3.825 \(\int (d \csc (e+f x))^n (a+b \sin (e+f x))^3 \, dx\)

Optimal. Leaf size=298 \[ \frac{b d^4 \left (3 a^2 (3-n)+b^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left (\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d^3 \left (a^2 (2-n)+3 b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)} \]

[Out]

(a^2*b*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^3*Cot[e + f*x]*(d*Cs
c[e + f*x])^(-3 + n)*(b + a*Csc[e + f*x]))/(f*(1 - n)) + (a*d^3*(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[C
os[e + f*x]^2]) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F
1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*(4 - n)*Sqrt[Cos[e + f*x]^2])

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Rubi [A]  time = 0.565934, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3238, 3842, 4047, 3772, 2643, 4046} \[ \frac{b d^4 \left (3 a^2 (3-n)+b^2 (2-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left (\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a d^3 \left (a^2 (2-n)+3 b^2 (1-n)\right ) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (a \csc (e+f x)+b) (d \csc (e+f x))^{n-3}}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*b*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^3*Cot[e + f*x]*(d*Cs
c[e + f*x])^(-3 + n)*(b + a*Csc[e + f*x]))/(f*(1 - n)) + (a*d^3*(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[C
os[e + f*x]^2]) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F
1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*(4 - 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 3842

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(b^2*
Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n)/(f*(m + n - 1)), x] + Dist[1/(d*(m + n - 1)), In
t[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n*Simp[a^3*d*(m + n - 1) + a*b^2*d*n + b*(b^2*d*(m + n - 2) +
3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2*d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f
, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] &&  !Integ
erQ[m])

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, 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))^3 \, dx &=d^3 \int (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))^3 \, dx\\ &=\frac{a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}-\frac{d^2 \int (d \csc (e+f x))^{-3+n} \left (-b d \left (b^2 (1-n)+a^2 (3-n)\right )-a d \left (3 b^2 (1-n)+a^2 (2-n)\right ) \csc (e+f x)-a^2 b d (1-2 n) \csc ^2(e+f x)\right ) \, dx}{1-n}\\ &=\frac{a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}-\frac{d^2 \int (d \csc (e+f x))^{-3+n} \left (-b d \left (b^2 (1-n)+a^2 (3-n)\right )-a^2 b d (1-2 n) \csc ^2(e+f x)\right ) \, dx}{1-n}+\frac{\left (a d^2 \left (3 b^2 (1-n)+a^2 (2-n)\right )\right ) \int (d \csc (e+f x))^{-2+n} \, dx}{1-n}\\ &=\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac{\left (b d^3 \left (b^2 (2-n)+3 a^2 (3-n)\right )\right ) \int (d \csc (e+f x))^{-3+n} \, dx}{2-n}+\frac{\left (a d^2 \left (3 b^2 (1-n)+a^2 (2-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}{1-n}\\ &=\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac{a \left (3 b^2 (1-n)+a^2 (2-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 (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{\left (b d^3 \left (b^2 (2-n)+3 a^2 (3-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 )^{3-n} \, dx}{2-n}\\ &=\frac{a^2 b d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{a^2 d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} (b+a \csc (e+f x))}{f (1-n)}+\frac{a \left (3 b^2 (1-n)+a^2 (2-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 (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{b \left (b^2 (2-n)+3 a^2 (3-n)\right ) \cos (e+f x) (d \csc (e+f x))^n \, _2F_1\left (\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right ) \sin ^4(e+f x)}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.543765, size = 167, normalized size = 0.56 \[ -\frac{d \cos (e+f x) \sin ^2(e+f x)^{\frac{n-1}{2}} (d \csc (e+f x))^{n-1} \left (b \sqrt{\sin ^2(e+f x)} \csc (e+f x) \left (3 a^2 \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{3}{2};\cos ^2(e+f x)\right )+b^2 \, _2F_1\left (\frac{1}{2},\frac{n-2}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )+a^3 \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(e+f x)\right )+3 a b^2 \, _2F_1\left (\frac{1}{2},\frac{n-1}{2};\frac{3}{2};\cos ^2(e+f x)\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*(Sin[e + f*x]^2)^((-1 + n)/2)*(3*a*b^2*Hypergeometric2F1[1/2, (-1
+ n)/2, 3/2, Cos[e + f*x]^2] + a^3*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[e + f*x]^2] + b*Csc[e + f*x]*(b^
2*Hypergeometric2F1[1/2, (-2 + n)/2, 3/2, Cos[e + f*x]^2] + 3*a^2*Hypergeometric2F1[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 = 2.937, 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 ) ^{3}\, 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))^3,x)

[Out]

int((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^3,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 )}^{3} \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))^3,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^3*(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 (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\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))^3,x, algorithm="fricas")

[Out]

integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*(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 )^{3}\, 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))**3,x)

[Out]

Integral((d*csc(e + f*x))**n*(a + b*sin(e + f*x))**3, 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 )}^{3} \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))^3,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^3*(d*csc(f*x + e))^n, x)