3.549 \(\int e^{m x} \tan ^2(x) \, dx\)

Optimal. Leaf size=58 \[ -\frac {e^{m x}}{m}+\frac {4 e^{(m+2 i) x} \, _2F_1\left (2,1-\frac {i m}{2};2-\frac {i m}{2};-e^{2 i x}\right )}{m+2 i} \]

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Rubi [A]  time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.47, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4442, 2194, 2251} \[ \frac {4 e^{m x} \text {Hypergeometric2F1}\left (1,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )}{m}-\frac {4 e^{m x} \text {Hypergeometric2F1}\left (2,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )}{m}-\frac {e^{m x}}{m} \]

Antiderivative was successfully verified.

[In]

Int[E^(m*x)*Tan[x]^2,x]

[Out]

-(E^(m*x)/m) + (4*E^(m*x)*Hypergeometric2F1[1, (-I/2)*m, 1 - (I/2)*m, -E^((2*I)*x)])/m - (4*E^(m*x)*Hypergeome
tric2F1[2, (-I/2)*m, 1 - (I/2)*m, -E^((2*I)*x)])/m

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rule 4442

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Tan[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Dist[I^n, Int[ExpandIntegran
d[(F^(c*(a + b*x))*(1 - E^(2*I*(d + e*x)))^n)/(1 + E^(2*I*(d + e*x)))^n, x], x], x] /; FreeQ[{F, a, b, c, d, e
}, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int e^{m x} \tan ^2(x) \, dx &=-\int \left (e^{m x}+\frac {4 e^{m x}}{\left (1+e^{2 i x}\right )^2}-\frac {4 e^{m x}}{1+e^{2 i x}}\right ) \, dx\\ &=-\left (4 \int \frac {e^{m x}}{\left (1+e^{2 i x}\right )^2} \, dx\right )+4 \int \frac {e^{m x}}{1+e^{2 i x}} \, dx-\int e^{m x} \, dx\\ &=-\frac {e^{m x}}{m}+\frac {4 e^{m x} \, _2F_1\left (1,-\frac {i m}{2};1-\frac {i m}{2};-e^{2 i x}\right )}{m}-\frac {4 e^{m x} \, _2F_1\left (2,-\frac {i m}{2};1-\frac {i m}{2};-e^{2 i x}\right )}{m}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 97, normalized size = 1.67 \[ \frac {e^{m x} \left (\frac {i m^2 e^{2 i x} \, _2F_1\left (1,1-\frac {i m}{2};2-\frac {i m}{2};-e^{2 i x}\right )}{m+2 i}-i m \, _2F_1\left (1,-\frac {i m}{2};1-\frac {i m}{2};-e^{2 i x}\right )+m \tan (x)-1\right )}{m} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(m*x)*Tan[x]^2,x]

[Out]

(E^(m*x)*(-1 + (I*E^((2*I)*x)*m^2*Hypergeometric2F1[1, 1 - (I/2)*m, 2 - (I/2)*m, -E^((2*I)*x)])/(2*I + m) - I*
m*Hypergeometric2F1[1, (-1/2*I)*m, 1 - (I/2)*m, -E^((2*I)*x)] + m*Tan[x]))/m

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \tan ^2(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[E^(m*x)*Tan[x]^2,x]

[Out]

Could not integrate

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fricas [F]  time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (m x\right )} \tan \relax (x)^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(m*x)*tan(x)^2,x, algorithm="fricas")

[Out]

integral(e^(m*x)*tan(x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (m x\right )} \tan \relax (x)^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(m*x)*tan(x)^2,x, algorithm="giac")

[Out]

integrate(e^(m*x)*tan(x)^2, x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{m x} \left (\tan ^{2}\relax (x )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(m*x)*tan(x)^2,x)

[Out]

int(exp(m*x)*tan(x)^2,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(m*x)*tan(x)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{m\,x}\,{\mathrm {tan}\relax (x)}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(m*x)*tan(x)^2,x)

[Out]

int(exp(m*x)*tan(x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \tan ^{2}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(m*x)*tan(x)**2,x)

[Out]

Integral(exp(m*x)*tan(x)**2, x)

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