\(\int F^{c (a+b x)} \sec ^3(d+e x) \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 141 \[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=-\frac {e^{i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right ) (i e-b c \log (F))}{e^2}-\frac {b c F^{c (a+b x)} \log (F) \sec (d+e x)}{2 e^2}+\frac {F^{c (a+b x)} \sec (d+e x) \tan (d+e x)}{2 e} \]

[Out]

-exp(I*(e*x+d))*F^(c*(b*x+a))*hypergeom([1, 1/2*(e-I*b*c*ln(F))/e],[3/2-1/2*I*b*c*ln(F)/e],-exp(2*I*(e*x+d)))*
(I*e-b*c*ln(F))/e^2-1/2*b*c*F^(c*(b*x+a))*ln(F)*sec(e*x+d)/e^2+1/2*F^(c*(b*x+a))*sec(e*x+d)*tan(e*x+d)/e

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4533, 4536} \[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=-\frac {e^{i (d+e x)} F^{c (a+b x)} (-b c \log (F)+i e) \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right )}{e^2}-\frac {b c \log (F) \sec (d+e x) F^{c (a+b x)}}{2 e^2}+\frac {\tan (d+e x) \sec (d+e x) F^{c (a+b x)}}{2 e} \]

[In]

Int[F^(c*(a + b*x))*Sec[d + e*x]^3,x]

[Out]

-((E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), (3 - (I*b*c*Log[F])/e)/2, -E
^((2*I)*(d + e*x))]*(I*e - b*c*Log[F]))/e^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sec[d + e*x])/(2*e^2) + (F^(c*(a +
b*x))*Sec[d + e*x]*Tan[d + e*x])/(2*e)

Rule 4533

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Sec[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2))), x] + (Dist[(e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)
*(n - 2)), Int[F^(c*(a + b*x))*Sec[d + e*x]^(n - 2), x], x] + Simp[F^(c*(a + b*x))*Sec[d + e*x]^(n - 1)*(Sin[d
+ e*x]/(e*(n - 1))), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n
, 1] && NeQ[n, 2]

Rule 4536

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n*E^(I*n*(d + e*x))*(
F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(Log[F]
/(2*e)), -E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {b c F^{c (a+b x)} \log (F) \sec (d+e x)}{2 e^2}+\frac {F^{c (a+b x)} \sec (d+e x) \tan (d+e x)}{2 e}+\frac {1}{2} \left (1+\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \sec (d+e x) \, dx \\ & = -\frac {e^{i (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right ),-e^{2 i (d+e x)}\right ) (i e-b c \log (F))}{e^2}-\frac {b c F^{c (a+b x)} \log (F) \sec (d+e x)}{2 e^2}+\frac {F^{c (a+b x)} \sec (d+e x) \tan (d+e x)}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (2 e^{i (d+e x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e-i b c \log (F)}{2 e},\frac {3}{2}-\frac {i b c \log (F)}{2 e},-e^{2 i (d+e x)}\right ) (-i e+b c \log (F))+\sec (d+e x) (-b c \log (F)+e \tan (d+e x))\right )}{2 e^2} \]

[In]

Integrate[F^(c*(a + b*x))*Sec[d + e*x]^3,x]

[Out]

(F^(c*(a + b*x))*(2*E^(I*(d + e*x))*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), 3/2 - ((I/2)*b*c*Log[F])/e,
-E^((2*I)*(d + e*x))]*((-I)*e + b*c*Log[F]) + Sec[d + e*x]*(-(b*c*Log[F]) + e*Tan[d + e*x])))/(2*e^2)

Maple [F]

\[\int F^{c \left (x b +a \right )} \sec \left (e x +d \right )^{3}d x\]

[In]

int(F^(c*(b*x+a))*sec(e*x+d)^3,x)

[Out]

int(F^(c*(b*x+a))*sec(e*x+d)^3,x)

Fricas [F]

\[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \sec \left (e x + d\right )^{3} \,d x } \]

[In]

integrate(F^(c*(b*x+a))*sec(e*x+d)^3,x, algorithm="fricas")

[Out]

integral(F^(b*c*x + a*c)*sec(e*x + d)^3, x)

Sympy [F]

\[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=\int F^{c \left (a + b x\right )} \sec ^{3}{\left (d + e x \right )}\, dx \]

[In]

integrate(F**(c*(b*x+a))*sec(e*x+d)**3,x)

[Out]

Integral(F**(c*(a + b*x))*sec(d + e*x)**3, x)

Maxima [F]

\[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \sec \left (e x + d\right )^{3} \,d x } \]

[In]

integrate(F^(c*(b*x+a))*sec(e*x+d)^3,x, algorithm="maxima")

[Out]

