3.457 \(\int \frac {\tan ^5(e+f x)}{a+b \sec ^3(e+f x)} \, dx\)

Optimal. Leaf size=219 \[ \frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} b^{4/3} f}-\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{4/3} f}-\frac {\left (a^{2/3}+2 b^{2/3}\right ) \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3} f}-\frac {\log \left (a \cos ^3(e+f x)+b\right )}{3 a f}+\frac {\sec (e+f x)}{b f} \]

[Out]

-1/3*(a^(2/3)-2*b^(2/3))*ln(b^(1/3)+a^(1/3)*cos(f*x+e))/a^(1/3)/b^(4/3)/f+1/6*(a^(2/3)-2*b^(2/3))*ln(b^(2/3)-a
^(1/3)*b^(1/3)*cos(f*x+e)+a^(2/3)*cos(f*x+e)^2)/a^(1/3)/b^(4/3)/f-1/3*ln(b+a*cos(f*x+e)^3)/a/f+sec(f*x+e)/b/f-
1/3*(a^(2/3)+2*b^(2/3))*arctan(1/3*(b^(1/3)-2*a^(1/3)*cos(f*x+e))/b^(1/3)*3^(1/2))/a^(1/3)/b^(4/3)/f*3^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4138, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (a^{2/3} \cos ^2(e+f x)-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+b^{2/3}\right )}{6 \sqrt [3]{a} b^{4/3} f}-\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (\sqrt [3]{a} \cos (e+f x)+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{4/3} f}-\frac {\left (a^{2/3}+2 b^{2/3}\right ) \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \cos (e+f x)}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3} f}-\frac {\log \left (a \cos ^3(e+f x)+b\right )}{3 a f}+\frac {\sec (e+f x)}{b f} \]

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^3),x]

[Out]

-(((a^(2/3) + 2*b^(2/3))*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*b^(4/3
)*f)) - ((a^(2/3) - 2*b^(2/3))*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*b^(4/3)*f) + ((a^(2/3) - 2*b^(2
/3))*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*b^(4/3)*f) - Log[b + a*C
os[e + f*x]^3]/(3*a*f) + Sec[e + f*x]/(b*f)

Rule 31

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

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 260

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

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 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]

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 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 4138

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*
(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\tan ^5(e+f x)}{a+b \sec ^3(e+f x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2 \left (b+a x^3\right )} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b x^2}+\frac {-2 b-a x+b x^2}{b \left (b+a x^3\right )}\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=\frac {\sec (e+f x)}{b f}-\frac {\operatorname {Subst}\left (\int \frac {-2 b-a x+b x^2}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{b f}\\ &=\frac {\sec (e+f x)}{b f}-\frac {\operatorname {Subst}\left (\int \frac {x^2}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{f}-\frac {\operatorname {Subst}\left (\int \frac {-2 b-a x}{b+a x^3} \, dx,x,\cos (e+f x)\right )}{b f}\\ &=-\frac {\log \left (b+a \cos ^3(e+f x)\right )}{3 a f}+\frac {\sec (e+f x)}{b f}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{b} \left (-a \sqrt [3]{b}-4 \sqrt [3]{a} b\right )+\sqrt [3]{a} \left (-a \sqrt [3]{b}+2 \sqrt [3]{a} b\right ) x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\cos (e+f x)\right )}{3 \sqrt [3]{a} b^{5/3} f}-\frac {\left (a^{2/3}-2 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,\cos (e+f x)\right )}{3 b^{4/3} f}\\ &=-\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} b^{4/3} f}-\frac {\log \left (b+a \cos ^3(e+f x)\right )}{3 a f}+\frac {\sec (e+f x)}{b f}+\frac {\left (a^{2/3}-2 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\cos (e+f x)\right )}{6 \sqrt [3]{a} b^{4/3} f}+\frac {\left (a^{2/3}+2 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,\cos (e+f x)\right )}{2 b f}\\ &=-\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} b^{4/3} f}+\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+a^{2/3} \cos ^2(e+f x)\right )}{6 \sqrt [3]{a} b^{4/3} f}-\frac {\log \left (b+a \cos ^3(e+f x)\right )}{3 a f}+\frac {\sec (e+f x)}{b f}+\frac {\left (a^{2/3}+2 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} \cos (e+f x)}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} b^{4/3} f}\\ &=-\frac {\left (a^{2/3}+2 b^{2/3}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} \cos (e+f x)}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} b^{4/3} f}-\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (\sqrt [3]{b}+\sqrt [3]{a} \cos (e+f x)\right )}{3 \sqrt [3]{a} b^{4/3} f}+\frac {\left (a^{2/3}-2 b^{2/3}\right ) \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \cos (e+f x)+a^{2/3} \cos ^2(e+f x)\right )}{6 \sqrt [3]{a} b^{4/3} f}-\frac {\log \left (b+a \cos ^3(e+f x)\right )}{3 a f}+\frac {\sec (e+f x)}{b f}\\ \end {align*}

