Integrand size = 23, antiderivative size = 262 \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=\frac {27 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac {d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (27+27 \csc (e+f x))}{f (1-n)}+\frac {27 d^3 (5-4 n) \cos (e+f x) (d \csc (e+f x))^{-3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {27 d^4 (11-4 n) \cos (e+f x) (d \csc (e+f x))^{-4+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-n}{2},\frac {6-n}{2},\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt {\cos ^2(e+f x)}} \]
a^3*d^3*(1-2*n)*cot(f*x+e)*(d*csc(f*x+e))^(-3+n)/f/(n^2-3*n+2)+d^3*cot(f*x +e)*(d*csc(f*x+e))^(-3+n)*(a^3+a^3*csc(f*x+e))/f/(1-n)+a^3*d^3*(5-4*n)*cos (f*x+e)*(d*csc(f*x+e))^(-3+n)*hypergeom([1/2, 3/2-1/2*n],[5/2-1/2*n],sin(f *x+e)^2)/f/(n^2-4*n+3)/(cos(f*x+e)^2)^(1/2)+a^3*d^4*(11-4*n)*cos(f*x+e)*(d *csc(f*x+e))^(-4+n)*hypergeom([1/2, 2-1/2*n],[3-1/2*n],sin(f*x+e)^2)/f/(n^ 2-6*n+8)/(cos(f*x+e)^2)^(1/2)
Time = 11.70 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.75 \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=\frac {27\ 2^{1-n} \csc ^{-n}(e+f x) (d \csc (e+f x))^n \tan \left (\frac {1}{2} (e+f x)\right ) \left (\cot \left (\frac {1}{2} (e+f x)\right )+\tan \left (\frac {1}{2} (e+f x)\right )\right )^n \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n} \left (\frac {\operatorname {Hypergeometric2F1}\left (4-n,\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{1-n}-\frac {6 \operatorname {Hypergeometric2F1}\left (4-n,1-\frac {n}{2},2-\frac {n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{-2+n}-\frac {15 \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},4-n,\frac {5-n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{-3+n}-\frac {20 \operatorname {Hypergeometric2F1}\left (4-n,2-\frac {n}{2},3-\frac {n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^3\left (\frac {1}{2} (e+f x)\right )}{-4+n}-\frac {15 \operatorname {Hypergeometric2F1}\left (4-n,\frac {5-n}{2},\frac {7-n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^4\left (\frac {1}{2} (e+f x)\right )}{-5+n}-\frac {6 \operatorname {Hypergeometric2F1}\left (4-n,3-\frac {n}{2},4-\frac {n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^5\left (\frac {1}{2} (e+f x)\right )}{-6+n}+\frac {\operatorname {Hypergeometric2F1}\left (4-n,\frac {7}{2}-\frac {n}{2},\frac {9}{2}-\frac {n}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^6\left (\frac {1}{2} (e+f x)\right )}{7-n}\right )}{f} \]
(27*2^(1 - n)*(d*Csc[e + f*x])^n*Tan[(e + f*x)/2]*(Cot[(e + f*x)/2] + Tan[ (e + f*x)/2])^n*(Hypergeometric2F1[4 - n, 1/2 - n/2, 3/2 - n/2, -Tan[(e + f*x)/2]^2]/(1 - n) - (6*Hypergeometric2F1[4 - n, 1 - n/2, 2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(-2 + n) - (15*Hypergeometric2F1[(3 - n)/2 , 4 - n, (5 - n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(-3 + n) - (2 0*Hypergeometric2F1[4 - n, 2 - n/2, 3 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(-4 + n) - (15*Hypergeometric2F1[4 - n, (5 - n)/2, (7 - n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4)/(-5 + n) - (6*Hypergeometric2F1[4 - n, 3 - n/2, 4 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^5)/(-6 + n) + (Hypergeometric2F1[4 - n, 7/2 - n/2, 9/2 - n/2, -Tan[(e + f*x)/2]^2]*Tan [(e + f*x)/2]^6)/(7 - n)))/(f*Csc[e + f*x]^n*(1 + Tan[(e + f*x)/2]^2)^n)
Time = 1.15 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3717, 3042, 4301, 25, 3042, 4485, 25, 3042, 4274, 3042, 4259, 3042, 3122}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (d \csc (e+f x))^n \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 (d \csc (e+f x))^ndx\) |
| \(\Big \downarrow \) 3717 |
| \(\displaystyle d^3 \int (d \csc (e+f x))^{n-3} (\csc (e+f x) a+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^3 \int (d \csc (e+f x))^{n-3} (\csc (e+f x) a+a)^3dx\) |
| \(\Big \downarrow \) 4301 |