8*(48*F^(b*c*x)*F^(a*c)*b*c*e^2*cos(e*x + d)*log(F) + 6*(F^(a*c)*b^2*c^2*e*log(F)^2 - 15*F^(a*c)*e^3)*F^(b*c*x
)*sin(e*x + d) + (48*F^(b*c*x)*F^(a*c)*b*c*e^2*cos(e*x + d)*log(F) + (F^(a*c)*b^3*c^3*log(F)^3 + 25*F^(a*c)*b*
c*e^2*log(F))*F^(b*c*x)*cos(3*e*x + 3*d) - 3*(F^(a*c)*b^2*c^2*e*log(F)^2 + 25*F^(a*c)*e^3)*F^(b*c*x)*sin(3*e*x
+ 3*d) + 6*(F^(a*c)*b^2*c^2*e*log(F)^2 - 15*F^(a*c)*e^3)*F^(b*c*x)*sin(e*x + d))*cos(6*e*x + 6*d) + 3*(48*F^(
b*c*x)*F^(a*c)*b*c*e^2*cos(e*x + d)*log(F) + (F^(a*c)*b^3*c^3*log(F)^3 + 25*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*
cos(3*e*x + 3*d) - 3*(F^(a*c)*b^2*c^2*e*log(F)^2 + 25*F^(a*c)*e^3)*F^(b*c*x)*sin(3*e*x + 3*d) + 6*(F^(a*c)*b^2
*c^2*e*log(F)^2 - 15*F^(a*c)*e^3)*F^(b*c*x)*sin(e*x + d))*cos(4*e*x + 4*d) + (3*(F^(a*c)*b^3*c^3*log(F)^3 + 25
*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*cos(2*e*x + 2*d) + 9*(F^(a*c)*b^2*c^2*e*log(F)^2 + 25*F^(a*c)*e^3)*F^(b*c*x
)*sin(2*e*x + 2*d) + (F^(a*c)*b^3*c^3*log(F)^3 + 25*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x))*cos(3*e*x + 3*d) + 18*(
8*F^(b*c*x)*F^(a*c)*b*c*e^2*cos(e*x + d)*log(F) + (F^(a*c)*b^2*c^2*e*log(F)^2 - 15*F^(a*c)*e^3)*F^(b*c*x)*sin(
e*x + d))*cos(2*e*x + 2*d) - 6*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F
^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F
) - 225*F^(a*c)*e^6*sin(d) + (F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(
a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F)
- 225*F^(a*c)*e^6*sin(d))*cos(6*e*x + 6*d)^2 + 9*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(
F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*
c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*cos(4*e*x + 4*d)^2 + 9*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a
*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin
(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*cos(2*e*x + 2*d)^2 + (F^(a*c)*b^5*c^5*e*cos(
d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^
2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*sin(6*e*x + 6*d)^2 + 9*(F^
(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3
- 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*sin(4*
e*x + 4*d)^2 + 18*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^
3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*
c)*e^6*sin(d))*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 9*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*
log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c
)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*sin(2*e*x + 2*d)^2 + 2*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 -
F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2
*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d) + 3*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 -
F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2
*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*cos(4*e*x + 4*d) + 3*(F^(a*c)*b^5*c^5*e*
cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^
2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*cos(2*e*x + 2*d))*cos(
6*e*x + 6*d) + 6*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3
*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c
)*e^6*sin(d) + 3*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3
*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c
)*e^6*sin(d))*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) + 6*(F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*
log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*sin(d) + 225*F^(a*c
)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*cos(2*e*x + 2*d) + 6*((F^(a*c)*b^5*c^5*e*cos(d)*log(F)^5 - F
^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c^2*e^4*log(F)^2*
sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*sin(4*e*x + 4*d) + (F^(a*c)*b^5*c^5*e*cos
(d)*log(F)^5 - F^(a*c)*b^4*c^4*e^2*log(F)^4*sin(d) + 34*F^(a*c)*b^3*c^3*e^3*cos(d)*log(F)^3 - 34*F^(a*c)*b^2*c
^2*e^4*log(F)^2*sin(d) + 225*F^(a*c)*b*c*e^5*cos(d)*log(F) - 225*F^(a*c)*e^6*sin(d))*sin(2*e*x + 2*d))*sin(6*e
*x + 6*d))*integrate((8*F^(b*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x) + (8*
F^(b*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*cos(8*e*x + 8*d) + 4*(8*F^(b
*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*cos(6*e*x + 6*d) + 6*(8*F^(b*c*x
)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*cos(4*e*x + 4*d) + 4*(8*F^(b*c*x)*b*
c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*cos(2*e*x + 2*d) + (8*F^(b*c*x)*b*c*e*lo
g(F)*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*sin(8*e*x + 8*d) + 4*(8*F^(b*c*x)*b*c*e*log(F)
*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*sin(6*e*x + 6*d) + 6*(8*F^(b*c*x)*b*c*e*log(F)*sin
(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*sin(4*e*x + 4*d) + 4*(8*F^(b*c*x)*b*c*e*log(F)*sin(e*x
) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*sin(2*e*x + 2*d))/(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F
)^2 + 225*e^4 + (b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(8*e*x + 8*d)^2 + 16*(b^4*c^4*log(F)
