3.699 \(\int \frac{\sec ^2(x)}{3-4 \tan ^3(x)} \, dx\)

Optimal. Leaf size=176 \[ \frac{x}{3\ 2^{2/3} \sqrt [6]{3}}+\frac{\log \left (2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}\right )}{6\ 6^{2/3}}-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}-\frac{\tan ^{-1}\left (\frac{-2\ 6^{2/3} \cos ^2(x)+2 \left (3-2 \sqrt [3]{6}\right ) \sin (x) \cos (x)+6^{2/3}}{\left (6-4 \sqrt [3]{6}\right ) \cos ^2(x)+2\ 6^{2/3} \sin (x) \cos (x)+4 \sqrt [3]{6}+3\ 2^{2/3} \sqrt [6]{3}}\right )}{3\ 2^{2/3} \sqrt [6]{3}} \]

[Out]

x/(3*2^(2/3)*3^(1/6)) - ArcTan[(6^(2/3) - 2*6^(2/3)*Cos[x]^2 + 2*(3 - 2*6^(1/3))*Cos[x]*Sin[x])/(3*2^(2/3)*3^(
1/6) + 4*6^(1/3) + (6 - 4*6^(1/3))*Cos[x]^2 + 2*6^(2/3)*Cos[x]*Sin[x])]/(3*2^(2/3)*3^(1/6)) - Log[3^(1/3) - 2^
(2/3)*Tan[x]]/(3*6^(2/3)) + Log[3^(2/3) + 2^(2/3)*3^(1/3)*Tan[x] + 2*2^(1/3)*Tan[x]^2]/(6*6^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 0.13967, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {3675, 200, 31, 634, 617, 204, 628} \[ \frac{x}{3\ 2^{2/3} \sqrt [6]{3}}+\frac{\log \left (2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}\right )}{6\ 6^{2/3}}-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}-\frac{\tan ^{-1}\left (\frac{-2\ 6^{2/3} \cos ^2(x)+2 \left (3-2 \sqrt [3]{6}\right ) \sin (x) \cos (x)+6^{2/3}}{\left (6-4 \sqrt [3]{6}\right ) \cos ^2(x)+2\ 6^{2/3} \sin (x) \cos (x)+4 \sqrt [3]{6}+3\ 2^{2/3} \sqrt [6]{3}}\right )}{3\ 2^{2/3} \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2/(3 - 4*Tan[x]^3),x]

[Out]

x/(3*2^(2/3)*3^(1/6)) - ArcTan[(6^(2/3) - 2*6^(2/3)*Cos[x]^2 + 2*(3 - 2*6^(1/3))*Cos[x]*Sin[x])/(3*2^(2/3)*3^(
1/6) + 4*6^(1/3) + (6 - 4*6^(1/3))*Cos[x]^2 + 2*6^(2/3)*Cos[x]*Sin[x])]/(3*2^(2/3)*3^(1/6)) - Log[3^(1/3) - 2^
(2/3)*Tan[x]]/(3*6^(2/3)) + Log[3^(2/3) + 2^(2/3)*3^(1/3)*Tan[x] + 2*2^(1/3)*Tan[x]^2]/(6*6^(2/3))

