3.9.29 \(\int \frac {(d \csc (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx\) [829]

Optimal. Leaf size=321 \[ -\frac {b^2 F_1\left (\frac {1}{2};\frac {1}{2} (-1+n),2;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{2+n} \sin ^3(e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-1+n)}}{\left (a^2-b^2\right )^2 d^2 f}-\frac {a^2 F_1\left (\frac {1}{2};\frac {1+n}{2},2;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{2+n} \sin (e+f x) \sin ^2(e+f x)^{\frac {1+n}{2}}}{\left (a^2-b^2\right )^2 d^2 f}+\frac {2 a b F_1\left (\frac {1}{2};\frac {n}{2},2;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right ) \cos (e+f x) (d \csc (e+f x))^{2+n} \sin ^2(e+f x)^{\frac {2+n}{2}}}{\left (a^2-b^2\right )^2 d^2 f} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3317, 3954, 2903, 3268, 440} \begin {gather*} -\frac {a^2 \sin (e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac {n+1}{2}} (d \csc (e+f x))^{n+2} F_1\left (\frac {1}{2};\frac {n+1}{2},2;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{d^2 f \left (a^2-b^2\right )^2}+\frac {2 a b \cos (e+f x) \sin ^2(e+f x)^{\frac {n+2}{2}} (d \csc (e+f x))^{n+2} F_1\left (\frac {1}{2};\frac {n}{2},2;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{d^2 f \left (a^2-b^2\right )^2}-\frac {b^2 \sin ^3(e+f x) \cos (e+f x) \sin ^2(e+f x)^{\frac {n-1}{2}} (d \csc (e+f x))^{n+2} F_1\left (\frac {1}{2};\frac {n-1}{2},2;\frac {3}{2};\cos ^2(e+f x),-\frac {b^2 \cos ^2(e+f x)}{a^2-b^2}\right )}{d^2 f \left (a^2-b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*AppellF1[1/2, (-1 + n)/2, 2, 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])^(2 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^((-1 + n)/2))/((a^2 - b^2)^2*d^2*f)) - (a^2*AppellF1[1/2
, (1 + n)/2, 2, 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])^(2 + n
)*Sin[e + f*x]*(Sin[e + f*x]^2)^((1 + n)/2))/((a^2 - b^2)^2*d^2*f) + (2*a*b*AppellF1[1/2, n/2, 2, 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])^(2 + n)*(Sin[e + f*x]^2)^((2 + n)/
2))/((a^2 - b^2)^2*d^2*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 2903

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

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*} \int \frac {(d \csc (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx &=\frac {\int \frac {(d \csc (e+f x))^{2+n}}{(b+a \csc (e+f x))^2} \, dx}{d^2}\\ &=\frac {\left ((d \csc (e+f x))^{2+n} \sin ^{2+n}(e+f x)\right ) \int \frac {\sin ^{-n}(e+f x)}{(a+b \sin (e+f x))^2} \, dx}{d^2}\\ &=\frac {\left ((d \csc (e+f x))^{2+n} \sin ^{2+n}(e+f x)\right ) \int \left (-\frac {2 a b \sin ^{1-n}(e+f x)}{\left (a^2-b^2 \sin ^2(e+f x)\right )^2}+\frac {a^2 \sin ^{-n}(e+f x)}{\left (a^2-b^2 \sin ^2(e+f x)\right )^2}+\frac {b^2 \sin ^{2-n}(e+f x)}{\left (-a^2+b^2 \sin ^2(e+f x)\right )^2}\right ) \, dx}{d^2}\\ &=\frac {\left (a^2 (d \csc (e+f x))^{2+n} \sin ^{2+n}(e+f x)\right ) \int \frac {\sin ^{-n}(e+f x)}{\left (a^2-b^2 \sin ^2(e+f x)\right )^2} \, dx}{d^2}-\frac {\left (2 a b (d \csc (e+f x))^{2+n} \sin ^{2+n}(e+f x)\right ) \int \frac {\sin ^{1-n}(e+f x)}{\left (a^2-b^2 \sin ^2(e+f x)\right )^2} \, dx}{d^2}+\frac {\left (b^2 (d \csc (e+f x))^{2+n} \sin ^{2+n}(e+f x)\right ) \int \frac {\sin ^{2-n}(e+f x)}{\left (-a^2+b^2 \sin ^2(e+f x)\right )^2} \, dx}{d^2}\\ &=-\frac {\left (b^2 (d \csc (e+f x))^{2+n} \sin ^{2+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-n}{2}}}{\left (-a^2+b^2-b^2 x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{d^2 f}-\frac {\left (a^2 (d \csc (e+f x))^{2+n} \sin ^{2+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)}}{\left (a^2-b^2+b^2 x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{d^2 f}+\frac {\left (2 a b (d \csc (e+f x))^{2+n} \sin ^2(e+f x)^{1+\frac {n}{2}}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{-n/2}}{\left (a^2-b^2+b^2 x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{d^2 f}\\ &=-\frac {b^2 F_1\left (\frac {1}{2};\frac {1}{2} (-1+n),2;\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))^{2+n} \sin ^3(e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-1+n)}}{\left (a^2-b^2\right )^2 d^2 f}-\frac {a^2 F_1\left (\frac {1}{2};\frac {1+n}{2},2;\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))^{2+n} \sin (e+f x) \sin ^2(e+f x)^{\frac {1+n}{2}}}{\left (a^2-b^2\right )^2 d^2 f}+\frac {2 a b F_1\left (\frac {1}{2};\frac {n}{2},2;\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))^{2+n} \sin ^2(e+f x)^{\frac {2+n}{2}}}{\left (a^2-b^2\right )^2 d^2 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1872\) vs. \(2(321)=642\).
time = 16.42, size = 1872, normalized size = 5.83 \begin {gather*} \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 (-a \left (a^2+b^2\right ) (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},1;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+2 b \left (a b (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},2;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+\left (a^2-b^2\right ) (-1+n) F_1\left (1-\frac {n}{2};\frac {1}{2} (-1-n),2;2-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan (e+f x)\right )\right )}{a^3 \left (a^2-b^2\right ) f (-2+n) (-1+n) (a+b \sin (e+f x))^2 \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 (-a \left (a^2+b^2\right ) (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},1;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+2 b \left (a b (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},2;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+\left (a^2-b^2\right ) (-1+n) F_1\left (1-\frac {n}{2};\frac {1}{2} (-1-n),2;2-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan (e+f x)\right )\right )}{a^3 \left (a^2-b^2\right ) (-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 (-a \left (a^2+b^2\right ) (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},1;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+2 b \left (a b (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},2;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+\left (a^2-b^2\right ) (-1+n) F_1\left (1-\frac {n}{2};\frac {1}{2} (-1-n),2;2-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan (e+f x)\right )\right )}{a^3 \left (a^2-b^2\right ) (-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 (-a \left (a^2+b^2\right ) (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},1;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+2 b \left (a b (-2+n) F_1\left (\frac {1-n}{2};-\frac {n}{2},2;\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right )+\left (a^2-b^2\right ) (-1+n) F_1\left (1-\frac {n}{2};\frac {1}{2} (-1-n),2;2-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan (e+f x)\right )\right )}{a^3 \left (a^2-b^2\right ) (-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 (-a \left (a^2+b^2\right ) (-2+n) \left (\frac {(1-n) n F_1\left (1+\frac {1-n}{2};1-\frac {n}{2},1;1+\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3-n}+\frac {2 \left (-1+\frac {b^2}{a^2}\right ) (1-n) F_1\left (1+\frac {1-n}{2};-\frac {n}{2},2;1+\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3-n}\right )+2 b \left (\left (a^2-b^2\right ) (-1+n) F_1\left (1-\frac {n}{2};\frac {1}{2} (-1-n),2;2-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x)+a b (-2+n) \left (\frac {(1-n) n F_1\left (1+\frac {1-n}{2};1-\frac {n}{2},2;1+\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3-n}+\frac {4 \left (-1+\frac {b^2}{a^2}\right ) (1-n) F_1\left (1+\frac {1-n}{2};-\frac {n}{2},3;1+\frac {3-n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3-n}\right )+\left (a^2-b^2\right ) (-1+n) \tan (e+f x) \left (-\frac {(-1-n) \left (1-\frac {n}{2}\right ) F_1\left (2-\frac {n}{2};1+\frac {1}{2} (-1-n),2;3-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{2-\frac {n}{2}}+\frac {4 \left (-1+\frac {b^2}{a^2}\right ) \left (1-\frac {n}{2}\right ) F_1\left (2-\frac {n}{2};\frac {1}{2} (-1-n),3;3-\frac {n}{2};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{2-\frac {n}{2}}\right )\right )\right )}{a^3 \left (a^2-b^2\right ) (-2+n) (-1+n)}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((d*Csc[e + f*x])^n*(Cot[e + f*x]*Sqrt[Sec[e + f*x]^2])^n*Tan[e + f*x]*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 -
n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1
- n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[
1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b
^2)*f*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2)*(a + b*Sin[e + f*x])^2*(((Sec[e + f*x]^2)^(1 - n/2)*(Cot[e + f*
x]*Sqrt[Sec[e + f*x]^2])^n*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2
, (-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^
2, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*
x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(-2 + n)*(-1 + n)) + (n*(Cot[e + f*x]*Sq
rt[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]*(-(a*(a
^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2])
+ 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]
+ (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^
2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(-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*(-(a*(a^2 + b^2)*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 1, (3 - n)/2, -Tan[e + f*x]^2,
(-1 + b^2/a^2)*Tan[e + f*x]^2]) + 2*b*(a*b*(-2 + n)*AppellF1[(1 - n)/2, -1/2*n, 2, (3 - n)/2, -Tan[e + f*x]^2
, (-1 + b^2/a^2)*Tan[e + f*x]^2] + (a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2, (-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x
]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x])))/(a^3*(a^2 - b^2)*(-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]*(-(a*(a^2 + b^2)*(-2 + n)*(((1 - n)*n*AppellF1[1 + (1 -
n)/2, 1 - n/2, 1, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])
/(3 - n) + (2*(-1 + b^2/a^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1/2*n, 2, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 +
b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n))) + 2*b*((a^2 - b^2)*(-1 + n)*AppellF1[1 - n/2,
(-1 - n)/2, 2, 2 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2 + a*b*(-2 + n)*(((1 - n
)*n*AppellF1[1 + (1 - n)/2, 1 - n/2, 2, 1 + (3 - n)/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e +
f*x]^2*Tan[e + f*x])/(3 - n) + (4*(-1 + b^2/a^2)*(1 - n)*AppellF1[1 + (1 - n)/2, -1/2*n, 3, 1 + (3 - n)/2, -T
an[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3 - n)) + (a^2 - b^2)*(-1 + n)*Tan
[e + f*x]*(-(((-1 - n)*(1 - n/2)*AppellF1[2 - n/2, 1 + (-1 - n)/2, 2, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)
*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)) + (4*(-1 + b^2/a^2)*(1 - n/2)*AppellF1[2 - n/2, (-1 -
n)/2, 3, 3 - n/2, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(2 - n/2)))))/
(a^3*(a^2 - b^2)*(-2 + n)*(-1 + n)*(Sec[e + f*x]^2)^(n/2))))

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Maple [F]
time = 0.51, size = 0, normalized size = 0.00 \[\int \frac {\left (d \csc \left (f x +e \right )\right )^{n}}{\left (a +b \sin \left (f x +e \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \csc {\left (e + f x \right )}\right )^{n}}{\left (a + b \sin {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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