^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(6*e*x + 6*d)^2 + 36*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 +
225*e^4)*cos(4*e*x + 4*d)^2 + 16*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d)^2 + (
b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(8*e*x + 8*d)^2 + 16*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e
^2*log(F)^2 + 225*e^4)*sin(6*e*x + 6*d)^2 + 36*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*
x + 4*d)^2 + 48*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 16*
(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(2*e*x + 2*d)^2 + 2*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e
^2*log(F)^2 + 225*e^4 + 4*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(6*e*x + 6*d) + 6*(b^4*c^4
*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d) + 4*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^
2 + 225*e^4)*cos(2*e*x + 2*d))*cos(8*e*x + 8*d) + 8*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4 + 6*
(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d) + 4*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2
*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 12*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225
*e^4 + 4*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) + 8*(b^4*c^
4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d) + 4*(2*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log
(F)^2 + 225*e^4)*sin(6*e*x + 6*d) + 3*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4*d)
+ 2*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(2*e*x + 2*d))*sin(8*e*x + 8*d) + 16*(3*(b^4*c^4
*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4*d) + 2*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^
2 + 225*e^4)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d)), x) + 6*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*
e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^
(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d) + (F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^
2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a
*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*cos(6*e*x + 6*d)^2 + 9*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d)
+ F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)
*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*cos(4*e*x + 4*d)^2 + 9*(F^(a*c)*b^5*c^
5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c
)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*cos(2*e*x + 2*d)^2
+ (F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*
sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*
sin(6*e*x + 6*d)^2 + 9*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b
^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*
F^(a*c)*e^6*cos(d))*sin(4*e*x + 4*d)^2 + 18*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*lo
g(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5
*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 9*(F^(a*c)*b^5*c^5*e*log(F)^5*sin
(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*co
s(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*sin(2*e*x + 2*d)^2 + 2*(F^(a*c)*b^
5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^
(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d) + 3*(F^(a*c)*b^
5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^
(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*cos(4*e*x + 4*
d) + 3*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F
)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(
d))*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 6*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log
(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*
log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d) + 3*(F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log
(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*
log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) + 6*(F^(a*c)*b^5*c^5*e*log(F)^5*sin
(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*co
s(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*cos(2*e*x + 2*d) + 6*((F^(a*c)*b^5
*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3*sin(d) + 34*F^(
a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))*sin(4*e*x + 4*d
) + (F^(a*c)*b^5*c^5*e*log(F)^5*sin(d) + F^(a*c)*b^4*c^4*e^2*cos(d)*log(F)^4 + 34*F^(a*c)*b^3*c^3*e^3*log(F)^3
*sin(d) + 34*F^(a*c)*b^2*c^2*e^4*cos(d)*log(F)^2 + 225*F^(a*c)*b*c*e^5*log(F)*sin(d) + 225*F^(a*c)*e^6*cos(d))
*sin(2*e*x + 2*d))*sin(6*e*x + 6*d))*integrate((8*F^(b*c*x)*b*c*e*log(F)*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2
)*F^(b*c*x)*cos(e*x) + (8*F^(b*c*x)*b*c*e*log(F)*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*co
s(8*e*x + 8*d) + 4*(8*F^(b*c*x)*b*c*e*log(F)*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*cos(6*
e*x + 6*d) + 6*(8*F^(b*c*x)*b*c*e*log(F)*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*cos(4*e*x
+ 4*d) + 4*(8*F^(b*c*x)*b*c*e*log(F)*sin(e*x) - (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*cos(e*x))*cos(2*e*x + 2*
