Integrand size = 21, antiderivative size = 73 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {(A+C) \cos (e+f x)}{f}+\frac {(2 A+3 C) \cos ^3(e+f x)}{3 f}-\frac {(A+3 C) \cos ^5(e+f x)}{5 f}+\frac {C \cos ^7(e+f x)}{7 f} \]
-(A+C)*cos(f*x+e)/f+1/3*(2*A+3*C)*cos(f*x+e)^3/f-1/5*(A+3*C)*cos(f*x+e)^5/ f+1/7*C*cos(f*x+e)^7/f
Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.49 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {5 A \cos (e+f x)}{8 f}-\frac {35 C \cos (e+f x)}{64 f}+\frac {5 A \cos (3 (e+f x))}{48 f}+\frac {7 C \cos (3 (e+f x))}{64 f}-\frac {A \cos (5 (e+f x))}{80 f}-\frac {7 C \cos (5 (e+f x))}{320 f}+\frac {C \cos (7 (e+f x))}{448 f} \]
(-5*A*Cos[e + f*x])/(8*f) - (35*C*Cos[e + f*x])/(64*f) + (5*A*Cos[3*(e + f *x)])/(48*f) + (7*C*Cos[3*(e + f*x)])/(64*f) - (A*Cos[5*(e + f*x)])/(80*f) - (7*C*Cos[5*(e + f*x)])/(320*f) + (C*Cos[7*(e + f*x)])/(448*f)
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3492, 290, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (e+f x)^5 \left (A+C \sin (e+f x)^2\right )dx\) |
\(\Big \downarrow \) 3492 |
\(\displaystyle -\frac {\int \left (1-\cos ^2(e+f x)\right )^2 \left (-C \cos ^2(e+f x)+A+C\right )d\cos (e+f x)}{f}\) |
\(\Big \downarrow \) 290 |
\(\displaystyle -\frac {\int \left (-C \cos ^6(e+f x)+(A+3 C) \cos ^4(e+f x)-(2 A+3 C) \cos ^2(e+f x)+A \left (\frac {C}{A}+1\right )\right )d\cos (e+f x)}{f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{5} (A+3 C) \cos ^5(e+f x)-\frac {1}{3} (2 A+3 C) \cos ^3(e+f x)+(A+C) \cos (e+f x)-\frac {1}{7} C \cos ^7(e+f x)}{f}\) |
-(((A + C)*Cos[e + f*x] - ((2*A + 3*C)*Cos[e + f*x]^3)/3 + ((A + 3*C)*Cos[ e + f*x]^5)/5 - (C*Cos[e + f*x]^7)/7)/f)
3.1.16.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Time = 1.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {\left (700 A +735 C \right ) \cos \left (3 f x +3 e \right )+\left (-84 A -147 C \right ) \cos \left (5 f x +5 e \right )+15 C \cos \left (7 f x +7 e \right )+\left (-4200 A -3675 C \right ) \cos \left (f x +e \right )-3584 A -3072 C}{6720 f}\) | \(73\) |
derivativedivides | \(\frac {-\frac {C \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-\frac {A \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\) | \(74\) |
default | \(\frac {-\frac {C \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-\frac {A \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\) | \(74\) |
parts | \(-\frac {A \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {C \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7 f}\) | \(76\) |
risch | \(-\frac {5 \cos \left (f x +e \right ) A}{8 f}-\frac {35 \cos \left (f x +e \right ) C}{64 f}+\frac {C \cos \left (7 f x +7 e \right )}{448 f}-\frac {\cos \left (5 f x +5 e \right ) A}{80 f}-\frac {7 \cos \left (5 f x +5 e \right ) C}{320 f}+\frac {5 \cos \left (3 f x +3 e \right ) A}{48 f}+\frac {7 \cos \left (3 f x +3 e \right ) C}{64 f}\) | \(101\) |
norman | \(\frac {-\frac {112 A +96 C}{105 f}-\frac {32 A \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (80 A +96 C \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (112 A +96 C \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f}-\frac {\left (112 A +96 C \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{7}}\) | \(116\) |
1/6720*((700*A+735*C)*cos(3*f*x+3*e)+(-84*A-147*C)*cos(5*f*x+5*e)+15*C*cos (7*f*x+7*e)+(-4200*A-3675*C)*cos(f*x+e)-3584*A-3072*C)/f
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {15 \, C \cos \left (f x + e\right )^{7} - 21 \, {\left (A + 3 \, C\right )} \cos \left (f x + e\right )^{5} + 35 \, {\left (2 \, A + 3 \, C\right )} \cos \left (f x + e\right )^{3} - 105 \, {\left (A + C\right )} \cos \left (f x + e\right )}{105 \, f} \]
1/105*(15*C*cos(f*x + e)^7 - 21*(A + 3*C)*cos(f*x + e)^5 + 35*(2*A + 3*C)* cos(f*x + e)^3 - 105*(A + C)*cos(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (61) = 122\).
Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.10 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=\begin {cases} - \frac {A \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 A \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {8 A \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {C \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 C \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 C \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {16 C \cos ^{7}{\left (e + f x \right )}}{35 f} & \text {for}\: f \neq 0 \\x \left (A + C \sin ^{2}{\left (e \right )}\right ) \sin ^{5}{\left (e \right )} & \text {otherwise} \end {cases} \]
Piecewise((-A*sin(e + f*x)**4*cos(e + f*x)/f - 4*A*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 8*A*cos(e + f*x)**5/(15*f) - C*sin(e + f*x)**6*cos(e + f* x)/f - 2*C*sin(e + f*x)**4*cos(e + f*x)**3/f - 8*C*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) - 16*C*cos(e + f*x)**7/(35*f), Ne(f, 0)), (x*(A + C*sin(e)* *2)*sin(e)**5, True))
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {15 \, C \cos \left (f x + e\right )^{7} - 21 \, {\left (A + 3 \, C\right )} \cos \left (f x + e\right )^{5} + 35 \, {\left (2 \, A + 3 \, C\right )} \cos \left (f x + e\right )^{3} - 105 \, {\left (A + C\right )} \cos \left (f x + e\right )}{105 \, f} \]
1/105*(15*C*cos(f*x + e)^7 - 21*(A + 3*C)*cos(f*x + e)^5 + 35*(2*A + 3*C)* cos(f*x + e)^3 - 105*(A + C)*cos(f*x + e))/f
Time = 0.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {C \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} - \frac {{\left (4 \, A + 7 \, C\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (20 \, A + 21 \, C\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} - \frac {5 \, {\left (8 \, A + 7 \, C\right )} \cos \left (f x + e\right )}{64 \, f} \]
1/448*C*cos(7*f*x + 7*e)/f - 1/320*(4*A + 7*C)*cos(5*f*x + 5*e)/f + 1/192* (20*A + 21*C)*cos(3*f*x + 3*e)/f - 5/64*(8*A + 7*C)*cos(f*x + e)/f
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {\frac {C\,{\cos \left (e+f\,x\right )}^7}{7}+\left (-\frac {A}{5}-\frac {3\,C}{5}\right )\,{\cos \left (e+f\,x\right )}^5+\left (\frac {2\,A}{3}+C\right )\,{\cos \left (e+f\,x\right )}^3+\left (-A-C\right )\,\cos \left (e+f\,x\right )}{f} \]