\(\int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx\) [828]

Optimal result

Integrand size = 23, antiderivative size = 195 \[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\frac {b \operatorname {AppellF1}\left (\frac {1}{2},\frac {n}{2},1,\frac {3}{2},\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{9-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{1+n} \sin (e+f x) \sin ^2(e+f x)^{n/2}}{\left (9-b^2\right ) d f}-\frac {3 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1+n}{2},1,\frac {3}{2},\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{9-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{1+n} \sin ^2(e+f x)^{\frac {1+n}{2}}}{\left (9-b^2\right ) d f} \]

[Out]

b*AppellF1(1/2,1/2*n,1,3/2,cos(f*x+e)^2,-b^2*cos(f*x+e)^2/(a^2-b^2))*cos(f*x+e)*(d*csc(f*x+e))^(1+n)*sin(f*x+e
)*(sin(f*x+e)^2)^(1/2*n)/(a^2-b^2)/d/f-a*AppellF1(1/2,1/2+1/2*n,1,3/2,cos(f*x+e)^2,-b^2*cos(f*x+e)^2/(a^2-b^2)
)*cos(f*x+e)*(d*csc(f*x+e))^(1+n)*(sin(f*x+e)^2)^(1/2+1/2*n)/(a^2-b^2)/d/f

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3317, 3954, 2902, 3268, 440} \[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\frac {b \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{n/2} (d \csc (e+f x))^{n+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {n}{2},1,\frac {3}{2},\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{d f \left (a^2-b^2\right )}-\frac {a \cos (e+f x) \sin ^2(e+f x)^{\frac {n+1}{2}} (d \csc (e+f x))^{n+1} \operatorname {AppellF1}\left (\frac {1}{2},\frac {n+1}{2},1,\frac {3}{2},\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{d f \left (a^2-b^2\right )} \]

[In]

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

[Out]

(b*AppellF1[1/2, n/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x
])^(1 + n)*Sin[e + f*x]*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)*d*f) - (a*AppellF1[1/2, (1 + n)/2, 1, 3/2, Cos[e
+ f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*(Sin[e + f*x]^2)^((1 + n)
/2))/((a^2 - b^2)*d*f)

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 2902

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[a, Int[(d*
Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x]^2), x], x] - Dist[b/d, Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e +
f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]

Rule 3268

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff
= FreeFactors[Cos[e + f*x], x]}, Dist[(-ff)*d^(2*IntPart[(m - 1)/2] + 1)*((d*Sin[e + f*x])^(2*FracPart[(m - 1
)/2])/(f*(Sin[e + f*x]^2)^FracPart[(m - 1)/2])), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p,
x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] &&  !IntegerQ[m]

Rule 3317

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[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 3954

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Dist[Sin[
e + f*x]^n*(d*Csc[e + f*x])^n, Int[(b + a*Sin[e + f*x])^m/Sin[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, d, e, f
, n}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(d \csc (e+f x))^{1+n}}{b+a \csc (e+f x)} \, dx}{d} \\ & = \frac {\left ((d \csc (e+f x))^{1+n} \sin ^{1+n}(e+f x)\right ) \int \frac {\sin ^{-n}(e+f x)}{a+b \sin (e+f x)} \, dx}{d} \\ & = \frac {\left (a (d \csc (e+f x))^{1+n} \sin ^{1+n}(e+f x)\right ) \int \frac {\sin ^{-n}(e+f x)}{a^2-b^2 \sin ^2(e+f x)} \, dx}{d}-\frac {\left (b (d \csc (e+f x))^{1+n} \sin ^{1+n}(e+f x)\right ) \int \frac {\sin ^{1-n}(e+f x)}{a^2-b^2 \sin ^2(e+f x)} \, dx}{d} \\ & = -\frac {\left (a (d \csc (e+f x))^{1+n} \sin ^{1+2 \left (-\frac {1}{2}-\frac {n}{2}\right )+n}(e+f x) \sin ^2(e+f x)^{\frac {1}{2}+\frac {n}{2}}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{\frac {1}{2} (-1-n)}}{a^2-b^2+b^2 x^2} \, dx,x,\cos (e+f x)\right )}{d f}+\frac {\left (b (d \csc (e+f x))^{1+n} \sin (e+f x) \sin ^2(e+f x)^{n/2}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{-n/2}}{a^2-b^2+b^2 x^2} \, dx,x,\cos (e+f x)\right )}{d f} \\ & = \frac {b \operatorname {AppellF1}\left (\frac {1}{2},\frac {n}{2},1,\frac {3}{2},\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{1+n} \sin (e+f x) \sin ^2(e+f x)^{n/2}}{\left (a^2-b^2\right ) d f}-\frac {a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1+n}{2},1,\frac {3}{2},\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{1+n} \sin ^2(e+f x)^{\frac {1+n}{2}}}{\left (a^2-b^2\right ) d f} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1581\) vs. \(2(195)=390\).

