3.286 \(\int F^{c (a+b x)} \cosh ^2(d+e x) \, dx\)

Optimal. Leaf size=132 \[ -\frac{b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \]

[Out]

(2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c*(a + b*x))*Cosh[d + e*x]^2*Log[F])
/(4*e^2 - b^2*c^2*Log[F]^2) + (2*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0528336, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {5477, 2194} \[ -\frac{b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Cosh[d + e*x]^2,x]

[Out]

(2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c*(a + b*x))*Cosh[d + e*x]^2*Log[F])
/(4*e^2 - b^2*c^2*Log[F]^2) + (2*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2)

Rule 5477

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx &=-\frac{b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{\left (2 e^2\right ) \int F^{c (a+b x)} \, dx}{4 e^2-b^2 c^2 \log ^2(F)}\\ &=\frac{2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac{b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac{2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}\\ \end{align*}

Mathematica [A]  time = 0.225002, size = 85, normalized size = 0.64 \[ \frac{F^{c (a+b x)} \left (b^2 c^2 \log ^2(F) \cosh (2 (d+e x))+b^2 c^2 \log ^2(F)-2 b c e \log (F) \sinh (2 (d+e x))-4 e^2\right )}{2 b^3 c^3 \log ^3(F)-8 b c e^2 \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^2,x]

[Out]

(F^(c*(a + b*x))*(-4*e^2 + b^2*c^2*Log[F]^2 + b^2*c^2*Cosh[2*(d + e*x)]*Log[F]^2 - 2*b*c*e*Log[F]*Sinh[2*(d +
e*x)]))/(-8*b*c*e^2*Log[F] + 2*b^3*c^3*Log[F]^3)

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 143, normalized size = 1.1 \begin{align*}{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{{\rm e}^{4\,ex+4\,d}}+2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{{\rm e}^{2\,ex+2\,d}}-2\,\ln \left ( F \right ) bce{{\rm e}^{4\,ex+4\,d}}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+2\,\ln \left ( F \right ) bce-8\,{e}^{2}{{\rm e}^{2\,ex+2\,d}} \right ){{\rm e}^{-2\,ex-2\,d}}{F}^{c \left ( bx+a \right ) }}{4\,bc\ln \left ( F \right ) \left ( bc\ln \left ( F \right ) -2\,e \right ) \left ( bc\ln \left ( F \right ) +2\,e \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*cosh(e*x+d)^2,x)

[Out]

1/4*(ln(F)^2*b^2*c^2*exp(4*e*x+4*d)+2*ln(F)^2*b^2*c^2*exp(2*e*x+2*d)-2*ln(F)*b*c*e*exp(4*e*x+4*d)+b^2*c^2*ln(F
)^2+2*ln(F)*b*c*e-8*e^2*exp(2*e*x+2*d))/ln(F)/b/c/(b*c*ln(F)-2*e)*exp(-2*e*x-2*d)/(b*c*ln(F)+2*e)*F^(c*(b*x+a)
)

________________________________________________________________________________________

Maxima [A]  time = 1.08939, size = 127, normalized size = 0.96 \begin{align*} \frac{F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{4 \,{\left (b c \log \left (F\right ) + 2 \, e\right )}} + \frac{F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{4 \,{\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}\right )}} + \frac{F^{b c x + a c}}{2 \, b c \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="maxima")

[Out]

1/4*F^(a*c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) + 1/4*F^(a*c)*e^(b*c*x*log(F) - 2*e*x)/(b*c*e^(2
*d)*log(F) - 2*e*e^(2*d)) + 1/2*F^(b*c*x + a*c)/(b*c*log(F))

________________________________________________________________________________________

Fricas [B]  time = 2.52552, size = 1783, normalized size = 13.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="fricas")

[Out]

