3.788 \(\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^3 \, dx\)

Optimal. Leaf size=139 \[ -\frac{4 i a^3 d^2 (d \cot (e+f x))^{n-2} \, _2F_1(1,n-2;n-1;-i \cot (e+f x))}{f (2-n)}+\frac{d^2 \left (a^3 \cot (e+f x)+i a^3\right ) (d \cot (e+f x))^{n-2}}{f (1-n)}+\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{n-2}}{f (1-n) (2-n)} \]

[Out]

(I*a^3*d^2*(1 - 2*n)*(d*Cot[e + f*x])^(-2 + n))/(f*(1 - n)*(2 - n)) + (d^2*(d*Cot[e + f*x])^(-2 + n)*(I*a^3 +
a^3*Cot[e + f*x]))/(f*(1 - n)) - ((4*I)*a^3*d^2*(d*Cot[e + f*x])^(-2 + n)*Hypergeometric2F1[1, -2 + n, -1 + n,
(-I)*Cot[e + f*x]])/(f*(2 - n))

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Rubi [A]  time = 0.364505, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3673, 3556, 3592, 3537, 12, 64} \[ -\frac{4 i a^3 d^2 (d \cot (e+f x))^{n-2} \, _2F_1(1,n-2;n-1;-i \cot (e+f x))}{f (2-n)}+\frac{d^2 \left (a^3 \cot (e+f x)+i a^3\right ) (d \cot (e+f x))^{n-2}}{f (1-n)}+\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{n-2}}{f (1-n) (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^3,x]

[Out]

(I*a^3*d^2*(1 - 2*n)*(d*Cot[e + f*x])^(-2 + n))/(f*(1 - n)*(2 - n)) + (d^2*(d*Cot[e + f*x])^(-2 + n)*(I*a^3 +
a^3*Cot[e + f*x]))/(f*(1 - n)) - ((4*I)*a^3*d^2*(d*Cot[e + f*x])^(-2 + n)*Hypergeometric2F1[1, -2 + n, -1 + n,
(-I)*Cot[e + f*x]])/(f*(2 - n))

Rule 3673

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rule 3556

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[a/(d*(m + n - 1
)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[b*c*(m - 2) + a*d*(m + 2*n) + (a*c*(m - 2) +
b*d*(3*m + 2*n - 4))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a
^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 1] && NeQ[m + n - 1, 0] && (IntegerQ[m] || Intege
rsQ[2*m, 2*n])

Rule 3592

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(B*d*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e
+ f*x])^m*Simp[A*c - B*d + (B*c + A*d)*Tan[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b
*c - a*d, 0] &&  !LeQ[m, -1]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int (d \cot (e+f x))^n (a+i a \tan (e+f x))^3 \, dx &=d^3 \int (d \cot (e+f x))^{-3+n} (i a+a \cot (e+f x))^3 \, dx\\ &=\frac{d^2 (d \cot (e+f x))^{-2+n} \left (i a^3+a^3 \cot (e+f x)\right )}{f (1-n)}+\frac{\left (i a d^2\right ) \int (d \cot (e+f x))^{-3+n} (i a+a \cot (e+f x)) (i a d (3-2 n)+a d (1-2 n) \cot (e+f x)) \, dx}{1-n}\\ &=\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac{d^2 (d \cot (e+f x))^{-2+n} \left (i a^3+a^3 \cot (e+f x)\right )}{f (1-n)}+\frac{\left (i a d^2\right ) \int (d \cot (e+f x))^{-3+n} \left (-4 a^2 d (1-n)+4 i a^2 d (1-n) \cot (e+f x)\right ) \, dx}{1-n}\\ &=\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac{d^2 (d \cot (e+f x))^{-2+n} \left (i a^3+a^3 \cot (e+f x)\right )}{f (1-n)}-\frac{\left (16 a^5 d^4 (1-n)\right ) \operatorname{Subst}\left (\int \frac{4^{3-n} \left (-\frac{i x}{a^2 (1-n)}\right )^{-3+n}}{-16 a^4 d^2 (1-n)^2-4 a^2 d (1-n) x} \, dx,x,4 i a^2 d (1-n) \cot (e+f x)\right )}{f}\\ &=\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac{d^2 (d \cot (e+f x))^{-2+n} \left (i a^3+a^3 \cot (e+f x)\right )}{f (1-n)}-\frac{\left (4^{5-n} a^5 d^4 (1-n)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i x}{a^2 (1-n)}\right )^{-3+n}}{-16 a^4 d^2 (1-n)^2-4 a^2 d (1-n) x} \, dx,x,4 i a^2 d (1-n) \cot (e+f x)\right )}{f}\\ &=\frac{i a^3 d^2 (1-2 n) (d \cot (e+f x))^{-2+n}}{f (1-n) (2-n)}+\frac{d^2 (d \cot (e+f x))^{-2+n} \left (i a^3+a^3 \cot (e+f x)\right )}{f (1-n)}-\frac{4 i a^3 d^2 (d \cot (e+f x))^{-2+n} \, _2F_1(1,-2+n;-1+n;-i \cot (e+f x))}{f (2-n)}\\ \end{align*}

Mathematica [A]  time = 3.57431, size = 234, normalized size = 1.68 \[ -\frac{e^{-3 i e} \left (1+e^{2 i (e+f x)}\right )^{-n-1} \left (\frac{i \left (1+e^{2 i (e+f x)}\right )}{-1+e^{2 i (e+f x)}}\right )^{n-1} \cos ^3(e+f x) (a+i a \tan (e+f x))^3 \left (2^{n+1} (n-2) \left (1+e^{2 i (e+f x)}\right )^2 \, _2F_1\left (1-n,1-n;2-n;\frac{1}{2} \left (1-e^{2 i (e+f x)}\right )\right )-\left (1+e^{2 i (e+f x)}\right )^n \left ((4 n-7) e^{2 i (e+f x)}+2 n-5\right )\right ) \cot ^{-n}(e+f x) (d \cot (e+f x))^n}{f (n-2) (n-1) (\cos (f x)+i \sin (f x))^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^3,x]

[Out]

-(((1 + E^((2*I)*(e + f*x)))^(-1 - n)*((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^(-1 + n)*Cos[
e + f*x]^3*(d*Cot[e + f*x])^n*(-((1 + E^((2*I)*(e + f*x)))^n*(-5 + 2*n + E^((2*I)*(e + f*x))*(-7 + 4*n))) + 2^
(1 + n)*(1 + E^((2*I)*(e + f*x)))^2*(-2 + n)*Hypergeometric2F1[1 - n, 1 - n, 2 - n, (1 - E^((2*I)*(e + f*x)))/
2])*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*f*(-2 + n)*(-1 + n)*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])^3))

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Maple [F]  time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^3,x)

[Out]

int((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} \left (d \cot \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

integrate((I*a*tan(f*x + e) + a)^3*(d*cot(f*x + e))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{8 \, a^{3} \left (\frac{i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} - 1}\right )^{n} e^{\left (6 i \, f x + 6 i \, e\right )}}{e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

integral(8*a^3*((I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) - 1))^n*e^(6*I*f*x + 6*I*e)/(e^(6*I*f*x +
6*I*e) + 3*e^(4*I*f*x + 4*I*e) + 3*e^(2*I*f*x + 2*I*e) + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))**n*(a+I*a*tan(f*x+e))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} \left (d \cot \left (f x + e\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((I*a*tan(f*x + e) + a)^3*(d*cot(f*x + e))^n, x)