3.8.88 \(\int (d \cot (e+f x))^n (a+i a \tan (e+f x))^3 \, dx\) [788]

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

[Out]

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

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Rubi [A]
time = 0.27, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 = {3754, 3637, 3673, 3618, 12, 66} \begin {gather*} -\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)} \end {gather*}

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 12

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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])))

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3637

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 3673

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 3754

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]

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 ) \text {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 ) \text {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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(925\) vs. \(2(139)=278\).
time = 8.36, size = 925, normalized size = 6.65 \begin {gather*} -\frac {4 i \left (1+e^{2 i (e+f x)}\right )^{-n} \left (\frac {i \left (1+e^{2 i (e+f x)}\right )}{-1+e^{2 i (e+f x)}}\right )^n \cos ^3(e+f x) \cot ^{-n}(e+f x) (d \cot (e+f x))^n \left (\left (1+e^{2 i (e+f x)}\right )^n \, _2F_1\left (1,-n;1-n;-\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right )-2^n \, _2F_1\left (-n,-n;1-n;\frac {1}{2} \left (1-e^{2 i (e+f x)}\right )\right )\right ) (a+i a \tan (e+f x))^3}{\left (e^{i e}+e^{3 i e}\right ) f n (\cos (f x)+i \sin (f x))^3}-\frac {4 i e^{-3 i e} \left (-1+e^{2 i (e+f x)}\right )^{-n} \left (\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^n \left (\frac {i \left (1+e^{2 i (e+f x)}\right )}{-1+e^{2 i (e+f x)}}\right )^n \cos ^3(e+f x) \cot ^{-n}(e+f x) (d \cot (e+f x))^n \left (\frac {\left (1+e^{2 i e}\right ) \left (-1+e^{2 i (e+f x)}\right ) \left (1+e^{2 i (e+f x)}\right )^{-1+n} \, _2F_1\left (1,1-n;2-n;\frac {1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right )}{-1+n}+\frac {\left (1+e^{2 i (e+f x)}\right )^n \, _2F_1\left (1,-n;1-n;\frac {1-e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}\right )}{n}-\frac {2^n \, _2F_1\left (-n,-n;1-n;\frac {1}{2} \left (1-e^{2 i (e+f x)}\right )\right )}{n}\right ) (a+i a \tan (e+f x))^3}{\left (1+e^{2 i e}\right ) f (\cos (f x)+i \sin (f x))^3}+\frac {\cos ^3(e+f x) (d \cot (e+f x))^n \left (\frac {(-3+2 n+\cos (2 e)) \sec ^2(e) \left (-\frac {1}{2} i \cos (3 e)-\frac {1}{2} \sin (3 e)\right )}{(-2+n) (-1+n)}+\frac {(-\cos (e-f x)+\cos (e+f x)) \sec ^2(e) \sec (e+f x) \left (\frac {1}{2} i \cos (3 e)+\frac {1}{2} \sin (3 e)\right )}{-1+n}+\frac {\sec ^2(e+f x) (i \cos (3 e)+\sin (3 e))}{-2+n}\right ) (a+i a \tan (e+f x))^3}{f (\cos (f x)+i \sin (f x))^3}+\frac {\cos ^3(e+f x) (d \cot (e+f x))^n \left (\frac {\sec ^2(e) (-1+\cos (2 e)+3 i \sin (2 e)) \left (-\frac {1}{2} i \cos (3 e)-\frac {1}{2} \sin (3 e)\right )}{-1+n}+\frac {\sec ^2(e) \sec (e+f x) \left (-\frac {1}{2} i \cos (3 e)-\frac {1}{2} \sin (3 e)\right ) (-\cos (e-f x)+\cos (e+f x)-3 i \sin (e-f x)+3 i \sin (e+f x))}{-1+n}\right ) (a+i a \tan (e+f x))^3}{f (\cos (f x)+i \sin (f x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*I)*((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]^3*(d*Cot[e + f*x])^n*((1 + E
^((2*I)*(e + f*x)))^n*Hypergeometric2F1[1, -n, 1 - n, -((-1 + E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x))))]
- 2^n*Hypergeometric2F1[-n, -n, 1 - n, (1 - E^((2*I)*(e + f*x)))/2])*(a + I*a*Tan[e + f*x])^3)/((E^(I*e) + E^
((3*I)*e))*(1 + E^((2*I)*(e + f*x)))^n*f*n*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])^3) - ((4*I)*((-1 + E^((2*I)*
(e + f*x)))/(1 + E^((2*I)*(e + f*x))))^n*((I*(1 + E^((2*I)*(e + f*x))))/(-1 + E^((2*I)*(e + f*x))))^n*Cos[e +
f*x]^3*(d*Cot[e + f*x])^n*(((1 + E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))*(1 + E^((2*I)*(e + f*x)))^(-1 + n)*Hy
pergeometric2F1[1, 1 - n, 2 - n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/(-1 + n) + ((1 + E^((2*
I)*(e + f*x)))^n*Hypergeometric2F1[1, -n, 1 - n, (1 - E^((2*I)*(e + f*x)))/(1 + E^((2*I)*(e + f*x)))])/n - (2^
n*Hypergeometric2F1[-n, -n, 1 - n, (1 - E^((2*I)*(e + f*x)))/2])/n)*(a + I*a*Tan[e + f*x])^3)/(E^((3*I)*e)*(1
+ E^((2*I)*e))*(-1 + E^((2*I)*(e + f*x)))^n*f*Cot[e + f*x]^n*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(d*C
ot[e + f*x])^n*(((-3 + 2*n + Cos[2*e])*Sec[e]^2*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/((-2 + n)*(-1 + n)) + ((-Cos
[e - f*x] + Cos[e + f*x])*Sec[e]^2*Sec[e + f*x]*((I/2)*Cos[3*e] + Sin[3*e]/2))/(-1 + n) + (Sec[e + f*x]^2*(I*C
os[3*e] + Sin[3*e]))/(-2 + n))*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3) + (Cos[e + f*x]^3*(d*Co
t[e + f*x])^n*((Sec[e]^2*(-1 + Cos[2*e] + (3*I)*Sin[2*e])*((-1/2*I)*Cos[3*e] - Sin[3*e]/2))/(-1 + n) + (Sec[e]
^2*Sec[e + f*x]*((-1/2*I)*Cos[3*e] - Sin[3*e]/2)*(-Cos[e - f*x] + Cos[e + f*x] - (3*I)*Sin[e - f*x] + (3*I)*Si
n[e + f*x]))/(-1 + n))*(a + I*a*Tan[e + f*x])^3)/(f*(Cos[f*x] + I*Sin[f*x])^3)

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Maple [F]
time = 0.78, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{3}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i a^{3} \left (\int i \left (d \cot {\left (e + f x \right )}\right )^{n}\, dx + \int \left (- 3 \left (d \cot {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx + \int \left (d \cot {\left (e + f x \right )}\right )^{n} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 3 i \left (d \cot {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \end {gather*}

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]

-I*a**3*(Integral(I*(d*cot(e + f*x))**n, x) + Integral(-3*(d*cot(e + f*x))**n*tan(e + f*x), x) + Integral((d*c
ot(e + f*x))**n*tan(e + f*x)**3, x) + Integral(-3*I*(d*cot(e + f*x))**n*tan(e + f*x)**2, x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(e + f*x))^n*(a + a*tan(e + f*x)*1i)^3,x)

[Out]

int((d*cot(e + f*x))^n*(a + a*tan(e + f*x)*1i)^3, x)

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