1/4*(((b^2*c^2*log(F)^2 - 2*b*c*e*log(F))*sinh(e*x + d)^4 - 8*e^2*cosh(e*x + d)^2 + 4*(b^2*c^2*cosh(e*x + d)*l
og(F)^2 - 2*b*c*e*cosh(e*x + d)*log(F))*sinh(e*x + d)^3 + (b^2*c^2*cosh(e*x + d)^4 + 2*b^2*c^2*cosh(e*x + d)^2
+ b^2*c^2)*log(F)^2 - 2*(6*b*c*e*cosh(e*x + d)^2*log(F) - (3*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 + 4*
e^2)*sinh(e*x + d)^2 - 2*(b*c*e*cosh(e*x + d)^4 - b*c*e)*log(F) - 4*(2*b*c*e*cosh(e*x + d)^3*log(F) + 4*e^2*co
sh(e*x + d) - (b^2*c^2*cosh(e*x + d)^3 + b^2*c^2*cosh(e*x + d))*log(F)^2)*sinh(e*x + d))*cosh((b*c*x + a*c)*lo
g(F)) + ((b^2*c^2*log(F)^2 - 2*b*c*e*log(F))*sinh(e*x + d)^4 - 8*e^2*cosh(e*x + d)^2 + 4*(b^2*c^2*cosh(e*x + d
)*log(F)^2 - 2*b*c*e*cosh(e*x + d)*log(F))*sinh(e*x + d)^3 + (b^2*c^2*cosh(e*x + d)^4 + 2*b^2*c^2*cosh(e*x + d
)^2 + b^2*c^2)*log(F)^2 - 2*(6*b*c*e*cosh(e*x + d)^2*log(F) - (3*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 +
4*e^2)*sinh(e*x + d)^2 - 2*(b*c*e*cosh(e*x + d)^4 - b*c*e)*log(F) - 4*(2*b*c*e*cosh(e*x + d)^3*log(F) + 4*e^2
*cosh(e*x + d) - (b^2*c^2*cosh(e*x + d)^3 + b^2*c^2*cosh(e*x + d))*log(F)^2)*sinh(e*x + d))*sinh((b*c*x + a*c)
*log(F)))/(b^3*c^3*cosh(e*x + d)^2*log(F)^3 - 4*b*c*e^2*cosh(e*x + d)^2*log(F) + (b^3*c^3*log(F)^3 - 4*b*c*e^2
*log(F))*sinh(e*x + d)^2 + 2*(b^3*c^3*cosh(e*x + d)*log(F)^3 - 4*b*c*e^2*cosh(e*x + d)*log(F))*sinh(e*x + d))

________________________________________________________________________________________

Sympy [A]  time = 110.87, size = 604, normalized size = 4.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*cosh(e*x+d)**2,x)

[Out]

Piecewise((-x*sinh(d + e*x)**2/2 + x*cosh(d + e*x)**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e), Eq(F, 1)), (zoo*e
**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*sinh(d + e*x)**2 + zoo*e**2*exp(-2*e/(b*c))**(a*c)*exp(-2*
e/(b*c))**(b*c*x)*sinh(d + e*x)*cosh(d + e*x) + zoo*e**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*cosh(
d + e*x)**2, Eq(F, exp(-2*e/(b*c)))), (zoo*e**2*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*sinh(d + e*x)**2
+ zoo*e**2*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*sinh(d + e*x)*cosh(d + e*x) + zoo*e**2*exp(2*e/(b*c)
)**(a*c)*exp(2*e/(b*c))**(b*c*x)*cosh(d + e*x)**2, Eq(F, exp(2*e/(b*c)))), (F**(a*c)*(-x*sinh(d + e*x)**2/2 +
x*cosh(d + e*x)**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e)), Eq(b, 0)), (-x*sinh(d + e*x)**2/2 + x*cosh(d + e*x)
**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e), Eq(c, 0)), (F**(a*c)*F**(b*c*x)*b**2*c**2*log(F)**2*cosh(d + e*x)**
2/(b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)) - 2*F**(a*c)*F**(b*c*x)*b*c*e*log(F)*sinh(d + e*x)*cosh(d + e*x)/(
b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)) + 2*F**(a*c)*F**(b*c*x)*e**2*sinh(d + e*x)**2/(b**3*c**3*log(F)**3 -
4*b*c*e**2*log(F)) - 2*F**(a*c)*F**(b*c*x)*e**2*cosh(d + e*x)**2/(b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)), Tr
ue))

________________________________________________________________________________________

Giac [C]  time = 1.45306, size = 1219, normalized size = 9.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="giac")

[Out]

(2*b*c*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)*log(abs(F))/(4*b^2*c^2*log(ab
s(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2
*pi*a*c*sgn(F) + 1/2*pi*a*c)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a*
c*log(abs(F))) - 1/2*I*(-2*I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(
2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F))) + 2*I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*p
i*a*c*sgn(F) + 1/2*I*pi*a*c)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)) + a*c
*log(abs(F))) + 1/2*(2*(b*c*log(abs(F)) + 2*e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1
/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b
*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F))
+ 2*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) - 1/2*I*(-2*I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*
I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) + 16*e) +
2*I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-4*I*pi*b*c*sgn(F) + 4*
I*pi*b*c + 8*b*c*log(abs(F)) + 16*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) + 1/2*(2*(b*c*log(
abs(F)) - 2*e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*
b*c)^2 + 4*(b*c*log(abs(F)) - 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2
*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) - 2*e)^2))*e^(a*c*log(abs(F)) +
(b*c*log(abs(F)) - 2*e)*x - 2*d) - 1/2*I*(-2*I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F)
- 1/2*I*pi*a*c)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*e) + 2*I*e^(-1/2*I*pi*b*c*x*sgn(F) +
1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) - 1
6*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - 2*e)*x - 2*d)