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Mathematica [C]  time = 0.36, size = 251, normalized size = 1.15 \[ \frac {-\text {RootSum}\left [\text {$\#$1}^3 a-\text {$\#$1}^3 b-6 \text {$\#$1}^2 a+12 \text {$\#$1} a-8 a\& ,\frac {\text {$\#$1}^2 a b \log \left (-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )-\text {$\#$1}^2 b^2 \log \left (-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )-4 a^2 \log \left (-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )+2 \text {$\#$1} a^2 \log \left (-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )+4 a b \log \left (-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )-8 \text {$\#$1} a b \log \left (-\text {$\#$1}+\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )}{\text {$\#$1}^2 a-\text {$\#$1}^2 b-4 \text {$\#$1} a+4 a}\& \right ]+3 a \sec (e+f x)+3 b \log \left (\sec ^2\left (\frac {1}{2} (e+f x)\right )\right )}{3 a b f} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^5/(a + b*Sec[e + f*x]^3),x]

[Out]

(3*b*Log[Sec[(e + f*x)/2]^2] - RootSum[-8*a + 12*a*#1 - 6*a*#1^2 + a*#1^3 - b*#1^3 & , (-4*a^2*Log[1 - #1 + Ta
n[(e + f*x)/2]^2] + 4*a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2] + 2*a^2*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 - 8*a*b
*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1 + a*b*Log[1 - #1 + Tan[(e + f*x)/2]^2]*#1^2 - b^2*Log[1 - #1 + Tan[(e + f
*x)/2]^2]*#1^2)/(4*a - 4*a*#1 + a*#1^2 - b*#1^2) & ] + 3*a*Sec[e + f*x])/(3*a*b*f)

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fricas [C]  time = 1.67, size = 4427, normalized size = 20.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^3),x, algorithm="fricas")

[Out]

-1/36*(2*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(
a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(
3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a
^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b*f*cos(f*x + e)*log(1/36*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*
a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3)
- 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)
/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))
^2*a^2*b^3*f^2 - ((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 +
8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*
(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a
^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^3*f + 4*a^2*b + 5*b^3 + (a^3 + 8*a*b^2)*cos(f*x + e)
) - (((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^
4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) +
1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b
^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b*f*cos(f*x + e) + 3*sqrt(1/3)*a*b*f*sqrt(-(((-I*sqrt(3) + 1)*(1/(
a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)
/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54
*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1
/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^
3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4
*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b
^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^
2))*cos(f*x + e) - 18*b*cos(f*x + e))*log(1/36*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(
-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 +
b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2
+ b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^3*f^2 - ((-I*sqrt(
3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(
2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3
*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*
b^4*f^3))^(1/3) + 6/(a*f))*a*b^3*f + 4*a^2*b + 5*b^3 - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2
+ b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) -
1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a
*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a^
2*b^3*f^2 + 18*a*b^3*f)*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3)
+ 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^
3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*
f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a
^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/
(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*
(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/
3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2)) - 2*(a^3 + 8*a*b^2)*cos(f*x + e)) - (((-I*sqrt(3) + 1
)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2
+ b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3)
+ 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^
3))^(1/3) + 6/(a*f))*a*b*f*cos(f*x + e) - 3*sqrt(1/3)*a*b*f*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b
^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54
*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4
*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*
b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^
2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*s
qrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 -
2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2))*cos(f*x + e) - 18
*b*cos(f*x + e))*log(-1/36*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/
54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^
(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3)
- 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^3*f^2 + ((-I*sqrt(3) + 1)*(1/(a^2*f^2)
- (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^
2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 +
8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/
(a*f))*a*b^3*f - 4*a^2*b - 5*b^3 - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2
))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^
2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*
a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a^2*b^3*f^2 + 18*a*b^3
*f)*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)
/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqr
t(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2
*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b
^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54
*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4
*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*
f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2)) + 2*(a^3 + 8*a*b^2)*cos(f*x + e)) - 36*a)/(a*b*f*cos(f*x + e))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (f x + e\right )^{5}}{b \sec \left (f x + e\right )^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^3),x, algorithm="giac")

