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

Optimal. Leaf size=251 \[ \frac {2 e f^2 \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c f^2 \log (F) \cosh ^2(d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {2 b c f^2 \log (F) \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 e f^2 \sinh (d+e x) \cosh (d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {f^2 F^{a c+b c x}}{b c \log (F)} \]

[Out]

f^2*F^(b*c*x+a*c)/b/c/ln(F)-2*b*c*f^2*F^(b*c*x+a*c)*cosh(e*x+d)*ln(F)/(e^2-b^2*c^2*ln(F)^2)+2*e^2*f^2*F^(b*c*x
+a*c)/b/c/ln(F)/(4*e^2-b^2*c^2*ln(F)^2)-b*c*f^2*F^(b*c*x+a*c)*cosh(e*x+d)^2*ln(F)/(4*e^2-b^2*c^2*ln(F)^2)+2*e*
f^2*F^(b*c*x+a*c)*sinh(e*x+d)/(e^2-b^2*c^2*ln(F)^2)+2*e*f^2*F^(b*c*x+a*c)*cosh(e*x+d)*sinh(e*x+d)/(4*e^2-b^2*c
^2*ln(F)^2)

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Rubi [A]  time = 0.31, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6741, 12, 6742, 2194, 5475, 5477} \[ \frac {2 e f^2 \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c f^2 \log (F) \cosh ^2(d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {2 b c f^2 \log (F) \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 e f^2 \sinh (d+e x) \cosh (d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {f^2 F^{a c+b c x}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 5475

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

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 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int F^{c (a+b x)} (f+f \cosh (d+e x))^2 \, dx &=\int f^2 F^{a c+b c x} (1+\cosh (d+e x))^2 \, dx\\ &=f^2 \int F^{a c+b c x} (1+\cosh (d+e x))^2 \, dx\\ &=f^2 \int \left (F^{a c+b c x}+2 F^{a c+b c x} \cosh (d+e x)+F^{a c+b c x} \cosh ^2(d+e x)\right ) \, dx\\ &=f^2 \int F^{a c+b c x} \, dx+f^2 \int F^{a c+b c x} \cosh ^2(d+e x) \, dx+\left (2 f^2\right ) \int F^{a c+b c x} \cosh (d+e x) \, dx\\ &=\frac {f^2 F^{a c+b c x}}{b c \log (F)}-\frac {2 b c f^2 F^{a c+b c x} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c f^2 F^{a c+b c x} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {\left (2 e^2 f^2\right ) \int F^{a c+b c x} \, dx}{4 e^2-b^2 c^2 \log ^2(F)}\\ &=\frac {f^2 F^{a c+b c x}}{b c \log (F)}-\frac {2 b c f^2 F^{a c+b c x} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c f^2 F^{a c+b c x} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 230, normalized size = 0.92 \[ \frac {f^2 F^{c (a+b x)} \left (3 b^4 c^4 \log ^4(F)-4 b^3 c^3 e \log ^3(F) \sinh (d+e x)-2 b^3 c^3 e \log ^3(F) \sinh (2 (d+e x))-15 b^2 c^2 e^2 \log ^2(F)+4 \cosh (d+e x) \left (b^4 c^4 \log ^4(F)-4 b^2 c^2 e^2 \log ^2(F)\right )+\cosh (2 (d+e x)) \left (b^4 c^4 \log ^4(F)-b^2 c^2 e^2 \log ^2(F)\right )+16 b c e^3 \log (F) \sinh (d+e x)+2 b c e^3 \log (F) \sinh (2 (d+e x))+12 e^4\right )}{2 \left (b^5 c^5 \log ^5(F)-5 b^3 c^3 e^2 \log ^3(F)+4 b c e^4 \log (F)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^2*F^(c*(a + b*x))*(12*e^4 - 15*b^2*c^2*e^2*Log[F]^2 + 3*b^4*c^4*Log[F]^4 + 4*Cosh[d + e*x]*(-4*b^2*c^2*e^2*
Log[F]^2 + b^4*c^4*Log[F]^4) + Cosh[2*(d + e*x)]*(-(b^2*c^2*e^2*Log[F]^2) + b^4*c^4*Log[F]^4) + 16*b*c*e^3*Log
[F]*Sinh[d + e*x] - 4*b^3*c^3*e*Log[F]^3*Sinh[d + e*x] + 2*b*c*e^3*Log[F]*Sinh[2*(d + e*x)] - 2*b^3*c^3*e*Log[
F]^3*Sinh[2*(d + e*x)]))/(2*(4*b*c*e^4*Log[F] - 5*b^3*c^3*e^2*Log[F]^3 + b^5*c^5*Log[F]^5))

