\(\int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx\) [230]

Optimal result

Integrand size = 23, antiderivative size = 213 \[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=\frac {a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac {2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}+\frac {d^3 \left (b^2 (1-n)+a^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{-3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}} \] Output:

a^2*d^2*cot(f*x+e)*(d*csc(f*x+e))^(-2+n)/f/(1-n)+2*a*b*d^2*cos(f*x+e)*(d*c 
sc(f*x+e))^(-2+n)*hypergeom([1/2, 1-1/2*n],[2-1/2*n],sin(f*x+e)^2)/f/(2-n) 
/(cos(f*x+e)^2)^(1/2)+d^3*(b^2*(1-n)+a^2*(2-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/(1-n)/(3-n)/ 
(cos(f*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.63 \[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=-\frac {d \cos (e+f x) (d \csc (e+f x))^{-1+n} \sin ^2(e+f x)^{\frac {1}{2} (-1+n)} \left (b^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1+n),\frac {3}{2},\cos ^2(e+f x)\right )+a \left (a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3}{2},\cos ^2(e+f x)\right )+2 b \csc (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {3}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}\right )\right )}{f} \] Input:

Integrate[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^2,x]
 

Output:

-((d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*(Sin[e + f*x]^2)^((-1 + n)/2)* 
(b^2*Hypergeometric2F1[1/2, (-1 + n)/2, 3/2, Cos[e + f*x]^2] + a*(a*Hyperg 
eometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + 2*b*Csc[e + f*x]*Hyperg 
eometric2F1[1/2, n/2, 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2])))/f)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 3717, 3042, 4275, 3042, 4259, 3042, 3122, 4534, 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.

\(\displaystyle \int (a+b \sin (e+f x))^2 (d \csc (e+f x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \sin (e+f x))^2 (d \csc (e+f x))^ndx\)

\(\Big \downarrow \) 3717

\(\displaystyle d^2 \int (d \csc (e+f x))^{n-2} (b+a \csc (e+f x))^2dx\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \int (d \csc (e+f x))^{n-2} (b+a \csc (e+f x))^2dx\)

\(\Big \downarrow \) 4275

\(\displaystyle d^2 \left (\int (d \csc (e+f x))^{n-2} \left (b^2+a^2 \csc ^2(e+f x)\right )dx+\frac {2 a b \int (d \csc (e+f x))^{n-1}dx}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\int (d \csc (e+f x))^{n-2} \left (b^2+a^2 \csc (e+f x)^2\right )dx+\frac {2 a b \int (d \csc (e+f x))^{n-1}dx}{d}\right )\)

\(\Big \downarrow \) 4259

\(\displaystyle d^2 \left (\int (d \csc (e+f x))^{n-2} \left (b^2+a^2 \csc (e+f x)^2\right )dx+\frac {2 a b \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{1-n}dx}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\int (d \csc (e+f x))^{n-2} \left (b^2+a^2 \csc (e+f x)^2\right )dx+\frac {2 a b \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{1-n}dx}{d}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle d^2 \left (\int (d \csc (e+f x))^{n-2} \left (b^2+a^2 \csc (e+f x)^2\right )dx+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}\right )\)

\(\Big \downarrow \) 4534

\(\displaystyle d^2 \left (\left (\frac {a^2 (2-n)}{1-n}+b^2\right ) \int (d \csc (e+f x))^{n-2}dx+\frac {a^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\left (\frac {a^2 (2-n)}{1-n}+b^2\right ) \int (d \csc (e+f x))^{n-2}dx+\frac {a^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}\right )\)

\(\Big \downarrow \) 4259

\(\displaystyle d^2 \left (\left (\frac {a^2 (2-n)}{1-n}+b^2\right ) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{2-n}dx+\frac {a^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle d^2 \left (\left (\frac {a^2 (2-n)}{1-n}+b^2\right ) \left (\frac {\sin (e+f x)}{d}\right )^n (d \csc (e+f x))^n \int \left (\frac {\sin (e+f x)}{d}\right )^{2-n}dx+\frac {a^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle d^2 \left (\frac {d \left (\frac {a^2 (2-n)}{1-n}+b^2\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right )}{f (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}\right )\)

Input:

Int[(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^2,x]
 

Output:

d^2*((a^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a*b*Cos 
[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - 
 n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (d*(b^2 + (a^2* 
(2 - n))/(1 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1 
[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(3 - n)*Sqrt[Cos[e + f*x]^ 
2]))
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

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]
 

rule 3717

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]
 

rule 4259

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]
 

rule 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^2, x_Symbol] :> Simp[2*a*(b/d)   Int[(d*Csc[e + f*x])^(n + 1), x], x] 
 + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b, d, 
 e, f, n}, x]
 

rule 4534

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) 
+ (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) 
)), x] + Simp[(C*m + A*(m + 1))/(m + 1)   Int[(b*Csc[e + f*x])^m, x], x] /; 
 FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]
 

Maple [F]

\[\int \left (d \csc \left (f x +e \right )\right )^{n} \left (a +b \sin \left (f x +e \right )\right )^{2}d x\]

Input:

int((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x)
 

Output:

int((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x)
 

Fricas [F]

\[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \] Input:

integrate((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x, algorithm="fricas")
 

Output:

integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*(d*csc(f*x 
 + e))^n, x)
 

Sympy [F]

\[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=\int \left (d \csc {\left (e + f x \right )}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{2}\, dx \] Input:

integrate((d*csc(f*x+e))**n*(a+b*sin(f*x+e))**2,x)
 

Output:

Integral((d*csc(e + f*x))**n*(a + b*sin(e + f*x))**2, x)
 

Maxima [F]

\[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \] Input:

integrate((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x, algorithm="maxima")
 

Output:

integrate((b*sin(f*x + e) + a)^2*(d*csc(f*x + e))^n, x)
 

Giac [F]

\[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \] Input:

integrate((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((b*sin(f*x + e) + a)^2*(d*csc(f*x + e))^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=\int {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2 \,d x \] Input:

int((d/sin(e + f*x))^n*(a + b*sin(e + f*x))^2,x)
 

Output:

int((d/sin(e + f*x))^n*(a + b*sin(e + f*x))^2, x)
 

Reduce [F]

\[ \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx=d^{n} \left (\left (\int \csc \left (f x +e \right )^{n}d x \right ) a^{2}+\left (\int \csc \left (f x +e \right )^{n} \sin \left (f x +e \right )^{2}d x \right ) b^{2}+2 \left (\int \csc \left (f x +e \right )^{n} \sin \left (f x +e \right )d x \right ) a b \right ) \] Input:

int((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x)
                                                                                    
                                                                                    
 

Output:

d**n*(int(csc(e + f*x)**n,x)*a**2 + int(csc(e + f*x)**n*sin(e + f*x)**2,x) 
*b**2 + 2*int(csc(e + f*x)**n*sin(e + f*x),x)*a*b)