Integrand size = 15, antiderivative size = 176 \[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=\frac {x}{3\ 2^{2/3} \sqrt [6]{3}}-\frac {\arctan \left (\frac {6^{2/3}-2\ 6^{2/3} \cos ^2(x)+2 \left (3-2 \sqrt [3]{6}\right ) \cos (x) \sin (x)}{3\ 2^{2/3} \sqrt [6]{3}+4 \sqrt [3]{6}+\left (6-4 \sqrt [3]{6}\right ) \cos ^2(x)+2\ 6^{2/3} \cos (x) \sin (x)}\right )}{3\ 2^{2/3} \sqrt [6]{3}}-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}+\frac {\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tan (x)+2 \sqrt [3]{2} \tan ^2(x)\right )}{6\ 6^{2/3}} \]
1/18*x*2^(1/3)*3^(5/6)-1/18*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)))*2^(1/3)*3^(5/6)-1/18*ln(3^(1/3)-2^(2/3)*tan(x))*6^ (1/3)+1/36*ln(3^(2/3)+2^(2/3)*3^(1/3)*tan(x)+2*2^(1/3)*tan(x)^2)*6^(1/3)
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.42 \[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {3+2\ 6^{2/3} \tan (x)}{3 \sqrt {3}}\right )-2 \log \left (3-6^{2/3} \tan (x)\right )+\log \left (3+6^{2/3} \tan (x)+2 \sqrt [3]{6} \tan ^2(x)\right )}{6\ 6^{2/3}} \]
(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))
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.65, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4158, 750, 16, 27, 1142, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (x)^2}{3-4 \tan (x)^3}dx\) |
| \(\Big \downarrow \) 4158 |
| \(\displaystyle \int \frac {1}{3-4 \tan ^3(x)}d\tan (x)\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\int \frac {2^{2/3} \left (\tan (x)+\sqrt [3]{6}\right )}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)}{3\ 3^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{3}-2^{2/3} \tan (x)}d\tan (x)}{3\ 3^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {2^{2/3} \left (\tan (x)+\sqrt [3]{6}\right )}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)}{3\ 3^{2/3}}-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3}\right )^{2/3} \int \frac {\tan (x)+\sqrt [3]{6}}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3}\right )^{2/3} \left (\frac {3 \sqrt [3]{3} \int \frac {1}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)}{2\ 2^{2/3}}+\frac {\int \frac {\sqrt [3]{2} \left (4 \tan (x)+\sqrt [3]{6}\right )}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)}{4 \sqrt [3]{2}}\right )-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3}\right )^{2/3} \left (\frac {3 \sqrt [3]{3} \int \frac {1}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)}{2\ 2^{2/3}}+\frac {1}{4} \int \frac {4 \tan (x)+\sqrt [3]{6}}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)\right )-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3}\right )^{2/3} \left (\frac {1}{4} \int \frac {4 \tan (x)+\sqrt [3]{6}}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)-\frac {3 \int \frac {1}{-\left (\frac {2\ 2^{2/3} \tan (x)}{\sqrt [3]{3}}+1\right )^2-3}d\left (\frac {2\ 2^{2/3} \tan (x)}{\sqrt [3]{3}}+1\right )}{2 \sqrt [3]{2}}\right )-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3}\right )^{2/3} \left (\frac {1}{4} \int \frac {4 \tan (x)+\sqrt [3]{6}}{2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}}d\tan (x)+\frac {\sqrt {3} \arctan \left (\frac {\frac {2\ 2^{2/3} \tan (x)}{\sqrt [3]{3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}\right )-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3}\right )^{2/3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2\ 2^{2/3} \tan (x)}{\sqrt [3]{3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (2 \sqrt [3]{2} \tan ^2(x)+2^{2/3} \sqrt [3]{3} \tan (x)+3^{2/3}\right )}{4 \sqrt [3]{2}}\right )-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tan (x)\right )}{3\ 6^{2/3}}\) |
-1/3*Log[3^(1/3) - 2^(2/3)*Tan[x]]/6^(2/3) + ((2/3)^(2/3)*((Sqrt[3]*ArcTan [(1 + (2*2^(2/3)*Tan[x])/3^(1/3))/Sqrt[3]])/(2*2^(1/3)) + Log[3^(2/3) + 2^ (2/3)*3^(1/3)*Tan[x] + 2*2^(1/3)*Tan[x]^2]/(4*2^(1/3))))/3
3.7.99.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]}, Sim p[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] && I ntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.22
| method | result | size |
| risch | \(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (62208 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {41472}{25}-\frac {31104 i}{25}\right ) \textit {\_R}^{2}+\left (\frac {864}{25}+\frac {1152 i}{25}\right ) \textit {\_R} -\frac {7}{25}+\frac {24 i}{25}\right )\right )\) | \(38\) |
| derivativedivides | \(-\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}} \ln \left (\tan \left (x \right )-\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}}}{4}\right )}{36}+\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}} \ln \left (\tan \left (x \right )^{2}+\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}} \tan \left (x \right )}{4}+\frac {3^{\frac {2}{3}} 4^{\frac {1}{3}}}{4}\right )}{72}+\frac {3^{\frac {5}{6}} 4^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} 4^{\frac {1}{3}} \tan \left (x \right )}{3}+1\right )}{3}\right )}{36}\) | \(80\) |
| default | \(-\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}} \ln \left (\tan \left (x \right )-\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}}}{4}\right )}{36}+\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}} \ln \left (\tan \left (x \right )^{2}+\frac {3^{\frac {1}{3}} 4^{\frac {2}{3}} \tan \left (x \right )}{4}+\frac {3^{\frac {2}{3}} 4^{\frac {1}{3}}}{4}\right )}{72}+\frac {3^{\frac {5}{6}} 4^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} 4^{\frac {1}{3}} \tan \left (x \right )}{3}+1\right )}{3}\right )}{36}\) | \(80\) |
4*sum(_R*ln(exp(2*I*x)+(41472/25-31104/25*I)*_R^2+(864/25+1152/25*I)*_R-7/ 25+24/25*I),_R=RootOf(62208*_Z^3+1))
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (131) = 262\).
