3.4.8 \(\int (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx\) [308]

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

[Out]

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

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Rubi [A]
time = 0.04, 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 (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;i \tan (c+d x))}{d e (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Tan[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]

[Out]

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

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 (e \tan (c+d x))^m (a+i a \tan (c+d x)) \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {\left (-\frac {i e x}{a}\right )^m}{-a^2+a x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {a \, _2F_1(1,1+m;2+m;i \tan (c+d x)) (e \tan (c+d x))^{1+m}}{d e (1+m)}\\ \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. \(159\) vs. \(2(43)=86\).
time = 0.87, size = 159, normalized size = 3.70 \begin {gather*} \frac {2^{-1-m} a e^{-i c} \left (-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}\right )^{1+m} \left (1+e^{2 i (c+d x)}\right )^{1+m} \cos (c+d x) \, _2F_1\left (1+m,1+m;2+m;\frac {1}{2} \left (1-e^{2 i (c+d x)}\right )\right ) (1+i \tan (c+d x)) \tan ^{-m}(c+d x) (e \tan (c+d x))^m}{d (1+m) (\cos (d x)+i \sin (d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*Tan[c + d*x])^m*(a + I*a*Tan[c + d*x]),x]

[Out]

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

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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \left (e \tan \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x)

[Out]

int((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),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((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)*(e*tan(d*x + c))^m, 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((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*tan(d*x+c))**m*(a+I*a*tan(d*x+c)),x)

[Out]

I*a*(Integral(-I*(e*tan(c + d*x))**m, x) + Integral((e*tan(c + d*x))**m*tan(c + d*x), 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((e*tan(d*x+c))^m*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)*(e*tan(d*x + c))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*tan(c + d*x))^m*(a + a*tan(c + d*x)*1i),x)

[Out]

int((e*tan(c + d*x))^m*(a + a*tan(c + d*x)*1i), x)

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