[Out]

integrate(tan(f*x + e)^5/(b*sec(f*x + e)^3 + a), x)

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maple [A]  time = 0.66, size = 274, normalized size = 1.25 \[ \frac {2 \ln \left (\cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 f a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (\cos ^{2}\left (f x +e \right )-\left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{3 f a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \cos \left (f x +e \right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 f a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (\cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 f b \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (\cos ^{2}\left (f x +e \right )-\left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x +e \right )+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 f b \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \cos \left (f x +e \right )}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 f b \left (\frac {b}{a}\right )^{\frac {1}{3}}}-\frac {\ln \left (b +a \left (\cos ^{3}\left (f x +e \right )\right )\right )}{3 a f}+\frac {1}{f b \cos \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^5/(a+b*sec(f*x+e)^3),x)

[Out]

2/3/f/a/(1/a*b)^(2/3)*ln(cos(f*x+e)+(1/a*b)^(1/3))-1/3/f/a/(1/a*b)^(2/3)*ln(cos(f*x+e)^2-(1/a*b)^(1/3)*cos(f*x
+e)+(1/a*b)^(2/3))+2/3/f/a/(1/a*b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/a*b)^(1/3)*cos(f*x+e)-1))-1/3/f/b/(1
/a*b)^(1/3)*ln(cos(f*x+e)+(1/a*b)^(1/3))+1/6/f/b/(1/a*b)^(1/3)*ln(cos(f*x+e)^2-(1/a*b)^(1/3)*cos(f*x+e)+(1/a*b
)^(2/3))+1/3/f/b*3^(1/2)/(1/a*b)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/a*b)^(1/3)*cos(f*x+e)-1))-1/3*ln(b+a*cos(f*x+e
)^3)/a/f+1/f/b/cos(f*x+e)

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maxima [A]  time = 0.44, size = 218, normalized size = 1.00 \[ \frac {\frac {2 \, \sqrt {3} {\left (2 \, a b {\left (3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {b}{a}\right )} + 3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}} + 2 \, b^{2}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {b}{a}\right )^{\frac {1}{3}} - 2 \, \cos \left (f x + e\right )\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a b^{2}} - \frac {3 \, {\left (2 \, b {\left (\left (\frac {b}{a}\right )^{\frac {2}{3}} + 1\right )} - a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \log \left (\cos \left (f x + e\right )^{2} - \left (\frac {b}{a}\right )^{\frac {1}{3}} \cos \left (f x + e\right ) + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{a b \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {6 \, {\left (b {\left (\left (\frac {b}{a}\right )^{\frac {2}{3}} - 2\right )} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {b}{a}\right )^{\frac {1}{3}} + \cos \left (f x + e\right )\right )}{a b \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {18}{b \cos \left (f x + e\right )}}{18 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^3),x, algorithm="maxima")

[Out]

1/18*(2*sqrt(3)*(2*a*b*(3*(b/a)^(1/3) - b/a) + 3*a^2*(b/a)^(2/3) + 2*b^2)*arctan(-1/3*sqrt(3)*((b/a)^(1/3) - 2
*cos(f*x + e))/(b/a)^(1/3))/(a*b^2) - 3*(2*b*((b/a)^(2/3) + 1) - a*(b/a)^(1/3))*log(cos(f*x + e)^2 - (b/a)^(1/
3)*cos(f*x + e) + (b/a)^(2/3))/(a*b*(b/a)^(2/3)) - 6*(b*((b/a)^(2/3) - 2) + a*(b/a)^(1/3))*log((b/a)^(1/3) + c
os(f*x + e))/(a*b*(b/a)^(2/3)) + 18/(b*cos(f*x + e)))/f

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mupad [B]  time = 7.26, size = 7402, normalized size = 33.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^5/(a + b/cos(e + f*x)^3),x)