d) - (8*F^(b*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*sin(8*e*x + 8*d) - 4
*(8*F^(b*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*sin(6*e*x + 6*d) - 6*(8*
F^(b*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*sin(4*e*x + 4*d) - 4*(8*F^(b
*c*x)*b*c*e*cos(e*x)*log(F) + (b^2*c^2*log(F)^2 - 15*e^2)*F^(b*c*x)*sin(e*x))*sin(2*e*x + 2*d))/(b^4*c^4*log(F
)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4 + (b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(8*e*x + 8
*d)^2 + 16*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(6*e*x + 6*d)^2 + 36*(b^4*c^4*log(F)^4 +
34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d)^2 + 16*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e
^4)*cos(2*e*x + 2*d)^2 + (b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(8*e*x + 8*d)^2 + 16*(b^4*c
^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(6*e*x + 6*d)^2 + 36*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*lo
g(F)^2 + 225*e^4)*sin(4*e*x + 4*d)^2 + 48*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4
*d)*sin(2*e*x + 2*d) + 16*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(2*e*x + 2*d)^2 + 2*(b^4*c
^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4 + 4*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos
(6*e*x + 6*d) + 6*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d) + 4*(b^4*c^4*log(F)^
4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d))*cos(8*e*x + 8*d) + 8*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e
^2*log(F)^2 + 225*e^4 + 6*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d) + 4*(b^4*c^4
*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 12*(b^4*c^4*log(F)^4 + 34*
b^2*c^2*e^2*log(F)^2 + 225*e^4 + 4*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d))*co
s(4*e*x + 4*d) + 8*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d) + 4*(2*(b^4*c^4*log
(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(6*e*x + 6*d) + 3*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 +
225*e^4)*sin(4*e*x + 4*d) + 2*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(2*e*x + 2*d))*sin(8*e
*x + 8*d) + 16*(3*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4*d) + 2*(b^4*c^4*log(F)^
4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d)), x) + (48*F^(b*c*x)*F^(a*c)*b*c*e^2
*log(F)*sin(e*x + d) + 3*(F^(a*c)*b^2*c^2*e*log(F)^2 + 25*F^(a*c)*e^3)*F^(b*c*x)*cos(3*e*x + 3*d) - 6*(F^(a*c)
*b^2*c^2*e*log(F)^2 - 15*F^(a*c)*e^3)*F^(b*c*x)*cos(e*x + d) + (F^(a*c)*b^3*c^3*log(F)^3 + 25*F^(a*c)*b*c*e^2*
log(F))*F^(b*c*x)*sin(3*e*x + 3*d))*sin(6*e*x + 6*d) + 3*(48*F^(b*c*x)*F^(a*c)*b*c*e^2*log(F)*sin(e*x + d) + 3
*(F^(a*c)*b^2*c^2*e*log(F)^2 + 25*F^(a*c)*e^3)*F^(b*c*x)*cos(3*e*x + 3*d) - 6*(F^(a*c)*b^2*c^2*e*log(F)^2 - 15
*F^(a*c)*e^3)*F^(b*c*x)*cos(e*x + d) + (F^(a*c)*b^3*c^3*log(F)^3 + 25*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*sin(3*
e*x + 3*d))*sin(4*e*x + 4*d) - 3*(3*(F^(a*c)*b^2*c^2*e*log(F)^2 + 25*F^(a*c)*e^3)*F^(b*c*x)*cos(2*e*x + 2*d) -
(F^(a*c)*b^3*c^3*log(F)^3 + 25*F^(a*c)*b*c*e^2*log(F))*F^(b*c*x)*sin(2*e*x + 2*d) + (F^(a*c)*b^2*c^2*e*log(F)
^2 + 25*F^(a*c)*e^3)*F^(b*c*x))*sin(3*e*x + 3*d) + 18*(8*F^(b*c*x)*F^(a*c)*b*c*e^2*log(F)*sin(e*x + d) - (F^(a
*c)*b^2*c^2*e*log(F)^2 - 15*F^(a*c)*e^3)*F^(b*c*x)*cos(e*x + d))*sin(2*e*x + 2*d))/(b^4*c^4*log(F)^4 + 34*b^2*
c^2*e^2*log(F)^2 + 225*e^4 + (b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(6*e*x + 6*d)^2 + 9*(b^
4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d)^2 + 9*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*
log(F)^2 + 225*e^4)*cos(2*e*x + 2*d)^2 + (b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(6*e*x + 6*
d)^2 + 9*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4*d)^2 + 18*(b^4*c^4*log(F)^4 + 34
*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 9*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(
F)^2 + 225*e^4)*sin(2*e*x + 2*d)^2 + 2*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4 + 3*(b^4*c^4*log(
F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(4*e*x + 4*d) + 3*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 2
25*e^4)*cos(2*e*x + 2*d))*cos(6*e*x + 6*d) + 6*(b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4 + 3*(b^4*
c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) + 6*(b^4*c^4*log(F)^4 + 3
4*b^2*c^2*e^2*log(F)^2 + 225*e^4)*cos(2*e*x + 2*d) + 6*((b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)
*sin(4*e*x + 4*d) + (b^4*c^4*log(F)^4 + 34*b^2*c^2*e^2*log(F)^2 + 225*e^4)*sin(2*e*x + 2*d))*sin(6*e*x + 6*d))

Giac [F]

\[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \sec \left (e x + d\right )^{3} \,d x } \]

[In]

integrate(F^(c*(b*x+a))*sec(e*x+d)^3,x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)*sec(e*x + d)^3, x)

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} \sec ^3(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\cos \left (d+e\,x\right )}^3} \,d x \]

[In]

int(F^(c*(a + b*x))/cos(d + e*x)^3,x)

[Out]

int(F^(c*(a + b*x))/cos(d + e*x)^3, x)