Time = 0.35 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.51 \[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=-\frac {1}{36} \cdot 36^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {36^{\frac {1}{6}} {\left (28 \, {\left (36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} - 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cos \left (x\right )^{6} - 4 \, {\left (14 \cdot 36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} + 36 \cdot 36^{\frac {1}{3}} \sqrt {3} - 63 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cos \left (x\right )^{4} + {\left (37 \cdot 36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} + 144 \cdot 36^{\frac {1}{3}} \sqrt {3} + 144 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cos \left (x\right )^{2} - 6 \, {\left (16 \, {\left (36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} - 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cos \left (x\right )^{5} - {\left (24 \cdot 36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} - 7 \cdot 36^{\frac {1}{3}} \sqrt {3} - 72 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cos \left (x\right )^{3} + 4 \, {\left (36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} - 4 \cdot 36^{\frac {1}{3}} \sqrt {3} + 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) - 18 \cdot 36^{\frac {1}{3}} \sqrt {3} - 144 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )}}{108 \, {\left (48 \, \cos \left (x\right )^{6} - 72 \, \cos \left (x\right )^{4} + 18 \, \cos \left (x\right )^{2} + 14 \, {\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right ) + 3\right )}}\right ) - \frac {1}{432} \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-3 \, {\left (2 \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 8 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 25\right )} \cos \left (x\right )^{4} + 3 \, {\left (3 \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 4 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 32\right )} \cos \left (x\right )^{2} - 2 \, {\left ({\left (4 \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 9 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}}\right )} \cos \left (x\right )^{3} - 4 \, {\left (36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 9\right )} \cos \left (x\right )\right )} \sin \left (x\right ) - 12 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 48\right ) + \frac {1}{216} \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (3 \, {\left (2 \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 8 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 7\right )} \cos \left (x\right )^{2} + 2 \, {\left (4 \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 9 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 36\right )} \cos \left (x\right ) \sin \left (x\right ) - 3 \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 12 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 48\right ) \]
-1/36*36^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(-1/108*36^(1/6)*(28*(36^(2/3)*sqr t(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*3 6^(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))*co s(x)^5 - (24*36^(2/3)*sqrt(3)*(-1)^(2/3) - 7*36^(1/3)*sqrt(3) - 72*sqrt(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 - c os(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)
\[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=- \int \frac {\sec ^{2}{\left (x \right )}}{4 \tan ^{3}{\left (x \right )} - 3}\, dx \]
Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.51 \[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=\frac {1}{36} \cdot 4^{\frac {2}{3}} 3^{\frac {5}{6}} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} 3^{\frac {1}{6}} {\left (2 \cdot 4^{\frac {2}{3}} \tan \left (x\right ) + 4^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{72} \cdot 4^{\frac {2}{3}} 3^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} \tan \left (x\right )^{2} + 4^{\frac {1}{3}} 3^{\frac {1}{3}} \tan \left (x\right ) + 3^{\frac {2}{3}}\right ) - \frac {1}{36} \cdot 4^{\frac {2}{3}} 3^{\frac {1}{3}} \log \left (\frac {1}{4} \cdot 4^{\frac {2}{3}} {\left (4^{\frac {1}{3}} \tan \left (x\right ) - 3^{\frac {1}{3}}\right )}\right ) \]
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)))
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.35 \[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=\frac {1}{9} \, \sqrt {3} \left (\frac {3}{4}\right )^{\frac {1}{3}} \arctan \left (\frac {4}{9} \, \sqrt {3} \left (\frac {3}{4}\right )^{\frac {2}{3}} {\left (\left (\frac {3}{4}\right )^{\frac {1}{3}} + 2 \, \tan \left (x\right )\right )}\right ) + \frac {1}{36} \cdot 6^{\frac {1}{3}} \log \left (\tan \left (x\right )^{2} + \left (\frac {3}{4}\right )^{\frac {1}{3}} \tan \left (x\right ) + \left (\frac {3}{4}\right )^{\frac {2}{3}}\right ) - \frac {1}{9} \, \left (\frac {3}{4}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {3}{4}\right )^{\frac {1}{3}} + \tan \left (x\right ) \right |}\right ) \]
1/9*sqrt(3)*(3/4)^(1/3)*arctan(4/9*sqrt(3)*(3/4)^(2/3)*((3/4)^(1/3) + 2*ta n(x))) + 1/36*6^(1/3)*log(tan(x)^2 + (3/4)^(1/3)*tan(x) + (3/4)^(2/3)) - 1 /9*(3/4)^(1/3)*log(abs(-(3/4)^(1/3) + tan(x)))
Time = 27.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.43 \[ \int \frac {\sec ^2(x)}{3-4 \tan ^3(x)} \, dx=-\frac {6^{1/3}\,\ln \left (\mathrm {tan}\left (x\right )-\frac {6^{1/3}}{2}\right )}{18}-\frac {6^{1/3}\,\ln \left (\mathrm {tan}\left (x\right )-\frac {6^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{36}+\frac {6^{1/3}\,\ln \left (\mathrm {tan}\left (x\right )+\frac {6^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{36} \]