[Out]

symsum(log(-(262144*(148*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a
^4, z, k)*b^17 - 1920*a*b^15 - 156*b^16*cos(e + f*x) + 300*b^16 + 16*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*
a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*b^18 + 5232*a^2*b^14 - 7872*a^3*b^13 + 7080*a^4*b^12 -
3840*a^5*b^11 + 1200*a^6*b^10 - 192*a^7*b^9 + 12*a^8*b^8 - 5916*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^
4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^2*b^15 + 4820*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b
^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^3*b^14 + 5933*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*
b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^4*b^13 - 12882*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*
a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^5*b^12 + 8891*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9
*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^6*b^11 - 2872*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 +
9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^7*b^10 + 447*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 +
9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^8*b^9 - 26*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*
a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^9*b^8 + root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4
*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^10*b^7 + 1396*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^
4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a*b^17 + 192*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^
4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a*b^18 - 3936*a^2*b^14*cos(e + f*x) + 7152*a^3*b^13*cos(e
+ f*x) - 7800*a^4*b^12*cos(e + f*x) + 5136*a^5*b^11*cos(e + f*x) - 1920*a^6*b^10*cos(e + f*x) + 336*a^7*b^9*co
s(e + f*x) - 12*a^8*b^8*cos(e + f*x) - 768*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2
*a^2*b^2 + b^4 + a^4, z, k)^2*a^2*b^16 + 4772*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z
- 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^3*b^15 - 13924*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^
2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^4*b^14 + 6927*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3
*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^5*b^13 + 5747*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*
a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^6*b^12 - 5944*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z +
18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^7*b^11 + 2004*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z
+ 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^8*b^10 - 239*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4
*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^9*b^9 + 13*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4
*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^10*b^8 + 4296*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*
b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^2*b^17 - 11856*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 +
9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^3*b^16 + 16956*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^
2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^4*b^15 - 17916*root(27*a^3*b^4*z^3 + 27*a^2*b^
4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^5*b^14 + 11175*root(27*a^3*b^4*z^3 + 27*a^
2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^6*b^13 - 4608*root(27*a^3*b^4*z^3 + 27
*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^7*b^12 + 2118*root(27*a^3*b^4*z^3 +
27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^8*b^11 - 372*root(27*a^3*b^4*z^3
+ 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^9*b^10 + 15*root(27*a^3*b^4*z^
3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^10*b^9 + 864*root(27*a^3*b^4*
z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^2*b^18 + 3240*root(27*a^3*b
^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^3*b^17 - 12996*root(27*a
^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^4*b^16 + 4140*root(2
7*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^5*b^15 + 16668*ro
ot(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^6*b^14 - 1601
1*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^7*b^13 +
4959*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^8*b^12
- 873*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^9*b^
11 + 9*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^10*b
^10 + 1728*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^5*a^
3*b^18 - 5724*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^5
*a^4*b^17 + 6912*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k
)^5*a^5*b^16 - 3024*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z
, k)^5*a^6*b^15 - 1080*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4
, z, k)^5*a^7*b^14 + 1836*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 +
a^4, z, k)^5*a^8*b^13 - 648*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4
+ a^4, z, k)^5*a^9*b^12 + 1296*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b
^4 + a^4, z, k)^6*a^4*b^18 - 7452*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2
+ b^4 + a^4, z, k)^6*a^5*b^17 + 14904*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*
b^2 + b^4 + a^4, z, k)^6*a^6*b^16 - 12960*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*
a^2*b^2 + b^4 + a^4, z, k)^6*a^7*b^15 + 4536*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z -
2*a^2*b^2 + b^4 + a^4, z, k)^6*a^8*b^14 - 324*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z
- 2*a^2*b^2 + b^4 + a^4, z, k)^6*a^9*b^13 + 1456*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^
2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a*b^16 - 52*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z