| \(\displaystyle d^3 \left (\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}-\frac {a \int -(d \csc (e+f x))^{n-3} (\csc (e+f x) a+a) (2 a (2-n)+a (1-2 n) \csc (e+f x))dx}{1-n}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^3 \left (\frac {a \int (d \csc (e+f x))^{n-3} (\csc (e+f x) a+a) (2 a (2-n)+a (1-2 n) \csc (e+f x))dx}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^3 \left (\frac {a \int (d \csc (e+f x))^{n-3} (\csc (e+f x) a+a) (2 a (2-n)+a (1-2 n) \csc (e+f x))dx}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}-\frac {\int -(d \csc (e+f x))^{n-3} \left ((11-4 n) (1-n) a^2+(5-4 n) (2-n) \csc (e+f x) a^2\right )dx}{2-n}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {\int (d \csc (e+f x))^{n-3} \left ((11-4 n) (1-n) a^2+(5-4 n) (2-n) \csc (e+f x) a^2\right )dx}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {\int (d \csc (e+f x))^{n-3} \left ((11-4 n) (1-n) a^2+(5-4 n) (2-n) \csc (e+f x) a^2\right )dx}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {a^2 (11-4 n) (1-n) \int (d \csc (e+f x))^{n-3}dx+\frac {a^2 (5-4 n) (2-n) \int (d \csc (e+f x))^{n-2}dx}{d}}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {a^2 (11-4 n) (1-n) \int (d \csc (e+f x))^{n-3}dx+\frac {a^2 (5-4 n) (2-n) \int (d \csc (e+f x))^{n-2}dx}{d}}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 4259 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {\frac {a^2 (5-4 n) (2-n) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{2-n}dx}{d}+a^2 (11-4 n) (1-n) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{3-n}dx}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^3 \left (\frac {a \left (\frac {\frac {a^2 (5-4 n) (2-n) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{2-n}dx}{d}+a^2 (11-4 n) (1-n) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{3-n}dx}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}+\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}\right )\) |
| \(\Big \downarrow \) 3122 |
| \(\displaystyle d^3 \left (\frac {\cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)}+\frac {a \left (\frac {\frac {a^2 d (11-4 n) (1-n) \cos (e+f x) (d \csc (e+f x))^{n-4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4-n}{2},\frac {6-n}{2},\sin ^2(e+f x)\right )}{f (4-n) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 (5-4 n) (2-n) \cos (e+f x) (d \csc (e+f x))^{n-3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right )}{f (3-n) \sqrt {\cos ^2(e+f x)}}}{2-n}+\frac {a^2 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (2-n)}\right )}{1-n}\right )\) |
d^3*((Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(a^3 + a^3*Csc[e + f*x]))/(f* (1 - n)) + (a*((a^2*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*( 2 - n)) + ((a^2*(5 - 4*n)*(2 - n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*H ypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(3 - n)*Sq rt[Cos[e + f*x]^2]) + (a^2*d*(11 - 4*n)*(1 - n)*Cos[e + f*x]*(d*Csc[e + f* x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2]) /(f*(4 - n)*Sqrt[Cos[e + f*x]^2]))/(2 - n)))/(1 - n))
3.9.14.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^(n - 1)*((Sin[c + d*x]/b)^(n - 1) Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-b^2)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*(m + n - 1))), x] + Simp[b/(m + n - 1) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^ 2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[ e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Simp[1/(n + 1) Int[(d*Csc [e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x ], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[ n, -1]
\[\int \left (d \csc \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )^{3}d x\]
\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]
integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin (f*x + e))*(d*csc(f*x + e))^n, x)
\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=a^{3} \left (\int \left (d \csc {\left (e + f x \right )}\right )^{n}\, dx + \int 3 \left (d \csc {\left (e + f x \right )}\right )^{n} \sin {\left (e + f x \right )}\, dx + \int 3 \left (d \csc {\left (e + f x \right )}\right )^{n} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (d \csc {\left (e + f x \right )}\right )^{n} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
a**3*(Integral((d*csc(e + f*x))**n, x) + Integral(3*(d*csc(e + f*x))**n*si n(e + f*x), x) + Integral(3*(d*csc(e + f*x))**n*sin(e + f*x)**2, x) + Inte gral((d*csc(e + f*x))**n*sin(e + f*x)**3, x))
\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]
Timed out. \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x))^3 \, dx=\int {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]