \right ) a^{2}-24 \sin \left (2 f x +2 e \right ) b^{2}-144 \cos \left (f x +e \right ) a b -128 a b}{96 f}\) | \(90\) |
risch | \(\frac {a^{2} x}{2}+\frac {3 b^{2} x}{8}-\frac {3 a b \cos \left (f x +e \right )}{2 f}+\frac {b^{2} \sin \left (4 f x +4 e \right )}{32 f}+\frac {a b \cos \left (3 f x +3 e \right )}{6 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) b^{2}}{4 f}\) | \(94\) |
parts | \(\frac {a^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {b^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {2 a b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}\) | \(94\) |
norman | \(\frac {\left (\frac {a^{2}}{2}+\frac {3 b^{2}}{8}\right ) x +\left (2 a^{2}+\frac {3 b^{2}}{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 a^{2}+\frac {3 b^{2}}{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 a^{2}+\frac {9 b^{2}}{4}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {a^{2}}{2}+\frac {3 b^{2}}{8}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {8 a b}{3 f}-\frac {\left (4 a^{2}+3 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {\left (4 a^{2}+3 b^{2}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {\left (4 a^{2}+11 b^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {\left (4 a^{2}+11 b^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {8 a b \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {32 a b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(276\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {16 \, a b \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} f x - 48 \, a b \cos \left (f x + e\right ) + 3 \, {\left (2 \, b^{2} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{2} + 5 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (92) = 184\).
Time = 0.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.09 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {a^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a b \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 a b \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 b^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 b^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 b^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {5 b^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {3 b^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} & \text {for}\: f \neq 0 \\x \left (a + b \sin {\left (e \right )}\right )^{2} \sin ^{2}{\left (e \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} + 64 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2}}{96 \, f} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {1}{8} \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} x + \frac {a b \cos \left (3 \, f x + 3 \, e\right )}{6 \, f} - \frac {3 \, a b \cos \left (f x + e\right )}{2 \, f} + \frac {b^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
[In]
[Out]
Time = 6.53 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.84 \[ \int \sin ^2(e+f x) (a+b \sin (e+f x))^2 \, dx=\frac {\frac {3\,b^2\,\sin \left (4\,e+4\,f\,x\right )}{4}-6\,b^2\,\sin \left (2\,e+2\,f\,x\right )-6\,a^2\,\sin \left (2\,e+2\,f\,x\right )-36\,a\,b\,\cos \left (e+f\,x\right )+4\,a\,b\,\cos \left (3\,e+3\,f\,x\right )+12\,a^2\,f\,x+9\,b^2\,f\,x}{24\,f} \]
[In]
[Out]