- 2*a^2*b^2 + b^4 + a^4, z, k)*b^17*cos(e + f*x) + 1200*a*b^15*cos(e + f*x) - 880*root(27*a^3*b^4*z^3 + 27*a^
2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a*b^16*cos(e + f*x) + 4764*root(27*a^3*b^4
*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^2*b^15*cos(e + f*x) - 6932*r
oot(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^3*b^14*cos(e +
f*x) - 1109*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^
4*b^13*cos(e + f*x) + 12234*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4
+ a^4, z, k)*a^5*b^12*cos(e + f*x) - 12299*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2
*a^2*b^2 + b^4 + a^4, z, k)*a^6*b^11*cos(e + f*x) + 5032*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18
*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^7*b^10*cos(e + f*x) - 807*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9
*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^8*b^9*cos(e + f*x) + 50*root(27*a^3*b^4*z^3 + 27*a^2*
b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^9*b^8*cos(e + f*x) - root(27*a^3*b^4*z^3 +
27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)*a^10*b^7*cos(e + f*x) - 548*root(27*
a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a*b^17*cos(e + f*x) +
160*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^2*b^16
*cos(e + f*x) - 1380*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4,
z, k)^2*a^3*b^15*cos(e + f*x) + 12140*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*
b^2 + b^4 + a^4, z, k)^2*a^4*b^14*cos(e + f*x) - 14767*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a
^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^5*b^13*cos(e + f*x) - 1659*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 +
9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^6*b^12*cos(e + f*x) + 9272*root(27*a^3*b^4*z^3 + 2
7*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^7*b^11*cos(e + f*x) - 3691*root(27
*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^8*b^10*cos(e + f*x
) + 510*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^2*a^9*b
^9*cos(e + f*x) - 38*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4,
z, k)^2*a^10*b^8*cos(e + f*x) + root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 +
b^4 + a^4, z, k)^2*a^11*b^7*cos(e + f*x) - 1992*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*
z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^2*b^17*cos(e + f*x) + 8112*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4
*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^3*b^16*cos(e + f*x) - 18300*root(27*a^3*b^4*z^3 + 27*a^2*
b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^4*b^15*cos(e + f*x) + 19788*root(27*a^3*
b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^5*b^14*cos(e + f*x) - 1
0095*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^6*b^13
*cos(e + f*x) + 6000*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4,
z, k)^3*a^7*b^12*cos(e + f*x) - 4134*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b
^2 + b^4 + a^4, z, k)^3*a^8*b^11*cos(e + f*x) + 660*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*
b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^9*b^10*cos(e + f*x) - 39*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b
^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^3*a^10*b^9*cos(e + f*x) - 2376*root(27*a^3*b^4*z^3 + 27*a^2
*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^3*b^17*cos(e + f*x) + 11124*root(27*a^3
*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^4*b^16*cos(e + f*x) -
10044*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^5*b^1
5*cos(e + f*x) - 10260*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4
, z, k)^4*a^6*b^14*cos(e + f*x) + 19899*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^
2*b^2 + b^4 + a^4, z, k)^4*a^7*b^13*cos(e + f*x) - 10287*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18
*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^8*b^12*cos(e + f*x) + 2025*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2
+ 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^9*b^11*cos(e + f*x) - 81*root(27*a^3*b^4*z^3 + 2
7*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^4*a^10*b^10*cos(e + f*x) + 1404*root(2
7*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^5*a^4*b^17*cos(e + f*
x) - 6048*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^5*a^5
*b^16*cos(e + f*x) + 8208*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 +
a^4, z, k)^5*a^6*b^15*cos(e + f*x) - 2376*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*
a^2*b^2 + b^4 + a^4, z, k)^5*a^7*b^14*cos(e + f*x) - 2700*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 1
8*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^5*a^8*b^13*cos(e + f*x) + 1512*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2
+ 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^5*a^9*b^12*cos(e + f*x) + 3564*root(27*a^3*b^4*z^3
+ 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^6*a^5*b^17*cos(e + f*x) - 12312*roo
t(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^6*a^6*b^16*cos(e +
f*x) + 15552*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k)^6
*a^7*b^15*cos(e + f*x) - 8424*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^
4 + a^4, z, k)^6*a^8*b^14*cos(e + f*x) + 1620*root(27*a^3*b^4*z^3 + 27*a^2*b^4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z
- 2*a^2*b^2 + b^4 + a^4, z, k)^6*a^9*b^13*cos(e + f*x)))/cos(e/2 + (f*x)/2)^2)*root(27*a^3*b^4*z^3 + 27*a^2*b^
4*z^2 + 9*a*b^4*z + 18*a^3*b^2*z - 2*a^2*b^2 + b^4 + a^4, z, k), k, 1, 3)/f + log(1/cos(e/2 + (f*x)/2)^2)/(a*f
) + 1/(b*f*cos(e + f*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{5}{\left (e + f x \right )}}{a + b \sec ^{3}{\left (e + f x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**5/(a+b*sec(f*x+e)**3),x)

[Out]

Integral(tan(e + f*x)**5/(a + b*sec(e + f*x)**3), x)

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