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\sec ^2(x)}{3-4 \tan ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{3-4 x^3} \, dx,x,\tan (x)\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{3}-2^{2/3} x} \, dx,x,\tan (x)\right )}{3\ 3^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{2 \sqrt [3]{3}+2^{2/3} x}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tan (x)\right )}{3\ 3^{2/3}}\\ &=-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tan (x)\right )}{2 \sqrt [3]{3}}+\frac{\operatorname{Subst}\left (\int \frac{2^{2/3} \sqrt [3]{3}+4 \sqrt [3]{2} x}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tan (x)\right )}{6\ 6^{2/3}}\\ &=-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}+\frac{\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tan (x)+2 \sqrt [3]{2} \tan ^2(x)\right )}{6\ 6^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2\ 2^{2/3} \tan (x)}{\sqrt [3]{3}}\right )}{6^{2/3}}\\ &=\frac{x}{3\ 2^{2/3} \sqrt [6]{3}}-\frac{\tan ^{-1}\left (\frac{6^{2/3}-2\ 6^{2/3} \cos ^2(x)+2 \left (3-2 \sqrt [3]{6}\right ) \cos (x) \sin (x)}{3\ 2^{2/3} \sqrt [6]{3}+4 \sqrt [3]{6}+2 \left (3-2 \sqrt [3]{6}\right ) \cos ^2(x)+2\ 6^{2/3} \cos (x) \sin (x)}\right )}{3\ 2^{2/3} \sqrt [6]{3}}-\frac{\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}+\frac{\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tan (x)+2 \sqrt [3]{2} \tan ^2(x)\right )}{6\ 6^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.121195, size = 74, normalized size = 0.42 \[ \frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2\ 6^{2/3} \tan (x)+3}{3 \sqrt{3}}\right )+\log \left (2 \sqrt [3]{6} \tan ^2(x)+6^{2/3} \tan (x)+3\right )-2 \log \left (3-6^{2/3} \tan (x)\right )}{6\ 6^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2/(3 - 4*Tan[x]^3),x]

[Out]

(2*Sqrt[3]*ArcTan[(3 + 2*6^(2/3)*Tan[x])/(3*Sqrt[3])] - 2*Log[3 - 6^(2/3)*Tan[x]] + Log[3 + 6^(2/3)*Tan[x] + 2
*6^(1/3)*Tan[x]^2])/(6*6^(2/3))

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 80, normalized size = 0.5 \begin{align*} -{\frac{\sqrt [3]{3}{4}^{{\frac{2}{3}}}}{36}\ln \left ( \tan \left ( x \right ) -{\frac{\sqrt [3]{3}{4}^{{\frac{2}{3}}}}{4}} \right ) }+{\frac{\sqrt [3]{3}{4}^{{\frac{2}{3}}}}{72}\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+{\frac{\sqrt [3]{3}{4}^{{\frac{2}{3}}}\tan \left ( x \right ) }{4}}+{\frac{{3}^{{\frac{2}{3}}}\sqrt [3]{4}}{4}} \right ) }+{\frac{{3}^{{\frac{5}{6}}}{4}^{{\frac{2}{3}}}}{36}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}\sqrt [3]{4}\tan \left ( x \right ) }{3}}+1 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2/(3-4*tan(x)^3),x)

[Out]

-1/36*3^(1/3)*4^(2/3)*ln(tan(x)-1/4*3^(1/3)*4^(2/3))+1/72*3^(1/3)*4^(2/3)*ln(tan(x)^2+1/4*3^(1/3)*4^(2/3)*tan(
x)+1/4*3^(2/3)*4^(1/3))+1/36*3^(5/6)*4^(2/3)*arctan(1/3*3^(1/2)*(2/3*3^(2/3)*4^(1/3)*tan(x)+1))

________________________________________________________________________________________

Maxima [A]  time = 1.45544, size = 120, normalized size = 0.68 \begin{align*} \frac{1}{36} \cdot 4^{\frac{2}{3}} 3^{\frac{5}{6}} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} 3^{\frac{1}{6}}{\left (2 \cdot 4^{\frac{2}{3}} \tan \left (x\right ) + 4^{\frac{1}{3}} 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{72} \cdot 4^{\frac{2}{3}} 3^{\frac{1}{3}} \log \left (4^{\frac{2}{3}} \tan \left (x\right )^{2} + 4^{\frac{1}{3}} 3^{\frac{1}{3}} \tan \left (x\right ) + 3^{\frac{2}{3}}\right ) - \frac{1}{36} \cdot 4^{\frac{2}{3}} 3^{\frac{1}{3}} \log \left (\frac{1}{4} \cdot 4^{\frac{2}{3}}{\left (4^{\frac{1}{3}} \tan \left (x\right ) - 3^{\frac{1}{3}}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(3-4*tan(x)^3),x, algorithm="maxima")

