3.193 \(\int \tan ^2(e+f x) (a+b \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=40 \[ \frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^3(e+f x)}{3 f} \]

[Out]

-((a - b)*x) + ((a - b)*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)

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Rubi [A]  time = 0.0336666, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3631, 3473, 8} \[ \frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^3(e+f x)}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]

[Out]

-((a - b)*x) + ((a - b)*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)

Rule 3631

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp
[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[A - C, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[
{a, b, e, f, A, C, m}, x] && NeQ[A*b^2 + a^2*C, 0] &&  !LeQ[m, -1]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac{b \tan ^3(e+f x)}{3 f}+(a-b) \int \tan ^2(e+f x) \, dx\\ &=\frac{(a-b) \tan (e+f x)}{f}+\frac{b \tan ^3(e+f x)}{3 f}+(-a+b) \int 1 \, dx\\ &=-(a-b) x+\frac{(a-b) \tan (e+f x)}{f}+\frac{b \tan ^3(e+f x)}{3 f}\\ \end{align*}

Mathematica [A]  time = 0.0238519, size = 65, normalized size = 1.62 \[ -\frac{a \tan ^{-1}(\tan (e+f x))}{f}+\frac{a \tan (e+f x)}{f}+\frac{b \tan ^3(e+f x)}{3 f}+\frac{b \tan ^{-1}(\tan (e+f x))}{f}-\frac{b \tan (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]

[Out]

-((a*ArcTan[Tan[e + f*x]])/f) + (b*ArcTan[Tan[e + f*x]])/f + (a*Tan[e + f*x])/f - (b*Tan[e + f*x])/f + (b*Tan[
e + f*x]^3)/(3*f)

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Maple [A]  time = 0.003, size = 64, normalized size = 1.6 \begin{align*}{\frac{b \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3\,f}}+{\frac{a\tan \left ( fx+e \right ) }{f}}-{\frac{b\tan \left ( fx+e \right ) }{f}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a}{f}}+{\frac{b\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^2*(a+b*tan(f*x+e)^2),x)

[Out]

1/3*b*tan(f*x+e)^3/f+1/f*a*tan(f*x+e)-b*tan(f*x+e)/f-1/f*arctan(tan(f*x+e))*a+b/f*arctan(tan(f*x+e))

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Maxima [A]  time = 1.63988, size = 55, normalized size = 1.38 \begin{align*} \frac{b \tan \left (f x + e\right )^{3} - 3 \,{\left (f x + e\right )}{\left (a - b\right )} + 3 \,{\left (a - b\right )} \tan \left (f x + e\right )}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

1/3*(b*tan(f*x + e)^3 - 3*(f*x + e)*(a - b) + 3*(a - b)*tan(f*x + e))/f

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Fricas [A]  time = 1.02826, size = 90, normalized size = 2.25 \begin{align*} \frac{b \tan \left (f x + e\right )^{3} - 3 \,{\left (a - b\right )} f x + 3 \,{\left (a - b\right )} \tan \left (f x + e\right )}{3 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

1/3*(b*tan(f*x + e)^3 - 3*(a - b)*f*x + 3*(a - b)*tan(f*x + e))/f

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Sympy [A]  time = 0.344046, size = 54, normalized size = 1.35 \begin{align*} \begin{cases} - a x + \frac{a \tan{\left (e + f x \right )}}{f} + b x + \frac{b \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac{b \tan{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \left (a + b \tan ^{2}{\left (e \right )}\right ) \tan ^{2}{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**2*(a+b*tan(f*x+e)**2),x)

[Out]

Piecewise((-a*x + a*tan(e + f*x)/f + b*x + b*tan(e + f*x)**3/(3*f) - b*tan(e + f*x)/f, Ne(f, 0)), (x*(a + b*ta
n(e)**2)*tan(e)**2, True))

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Giac [B]  time = 1.56232, size = 390, normalized size = 9.75 \begin{align*} -\frac{3 \, a f x \tan \left (f x\right )^{3} \tan \left (e\right )^{3} - 3 \, b f x \tan \left (f x\right )^{3} \tan \left (e\right )^{3} - 9 \, a f x \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 9 \, b f x \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 3 \, a \tan \left (f x\right )^{3} \tan \left (e\right )^{2} - 3 \, b \tan \left (f x\right )^{3} \tan \left (e\right )^{2} + 3 \, a \tan \left (f x\right )^{2} \tan \left (e\right )^{3} - 3 \, b \tan \left (f x\right )^{2} \tan \left (e\right )^{3} + 9 \, a f x \tan \left (f x\right ) \tan \left (e\right ) - 9 \, b f x \tan \left (f x\right ) \tan \left (e\right ) + b \tan \left (f x\right )^{3} - 6 \, a \tan \left (f x\right )^{2} \tan \left (e\right ) + 9 \, b \tan \left (f x\right )^{2} \tan \left (e\right ) - 6 \, a \tan \left (f x\right ) \tan \left (e\right )^{2} + 9 \, b \tan \left (f x\right ) \tan \left (e\right )^{2} + b \tan \left (e\right )^{3} - 3 \, a f x + 3 \, b f x + 3 \, a \tan \left (f x\right ) - 3 \, b \tan \left (f x\right ) + 3 \, a \tan \left (e\right ) - 3 \, b \tan \left (e\right )}{3 \,{\left (f \tan \left (f x\right )^{3} \tan \left (e\right )^{3} - 3 \, f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 3 \, f \tan \left (f x\right ) \tan \left (e\right ) - f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm="giac")

[Out]

-1/3*(3*a*f*x*tan(f*x)^3*tan(e)^3 - 3*b*f*x*tan(f*x)^3*tan(e)^3 - 9*a*f*x*tan(f*x)^2*tan(e)^2 + 9*b*f*x*tan(f*
x)^2*tan(e)^2 + 3*a*tan(f*x)^3*tan(e)^2 - 3*b*tan(f*x)^3*tan(e)^2 + 3*a*tan(f*x)^2*tan(e)^3 - 3*b*tan(f*x)^2*t
an(e)^3 + 9*a*f*x*tan(f*x)*tan(e) - 9*b*f*x*tan(f*x)*tan(e) + b*tan(f*x)^3 - 6*a*tan(f*x)^2*tan(e) + 9*b*tan(f
*x)^2*tan(e) - 6*a*tan(f*x)*tan(e)^2 + 9*b*tan(f*x)*tan(e)^2 + b*tan(e)^3 - 3*a*f*x + 3*b*f*x + 3*a*tan(f*x) -
3*b*tan(f*x) + 3*a*tan(e) - 3*b*tan(e))/(f*tan(f*x)^3*tan(e)^3 - 3*f*tan(f*x)^2*tan(e)^2 + 3*f*tan(f*x)*tan(e
) - f)