Integrand size = 19, antiderivative size = 79 \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^3(e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-2+n)}}{d f (1+n)} \]
-(d*cot(f*x+e))^(1+n)*hypergeom([-1+1/2*n, 1/2+1/2*n],[3/2+1/2*n],cos(f*x+ e)^2)*sin(f*x+e)^3*(sin(f*x+e)^2)^(-1+1/2*n)/d/f/(1+n)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 3.10 (sec) , antiderivative size = 477, normalized size of antiderivative = 6.04 \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {4 (-4+n) \left (\operatorname {AppellF1}\left (1-\frac {n}{2},-n,3,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-\operatorname {AppellF1}\left (1-\frac {n}{2},-n,4,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x)}{f (-2+n) \left (2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,3,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,4,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 \left (n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,3,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,4,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+3 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,4,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,5,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))\right )} \]
(-4*(-4 + n)*(AppellF1[1 - n/2, -n, 3, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[( e + f*x)/2]^2] - AppellF1[1 - n/2, -n, 4, 2 - n/2, Tan[(e + f*x)/2]^2, -Ta n[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^3*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]* Sin[e + f*x]^3)/(f*(-2 + n)*(2*(-4 + n)*AppellF1[1 - n/2, -n, 3, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*(-4 + n)*A ppellF1[1 - n/2, -n, 4, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]* Cos[(e + f*x)/2]^2 - 2*(n*AppellF1[2 - n/2, 1 - n, 3, 3 - n/2, Tan[(e + f* x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 - n/2, 1 - n, 4, 3 - n/2, Tan [(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 3*AppellF1[2 - n/2, -n, 4, 3 - n/2 , Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[2 - n/2, -n, 5, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3042, 3097}
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.
\(\displaystyle \int \sin ^3(e+f x) (d \cot (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (-d \tan \left (e+f x-\frac {\pi }{2}\right )\right )^n}{\sec \left (e+f x-\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 3097 |
\(\displaystyle -\frac {\sin ^3(e+f x) \sin ^2(e+f x)^{\frac {n-2}{2}} (d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n-2}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)}\) |
-(((d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[(-2 + n)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x]^3*(Sin[e + f*x]^2)^((-2 + n)/2))/(d*f*( 1 + n)))
3.1.51.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[(n - 1)/2] && !IntegerQ[m/2]
\[\int \left (d \cot \left (f x +e \right )\right )^{n} \sin \left (f x +e \right )^{3}d x\]
\[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\text {Timed out} \]
\[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
\[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^3\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]