3.4.13 \(\int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx\) [313]

Optimal. Leaf size=43 \[ \frac {a \, _2F_1(1,1+n;2+n;i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]

[Out]

a*hypergeom([1, 1+n],[2+n],I*tan(f*x+e))*(d*tan(f*x+e))^(1+n)/d/f/(1+n)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3618, 66} \begin {gather*} \frac {a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(159\) vs. \(2(43)=86\).
time = 0.81, size = 159, normalized size = 3.70 \begin {gather*} \frac {2^{-1-n} a e^{-i e} \left (-\frac {i \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}\right )^{1+n} \left (1+e^{2 i (e+f x)}\right )^{1+n} \cos (e+f x) \, _2F_1\left (1+n,1+n;2+n;\frac {1}{2} \left (1-e^{2 i (e+f x)}\right )\right ) (1+i \tan (e+f x)) \tan ^{-n}(e+f x) (d \tan (e+f x))^n}{f (1+n) (\cos (f x)+i \sin (f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*a*(Integral(-I*(d*tan(e + f*x))**n, x) + Integral((d*tan(e + f*x))**n*tan(e + f*x), x))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________