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

Optimal. Leaf size=213 \[ \frac{d^3 \left (a^2 (2-n)+b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}} \]

[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])

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Rubi [A]  time = 0.269088, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3238, 3788, 3772, 2643, 4046} \[ \frac{d^3 \left (a^2 (2-n)+b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)}+\frac{2 a b d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[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 3238

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 3788

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 3772

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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 4046

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*} \int (d \csc (e+f x))^n (a+b \sin (e+f x))^2 \, dx &=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 \, _2F_1\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 \, _2F_1\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 \, _2F_1\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]  time = 0.374521, size = 135, normalized size = 0.63 \[ -\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left (a \left (a \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right )+2 b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )+b^2 \, _2F_1\left (\frac{1}{2},\frac{n-1}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[e + f*x])^n*(a + 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] + a*(a*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + 2*b*Csc[e + f*x]*Hype
rgeometric2F1[1/2, n/2, 3/2, Cos[e + f*x]^2]*Sqrt[Sin[e + f*x]^2])))/f)

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Maple [F]  time = 3.652, size = 0, normalized size = 0. \begin{align*} \int \left ( d\csc \left ( fx+e \right ) \right ) ^{n} \left ( a+b\sin \left ( fx+e \right ) \right ) ^{2}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \left (d \csc \left (f x + e\right )\right )^{n}, x\right ) \end{align*}

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(-(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)

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

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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \csc \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

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