Time = 16.38 (sec) , antiderivative size = 1581, normalized size of antiderivative = 8.11 \[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\frac {(d \csc (e+f x))^n \sec ^2(e+f x)^{-n/2} \left (\cot (e+f x) \sqrt {\sec ^2(e+f x)}\right )^n \tan (e+f x) \left (-3 b (-2+n) \operatorname {AppellF1}\left (\frac {1-n}{2},-\frac {n}{2},1,\frac {3-n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+(-1+n) \left (\left (-9+b^2\right ) \operatorname {AppellF1}\left (1-\frac {n}{2},\frac {1}{2} (-1-n),1,2-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+9 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},-\tan ^2(e+f x)\right )\right ) \tan (e+f x)\right )}{9 b f (-2+n) (-1+n) (3+b \sin (e+f x)) \left (\frac {\sec ^2(e+f x)^{1-\frac {n}{2}} \left (\cot (e+f x) \sqrt {\sec ^2(e+f x)}\right )^n \left (-3 b (-2+n) \operatorname {AppellF1}\left (\frac {1-n}{2},-\frac {n}{2},1,\frac {3-n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+(-1+n) \left (\left (-9+b^2\right ) \operatorname {AppellF1}\left (1-\frac {n}{2},\frac {1}{2} (-1-n),1,2-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+9 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},-\tan ^2(e+f x)\right )\right ) \tan (e+f x)\right )}{9 b (-2+n) (-1+n)}+\frac {n \sec ^2(e+f x)^{-n/2} \left (\cot (e+f x) \sqrt {\sec ^2(e+f x)}\right )^{-1+n} \left (\sqrt {\sec ^2(e+f x)}-\csc ^2(e+f x) \sqrt {\sec ^2(e+f x)}\right ) \tan (e+f x) \left (-3 b (-2+n) \operatorname {AppellF1}\left (\frac {1-n}{2},-\frac {n}{2},1,\frac {3-n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+(-1+n) \left (\left (-9+b^2\right ) \operatorname {AppellF1}\left (1-\frac {n}{2},\frac {1}{2} (-1-n),1,2-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+9 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},-\tan ^2(e+f x)\right )\right ) \tan (e+f x)\right )}{9 b (-2+n) (-1+n)}-\frac {n \sec ^2(e+f x)^{-n/2} \left (\cot (e+f x) \sqrt {\sec ^2(e+f x)}\right )^n \tan ^2(e+f x) \left (-3 b (-2+n) \operatorname {AppellF1}\left (\frac {1-n}{2},-\frac {n}{2},1,\frac {3-n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+(-1+n) \left (\left (-9+b^2\right ) \operatorname {AppellF1}\left (1-\frac {n}{2},\frac {1}{2} (-1-n),1,2-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+9 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},-\tan ^2(e+f x)\right )\right ) \tan (e+f x)\right )}{9 b (-2+n) (-1+n)}+\frac {\sec ^2(e+f x)^{-n/2} \left (\cot (e+f x) \sqrt {\sec ^2(e+f x)}\right )^n \tan (e+f x) \left ((-1+n) \left (\left (-9+b^2\right ) \operatorname {AppellF1}\left (1-\frac {n}{2},\frac {1}{2} (-1-n),1,2-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right )+9 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},-\tan ^2(e+f x)\right )\right ) \sec ^2(e+f x)-3 b (-2+n) \left (\frac {(1-n) n \operatorname {AppellF1}\left (1+\frac {1-n}{2},1-\frac {n}{2},1,1+\frac {3-n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3-n}+\frac {2 \left (-9+b^2\right ) (1-n) \operatorname {AppellF1}\left (1+\frac {1-n}{2},-\frac {n}{2},2,1+\frac {3-n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{9 (3-n)}\right )+(-1+n) \tan (e+f x) \left (\left (-9+b^2\right ) \left (-\frac {(-1-n) \left (1-\frac {n}{2}\right ) \operatorname {AppellF1}\left (2-\frac {n}{2},1+\frac {1}{2} (-1-n),1,3-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{2-\frac {n}{2}}+\frac {2 \left (-9+b^2\right ) \left (1-\frac {n}{2}\right ) \operatorname {AppellF1}\left (2-\frac {n}{2},\frac {1}{2} (-1-n),2,3-\frac {n}{2},-\tan ^2(e+f x),\frac {1}{9} \left (-9+b^2\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{9 \left (2-\frac {n}{2}\right )}\right )+18 \left (1-\frac {n}{2}\right ) \csc (e+f x) \sec (e+f x) \left (-\operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},-\tan ^2(e+f x)\right )+\left (1+\tan ^2(e+f x)\right )^{-\frac {1}{2}+\frac {n}{2}}\right )\right )\right )}{9 b (-2+n) (-1+n)}\right )} \]