[Out]

1/36*4^(2/3)*3^(5/6)*arctan(1/12*4^(2/3)*3^(1/6)*(2*4^(2/3)*tan(x) + 4^(1/3)*3^(1/3))) + 1/72*4^(2/3)*3^(1/3)*
log(4^(2/3)*tan(x)^2 + 4^(1/3)*3^(1/3)*tan(x) + 3^(2/3)) - 1/36*4^(2/3)*3^(1/3)*log(1/4*4^(2/3)*(4^(1/3)*tan(x
) - 3^(1/3)))

________________________________________________________________________________________

Fricas [B]  time = 3.06919, size = 1635, normalized size = 9.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(3-4*tan(x)^3),x, algorithm="fricas")

[Out]

-1/36*36^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(-1/108*36^(1/6)*(28*(36^(2/3)*sqrt(3)*(-1)^(2/3) - 9*sqrt(3)*(-1)^(1/
3))*cos(x)^6 - 4*(14*36^(2/3)*sqrt(3)*(-1)^(2/3) + 36*36^(1/3)*sqrt(3) - 63*sqrt(3)*(-1)^(1/3))*cos(x)^4 + (37
*36^(2/3)*sqrt(3)*(-1)^(2/3) + 144*36^(1/3)*sqrt(3) + 144*sqrt(3)*(-1)^(1/3))*cos(x)^2 - 6*(16*(36^(2/3)*sqrt(
3)*(-1)^(2/3) - 9*sqrt(3)*(-1)^(1/3))*cos(x)^5 - (24*36^(2/3)*sqrt(3)*(-1)^(2/3) - 7*36^(1/3)*sqrt(3) - 72*sqr
t(3)*(-1)^(1/3))*cos(x)^3 + 4*(36^(2/3)*sqrt(3)*(-1)^(2/3) - 4*36^(1/3)*sqrt(3) + 9*sqrt(3)*(-1)^(1/3))*cos(x)
)*sin(x) - 18*36^(1/3)*sqrt(3) - 144*sqrt(3)*(-1)^(1/3))/(48*cos(x)^6 - 72*cos(x)^4 + 18*cos(x)^2 + 14*(cos(x)
^5 - cos(x)^3)*sin(x) + 3)) - 1/432*36^(2/3)*(-1)^(1/3)*log(-3*(2*36^(2/3)*(-1)^(1/3) - 8*36^(1/3)*(-1)^(2/3)
+ 25)*cos(x)^4 + 3*(3*36^(2/3)*(-1)^(1/3) - 4*36^(1/3)*(-1)^(2/3) + 32)*cos(x)^2 - 2*((4*36^(2/3)*(-1)^(1/3) +
9*36^(1/3)*(-1)^(2/3))*cos(x)^3 - 4*(36^(2/3)*(-1)^(1/3) - 9)*cos(x))*sin(x) - 12*36^(1/3)*(-1)^(2/3) - 48) +
1/216*36^(2/3)*(-1)^(1/3)*log(3*(2*36^(2/3)*(-1)^(1/3) + 8*36^(1/3)*(-1)^(2/3) - 7)*cos(x)^2 + 2*(4*36^(2/3)*
(-1)^(1/3) - 9*36^(1/3)*(-1)^(2/3) + 36)*cos(x)*sin(x) - 3*36^(2/3)*(-1)^(1/3) - 12*36^(1/3)*(-1)^(2/3) + 48)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sec ^{2}{\left (x \right )}}{4 \tan ^{3}{\left (x \right )} - 3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2/(3-4*tan(x)**3),x)

[Out]

-Integral(sec(x)**2/(4*tan(x)**3 - 3), x)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2/(3-4*tan(x)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError