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

Optimal. Leaf size=272 \[ \frac{a^3 d^4 (11-4 n) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left (\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (5-4 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)} \]

[Out]

(a^3*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (d^3*Cot[e + f*x]*(d*Csc[e +
f*x])^(-3 + n)*(a^3 + a^3*Csc[e + f*x]))/(f*(1 - n)) + (a^3*d^3*(5 - 4*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]) + (a
^3*d^4*(11 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e +
f*x]^2])/(f*(2 - n)*(4 - n)*Sqrt[Cos[e + f*x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.458791, antiderivative size = 272, 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, 3814, 3997, 3787, 3772, 2643} \[ \frac{a^3 d^4 (11-4 n) \cos (e+f x) (d \csc (e+f x))^{n-4} \, _2F_1\left (\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\sin ^2(e+f x)\right )}{f (2-n) (4-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (5-4 n) \cos (e+f x) (d \csc (e+f x))^{n-3} \, _2F_1\left (\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\sin ^2(e+f x)\right )}{f (1-n) (3-n) \sqrt{\cos ^2(e+f x)}}+\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{n-3}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) \left (a^3 \csc (e+f x)+a^3\right ) (d \csc (e+f x))^{n-3}}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (d^3*Cot[e + f*x]*(d*Csc[e +
f*x])^(-3 + n)*(a^3 + a^3*Csc[e + f*x]))/(f*(1 - n)) + (a^3*d^3*(5 - 4*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]) + (a
^3*d^4*(11 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[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 3814

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[b/(m + n - 1), Int[(a
+ b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /; Fr
eeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]

Rule 3997

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*C
sc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f
, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], 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]

Rubi steps

\begin{align*} \int (d \csc (e+f x))^n (a+a \sin (e+f x))^3 \, dx &=d^3 \int (d \csc (e+f x))^{-3+n} (a+a \csc (e+f x))^3 \, dx\\ &=\frac{d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} \left (a^3+a^3 \csc (e+f x)\right )}{f (1-n)}-\frac{\left (a d^3\right ) \int (d \csc (e+f x))^{-3+n} (a+a \csc (e+f x)) (a (2+2 (-3+n))+a (5+2 (-3+n)) \csc (e+f x)) \, dx}{1-n}\\ &=\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} \left (a^3+a^3 \csc (e+f x)\right )}{f (1-n)}+\frac{\left (a d^3\right ) \int (d \csc (e+f x))^{-3+n} \left (a^2 (11-4 n) (1-n)+a^2 (5-4 n) (2-n) \csc (e+f x)\right ) \, dx}{2-3 n+n^2}\\ &=\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} \left (a^3+a^3 \csc (e+f x)\right )}{f (1-n)}+\frac{\left (a^3 d^2 (5-4 n)\right ) \int (d \csc (e+f x))^{-2+n} \, dx}{1-n}+\frac{\left (a^3 d^3 (11-4 n)\right ) \int (d \csc (e+f x))^{-3+n} \, dx}{2-n}\\ &=\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} \left (a^3+a^3 \csc (e+f x)\right )}{f (1-n)}+\frac{\left (a^3 d^2 (5-4 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{\left (a^3 d^3 (11-4 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 )^{3-n} \, dx}{2-n}\\ &=\frac{a^3 d^3 (1-2 n) \cot (e+f x) (d \csc (e+f x))^{-3+n}}{f (1-n) (2-n)}+\frac{d^3 \cot (e+f x) (d \csc (e+f x))^{-3+n} \left (a^3+a^3 \csc (e+f x)\right )}{f (1-n)}+\frac{a^3 (5-4 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)}}+\frac{a^3 (11-4 n) \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 = 11.5614, size = 493, normalized size = 1.81 \[ \frac{2^{1-n} \tan \left (\frac{1}{2} (e+f x)\right ) (a \sin (e+f x)+a)^3 \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )^{-n} \csc ^{-n}(e+f x) (d \csc (e+f x))^n \left (\frac{\tan ^6\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (4-n,\frac{7}{2}-\frac{n}{2};\frac{9}{2}-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{7-n}-\frac{6 \tan ^5\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (4-n,3-\frac{n}{2};4-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-6}-\frac{15 \tan ^4\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (4-n,\frac{5-n}{2};\frac{7-n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-5}-\frac{20 \tan ^3\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (4-n,2-\frac{n}{2};3-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-4}-\frac{15 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (\frac{3-n}{2},4-n;\frac{5-n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-3}-\frac{6 \tan \left (\frac{1}{2} (e+f x)\right ) \, _2F_1\left (4-n,1-\frac{n}{2};2-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{n-2}+\frac{\, _2F_1\left (4-n,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{1-n}\right ) \left (\tan \left (\frac{1}{2} (e+f x)\right )+\cot \left (\frac{1}{2} (e+f x)\right )\right )^n}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2^(1 - n)*(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^3*Tan[(e + f*x)/2]*(Cot[(e + f*x)/2] + Tan[(e + f*x)/2])^n*
(Hypergeometric2F1[4 - n, 1/2 - n/2, 3/2 - n/2, -Tan[(e + f*x)/2]^2]/(1 - n) - (6*Hypergeometric2F1[4 - n, 1 -
n/2, 2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2])/(-2 + n) - (15*Hypergeometric2F1[(3 - n)/2, 4 - n, (5 -
n)/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)/(-3 + n) - (20*Hypergeometric2F1[4 - n, 2 - n/2, 3 - n/2, -Tan[
(e + f*x)/2]^2]*Tan[(e + f*x)/2]^3)/(-4 + n) - (15*Hypergeometric2F1[4 - n, (5 - n)/2, (7 - n)/2, -Tan[(e + f*
x)/2]^2]*Tan[(e + f*x)/2]^4)/(-5 + n) - (6*Hypergeometric2F1[4 - n, 3 - n/2, 4 - n/2, -Tan[(e + f*x)/2]^2]*Tan
[(e + f*x)/2]^5)/(-6 + n) + (Hypergeometric2F1[4 - n, 7/2 - n/2, 9/2 - n/2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)
/2]^6)/(7 - n)))/(f*Csc[e + f*x]^n*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(1 + Tan[(e + f*x)/2]^2)^n)

________________________________________________________________________________________

Maple [F]  time = 2.757, 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 ) ^{3}\, 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))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \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+a*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{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+a*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*(d*csc(f*x + e))^n, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \left (d \csc{\left (e + f x \right )}\right )^{n}\, dx + \int 3 \left (d \csc{\left (e + f x \right )}\right )^{n} \sin{\left (e + f x \right )}\, dx + \int 3 \left (d \csc{\left (e + f x \right )}\right )^{n} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (d \csc{\left (e + f x \right )}\right )^{n} \sin ^{3}{\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))**3,x)

[Out]

a**3*(Integral((d*csc(e + f*x))**n, x) + Integral(3*(d*csc(e + f*x))**n*sin(e + f*x), x) + Integral(3*(d*csc(e
+ f*x))**n*sin(e + f*x)**2, x) + Integral((d*csc(e + f*x))**n*sin(e + f*x)**3, x))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \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+a*sin(f*x+e))^3,x, algorithm="giac")

[Out]

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