∫ sec 2 (x) tan(x)(1+ 3∘________1−8 tan 2 (x) ) ___________________________________________

Optimal. Leaf size=20 \[ -\frac {3}{32} \left (\sqrt [3]{1-8 \tan ^2(x)}+1\right )^2 \]

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Rubi [A] time = 0.23, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4342, 6686} \[ -\frac {3}{32} \left (\sqrt [3]{1-8 \tan ^2(x)}+1\right )^2 \]

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Tan[c*(a + b*x)], x]}, Dist[d/

(b*c), Subst[Int[SubstFor[1, Tan[c*(a + b*x)]/d, u, x], x], x, Tan[c*(a + b*x)]/d], x] /; FunctionOfQ[Tan[c*(a

+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Sec] || EqQ[F, sec])

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /; !F

\begin {align*} \int \frac {\sec ^2(x) \tan (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {x \left (1+\sqrt [3]{1-8 x^2}\right )}{\left (1-8 x^2\right )^{2/3}} \, dx,x,\tan (x)\right )\\ &=-\frac {3}{32} \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )^2\\ \end {align*}

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Mathematica [B] time = 0.21, size = 42, normalized size = 2.10 \[ -\frac {3 (9 \cos (2 x)-7) \left (\sqrt [3]{1-8 \tan ^2(x)}+2\right ) \sec ^2(x)}{64 \left (1-8 \tan ^2(x)\right )^{2/3}} \]

Integrate[(Sec[x]^2*Tan[x]*(1 + (1 - 8*Tan[x]^2)^(1/3)))/(1 - 8*Tan[x]^2)^(2/3),x]

(-3*(-7 + 9*Cos[2*x])*Sec[x]^2*(2 + (1 - 8*Tan[x]^2)^(1/3)))/(64*(1 - 8*Tan[x]^2)^(2/3))

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^2(x) \tan (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx \]

IntegrateAlgebraic[(Sec[x]^2*Tan[x]*(1 + (1 - 8*Tan[x]^2)^(1/3)))/(1 - 8*Tan[x]^2)^(2/3),x]

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fricas [B] time = 1.56, size = 35, normalized size = 1.75 \[ -\frac {3}{32} \, \left (\frac {9 \, \cos \relax (x)^{2} - 8}{\cos \relax (x)^{2}}\right )^{\frac {2}{3}} - \frac {3}{16} \, \left (\frac {9 \, \cos \relax (x)^{2} - 8}{\cos \relax (x)^{2}}\right )^{\frac {1}{3}} \]

integrate(tan(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x, algorithm="fricas")

-3/32*((9*cos(x)^2 - 8)/cos(x)^2)^(2/3) - 3/16*((9*cos(x)^2 - 8)/cos(x)^2)^(1/3)

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giac [A] time = 0.88, size = 25, normalized size = 1.25 \[ -\frac {3}{32} \, {\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} - \frac {3}{16} \, {\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} \]

integrate(tan(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x, algorithm="giac")

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method	result	size

derivativedivides	\(-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {1}{3}}}{16}-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}{32}\)	\(26\)
default	\(-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {1}{3}}}{16}-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}{32}\)	\(26\)

int(tan(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x,method=_RETURNVERBOSE)

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maxima [B] time = 0.69, size = 86, normalized size = 4.30 \[ -\frac {3 \, {\left (\frac {{\left (9 \, \sin \relax (x)^{2} - 1\right )} {\left (3 \, \sin \relax (x) - 1\right )}^{\frac {1}{3}} {\left (\sin \relax (x) + 1\right )}^{\frac {1}{3}} {\left (\sin \relax (x) - 1\right )}^{\frac {1}{3}}}{{\left (3 \, \sin \relax (x) + 1\right )}^{\frac {1}{3}}} + \frac {2 \, {\left (9 \, \sin \relax (x)^{2} - 1\right )} {\left (\sin \relax (x) + 1\right )}^{\frac {2}{3}} {\left (\sin \relax (x) - 1\right )}^{\frac {2}{3}}}{{\left (3 \, \sin \relax (x) + 1\right )}^{\frac {2}{3}}}\right )}}{32 \, {\left (\sin \relax (x)^{2} - 1\right )} {\left (3 \, \sin \relax (x) - 1\right )}^{\frac {2}{3}}} \]

integrate(tan(x)*(1+(1-8*tan(x)^2)^(1/3))/cos(x)^2/(1-8*tan(x)^2)^(2/3),x, algorithm="maxima")

-3/32*((9*sin(x)^2 - 1)*(3*sin(x) - 1)^(1/3)*(sin(x) + 1)^(1/3)*(sin(x) - 1)^(1/3)/(3*sin(x) + 1)^(1/3) + 2*(9

*sin(x)^2 - 1)*(sin(x) + 1)^(2/3)*(sin(x) - 1)^(2/3)/(3*sin(x) + 1)^(2/3))/((sin(x)^2 - 1)*(3*sin(x) - 1)^(2/3

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mupad [B] time = 0.62, size = 43, normalized size = 2.15 \[ -\frac {3\,\left ({\left (18\,{\cos \relax (x)}^2-16\right )}^{1/3}+2\,{\left (2\,{\cos \relax (x)}^2\right )}^{1/3}\right )\,{\left (18\,{\cos \relax (x)}^2-16\right )}^{1/3}}{32\,{\left (2\,{\cos \relax (x)}^2\right )}^{2/3}} \]

-(3*((18*cos(x)^2 - 16)^(1/3) + 2*(2*cos(x)^2)^(1/3))*(18*cos(x)^2 - 16)^(1/3))/(32*(2*cos(x)^2)^(2/3))

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sqrt [3]{1 - 8 \tan ^{2}{\relax (x )}} + 1\right ) \tan {\relax (x )}}{\left (1 - 8 \tan ^{2}{\relax (x )}\right )^{\frac {2}{3}} \cos ^{2}{\relax (x )}}\, dx \]

Integral(((1 - 8*tan(x)**2)**(1/3) + 1)*tan(x)/((1 - 8*tan(x)**2)**(2/3)*cos(x)**2), x)

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3.451 \(\int \frac {\sec ^2(x) \tan (x) (1+\sqrt [3]{1-8 \tan ^2(x)})}{(1-8 \tan ^2(x))^{2/3}} \, dx\)