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

Optimal. Leaf size=203 \[ \frac{a^2 d^3 (3-2 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{2 a^2 d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)} \]

[Out]

(a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a^2*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]) + (a^2*d^3*(
3 - 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 = 0.256929, antiderivative size = 203, 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} \[ \frac{a^2 d^3 (3-2 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{2 a^2 d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\sin ^2(e+f x)\right )}{f (2-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{n-2}}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[e + f*x])^n*(a + a*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^2*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]) + (a^2*d^3*(
3 - 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+a \sin (e+f x))^2 \, dx &=d^2 \int (d \csc (e+f x))^{-2+n} (a+a \csc (e+f x))^2 \, dx\\ &=\left (2 a^2 d\right ) \int (d \csc (e+f x))^{-1+n} \, dx+d^2 \int (d \csc (e+f x))^{-2+n} \left (a^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)}+\frac{\left (a^2 d^2 (3-2 n)\right ) \int (d \csc (e+f x))^{-2+n} \, dx}{1-n}+\left (2 a^2 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^2 \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 (a^2 d^2 (3-2 n) (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}{1-n}\\ &=\frac{a^2 d^2 \cot (e+f x) (d \csc (e+f x))^{-2+n}}{f (1-n)}+\frac{2 a^2 \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{a^2 (3-2 n) \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 (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.11137, size = 342, normalized size = 1.68 \[ \frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) (a \sin (e+f x)+a)^2 \sec ^2\left (\frac{1}{2} (e+f x)\right )^{-n} (d \csc (e+f x))^n \left (\frac{\, _2F_1\left (3-n,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{1-n}+\tan \left (\frac{1}{2} (e+f x)\right ) \left (\tan \left (\frac{1}{2} (e+f x)\right ) \left (\frac{\tan ^2\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (3-n,\frac{5}{2}-\frac{n}{2};\frac{7}{2}-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{5-n}-\frac{4 \tan \left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (3-n,2-\frac{n}{2};3-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-4}-\frac{6 \, _2F_1\left (\frac{3-n}{2},3-n;\frac{5-n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-3}\right )-\frac{4 \, _2F_1\left (3-n,1-\frac{n}{2};2-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-2}\right )\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^2*Tan[(e + f*x)/2]*(Hypergeometric2F1[3 - n, 1/2 - n/2, 3/2 - n/2,
-Tan[(e + f*x)/2]^2]/(1 - n) + Tan[(e + f*x)/2]*((-4*Hypergeometric2F1[3 - n, 1 - n/2, 2 - n/2, -Tan[(e + f*x)
/2]^2])/(-2 + n) + Tan[(e + f*x)/2]*((-6*Hypergeometric2F1[(3 - n)/2, 3 - n, (5 - n)/2, -Tan[(e + f*x)/2]^2])/
(-3 + n) - (4*Hypergeometric2F1[3 - n, 2 - n/2, 3 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(-4 + n) + (Hy
pergeometric2F1[3 - n, 5/2 - n/2, 7/2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(5 - n)))))/(f*(Sec[(e +
f*x)/2]^2)^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

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Maple [F]  time = 3.602, size = 0, normalized size = 0. \begin{align*} \int \left ( d\csc \left ( fx+e \right ) \right ) ^{n} \left ( a+a\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+a*sin(f*x+e))^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \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+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((a*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 (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{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+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \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+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

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