\(\int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx\) [826]

Optimal result

Integrand size = 23, antiderivative size = 208 \[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\frac {9 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac {6 b d^2 \cos (e+f x) (d \csc (e+f x))^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}+\frac {d^3 \left (b^2 (1-n)+9 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{-3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}} \]

[Out]

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*csc(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/(n^2-4*n+3)/(cos(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3317, 3873, 3857, 2722, 4131} \[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\frac {d^3 \left (a^2 (2-n)+b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt {\cos ^2(e+f x)}}+\frac {a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac {2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}} \]

[In]

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

[Out]

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

Rule 2722

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]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

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 3857

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 3873

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Dist[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 4131

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] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = d^2 \int (d \csc (e+f x))^{-2+n} (b+a \csc (e+f x))^2 \, dx \\ & = (2 a b d) \int (d \csc (e+f x))^{-1+n} \, dx+d^2 \int (d \csc (e+f x))^{-2+n} \left (b^2+a^2 \csc ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\left (d^2 \left (b^2+\frac {a^2 (2-n)}{1-n}\right )\right ) \int (d \csc (e+f x))^{-2+n} \, dx+\left (2 a b d (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{1-n} \, dx \\ & = \frac {a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt {\cos ^2(e+f x)}}+\left (d^2 \left (b^2+\frac {a^2 (2-n)}{1-n}\right ) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{2-n} \, dx \\ & = \frac {a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac {2 a b \cos (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt {\cos ^2(e+f x)}}+\frac {\left (b^2+\frac {a^2 (2-n)}{1-n}\right ) \cos (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3-n}{2},\frac {5-n}{2},\sin ^2(e+f x)\right ) \sin ^3(e+f x)}{f (3-n) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.63 \[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=-\frac {d \cos (e+f x) (d \csc (e+f x))^{-1+n} \sin ^2(e+f x)^{\frac {1}{2} (-1+n)} \left (b^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1+n),\frac {3}{2},\cos ^2(e+f x)\right )+9 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3}{2},\cos ^2(e+f x)\right )+6 b \csc (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {3}{2},\cos ^2(e+f x)\right ) \sqrt {\sin ^2(e+f x)}\right )}{f} \]

[In]

Integrate[(d*Csc[e + f*x])^n*(3 + b*Sin[e + f*x])^2,x]

[Out]

-((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] + 9*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + 6*b*Csc[e + f*x]*Hyperge
ometric2F1[1/2, n/2, 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2]))/f)

Maple [F]

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

[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)

Fricas [F]

\[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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]

\[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\int \left (d \csc {\left (e + f x \right )}\right )^{n} \left (a + b \sin {\left (e + f x \right )}\right )^{2}\, dx \]

[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)

Maxima [F]

\[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d \csc (e+f x))^n (3+b \sin (e+f x))^2 \, dx=\int {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]

[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)