[In]

Integrate[(d*Csc[e + f*x])^n/(3 + b*Sin[e + f*x]),x]

[Out]

((d*Csc[e + f*x])^n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(-3*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2
*n, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + (-1 + n)*((-9 + b^2)*AppellF1[1 - n/2, (-1
- n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + 9*Hypergeometric2F1[1/2 - n/2, 1 - n/2,
2 - n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(9*b*f*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)*(3 + b*Sin[e + f*x]
)*(((Sec[e + f*x]^2)^(1 - n/2)*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*(-3*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n
, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + (-1 + n)*((-9 + b^2)*AppellF1[1 - n/2, (-1 -
n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + 9*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2
- n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(9*b*(-2 + n)*(-1 + n)) + (n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^(-1
+ n)*(Sqrt[Sec[e + f*x]^2] - Csc[e + f*x]^2*Sqrt[Sec[e + f*x]^2])*Tan[e + f*x]*(-3*b*(-2 + n)*AppellF1[(1 - n)
/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + (-1 + n)*((-9 + b^2)*AppellF1[1 -
n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + 9*Hypergeometric2F1[1/2 - n/2,
1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Tan[e + f*x]))/(9*b*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) - (n*(Cot[e
+ f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]^2*(-3*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e
+ f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + (-1 + n)*((-9 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan
[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9] + 9*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2
])*Tan[e + f*x]))/(9*b*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)) + ((Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[
e + f*x]*((-1 + n)*((-9 + b^2)*AppellF1[1 - n/2, (-1 - n)/2, 1, 2 - n/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e +
f*x]^2)/9] + 9*Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2])*Sec[e + f*x]^2 - 3*b*(-2 + n)*
(((1 - n)*n*AppellF1[1 + (1 - n)/2, 1 - n/2, 1, 1 + (3 - n)/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9]
*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n) + (2*(-9 + b^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1/2*n, 2, 1 + (3 - n)/2
, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9]*Sec[e + f*x]^2*Tan[e + f*x])/(9*(3 - n))) + (-1 + n)*Tan[e +
f*x]*((-9 + b^2)*(-(((-1 - n)*(1 - n/2)*AppellF1[2 - n/2, 1 + (-1 - n)/2, 1, 3 - n/2, -Tan[e + f*x]^2, ((-9 +
b^2)*Tan[e + f*x]^2)/9]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) + (2*(-9 + b^2)*(1 - n/2)*AppellF1[2 - n/2, (
-1 - n)/2, 2, 3 - n/2, -Tan[e + f*x]^2, ((-9 + b^2)*Tan[e + f*x]^2)/9]*Sec[e + f*x]^2*Tan[e + f*x])/(9*(2 - n/
2))) + 18*(1 - n/2)*Csc[e + f*x]*Sec[e + f*x]*(-Hypergeometric2F1[1/2 - n/2, 1 - n/2, 2 - n/2, -Tan[e + f*x]^2
] + (1 + Tan[e + f*x]^2)^(-1/2 + n/2)))))/(9*b*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2))))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{b \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\int \frac {\left (d \csc {\left (e + f x \right )}\right )^{n}}{a + b \sin {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{b \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{b \sin \left (f x + e\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \csc (e+f x))^n}{3+b \sin (e+f x)} \, dx=\int \frac {{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n}{a+b\,\sin \left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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