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fricas [B]  time = 0.49, size = 2340, normalized size = 9.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((24*e^4*f^2*cosh(e*x + d)^2 + (b^4*c^4*f^2*cosh(e*x + d)^4 + 4*b^4*c^4*f^2*cosh(e*x + d)^3 + 6*b^4*c^4*f^
2*cosh(e*x + d)^2 + 4*b^4*c^4*f^2*cosh(e*x + d) + b^4*c^4*f^2)*log(F)^4 + (b^4*c^4*f^2*log(F)^4 - 2*b^3*c^3*e*
f^2*log(F)^3 - b^2*c^2*e^2*f^2*log(F)^2 + 2*b*c*e^3*f^2*log(F))*sinh(e*x + d)^4 - 2*(b^3*c^3*e*f^2*cosh(e*x +
d)^4 + 2*b^3*c^3*e*f^2*cosh(e*x + d)^3 - 2*b^3*c^3*e*f^2*cosh(e*x + d) - b^3*c^3*e*f^2)*log(F)^3 + 4*((b^4*c^4
*f^2*cosh(e*x + d) + b^4*c^4*f^2)*log(F)^4 - (2*b^3*c^3*e*f^2*cosh(e*x + d) + b^3*c^3*e*f^2)*log(F)^3 - (b^2*c
^2*e^2*f^2*cosh(e*x + d) + 4*b^2*c^2*e^2*f^2)*log(F)^2 + 2*(b*c*e^3*f^2*cosh(e*x + d) + 2*b*c*e^3*f^2)*log(F))
*sinh(e*x + d)^3 - (b^2*c^2*e^2*f^2*cosh(e*x + d)^4 + 16*b^2*c^2*e^2*f^2*cosh(e*x + d)^3 + 30*b^2*c^2*e^2*f^2*
cosh(e*x + d)^2 + 16*b^2*c^2*e^2*f^2*cosh(e*x + d) + b^2*c^2*e^2*f^2)*log(F)^2 + 6*(4*e^4*f^2 + (b^4*c^4*f^2*c
osh(e*x + d)^2 + 2*b^4*c^4*f^2*cosh(e*x + d) + b^4*c^4*f^2)*log(F)^4 - 2*(b^3*c^3*e*f^2*cosh(e*x + d)^2 + b^3*
c^3*e*f^2*cosh(e*x + d))*log(F)^3 - (b^2*c^2*e^2*f^2*cosh(e*x + d)^2 + 8*b^2*c^2*e^2*f^2*cosh(e*x + d) + 5*b^2
*c^2*e^2*f^2)*log(F)^2 + 2*(b*c*e^3*f^2*cosh(e*x + d)^2 + 4*b*c*e^3*f^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^2
+ 2*(b*c*e^3*f^2*cosh(e*x + d)^4 + 8*b*c*e^3*f^2*cosh(e*x + d)^3 - 8*b*c*e^3*f^2*cosh(e*x + d) - b*c*e^3*f^2)
*log(F) + 4*(12*e^4*f^2*cosh(e*x + d) + (b^4*c^4*f^2*cosh(e*x + d)^3 + 3*b^4*c^4*f^2*cosh(e*x + d)^2 + 3*b^4*c
^4*f^2*cosh(e*x + d) + b^4*c^4*f^2)*log(F)^4 - (2*b^3*c^3*e*f^2*cosh(e*x + d)^3 + 3*b^3*c^3*e*f^2*cosh(e*x + d
)^2 - b^3*c^3*e*f^2)*log(F)^3 - (b^2*c^2*e^2*f^2*cosh(e*x + d)^3 + 12*b^2*c^2*e^2*f^2*cosh(e*x + d)^2 + 15*b^2
*c^2*e^2*f^2*cosh(e*x + d) + 4*b^2*c^2*e^2*f^2)*log(F)^2 + 2*(b*c*e^3*f^2*cosh(e*x + d)^3 + 6*b*c*e^3*f^2*cosh
(e*x + d)^2 - 2*b*c*e^3*f^2)*log(F))*sinh(e*x + d))*cosh((b*c*x + a*c)*log(F)) + (24*e^4*f^2*cosh(e*x + d)^2 +
(b^4*c^4*f^2*cosh(e*x + d)^4 + 4*b^4*c^4*f^2*cosh(e*x + d)^3 + 6*b^4*c^4*f^2*cosh(e*x + d)^2 + 4*b^4*c^4*f^2*
cosh(e*x + d) + b^4*c^4*f^2)*log(F)^4 + (b^4*c^4*f^2*log(F)^4 - 2*b^3*c^3*e*f^2*log(F)^3 - b^2*c^2*e^2*f^2*log
(F)^2 + 2*b*c*e^3*f^2*log(F))*sinh(e*x + d)^4 - 2*(b^3*c^3*e*f^2*cosh(e*x + d)^4 + 2*b^3*c^3*e*f^2*cosh(e*x +
d)^3 - 2*b^3*c^3*e*f^2*cosh(e*x + d) - b^3*c^3*e*f^2)*log(F)^3 + 4*((b^4*c^4*f^2*cosh(e*x + d) + b^4*c^4*f^2)*
log(F)^4 - (2*b^3*c^3*e*f^2*cosh(e*x + d) + b^3*c^3*e*f^2)*log(F)^3 - (b^2*c^2*e^2*f^2*cosh(e*x + d) + 4*b^2*c
^2*e^2*f^2)*log(F)^2 + 2*(b*c*e^3*f^2*cosh(e*x + d) + 2*b*c*e^3*f^2)*log(F))*sinh(e*x + d)^3 - (b^2*c^2*e^2*f^
2*cosh(e*x + d)^4 + 16*b^2*c^2*e^2*f^2*cosh(e*x + d)^3 + 30*b^2*c^2*e^2*f^2*cosh(e*x + d)^2 + 16*b^2*c^2*e^2*f
^2*cosh(e*x + d) + b^2*c^2*e^2*f^2)*log(F)^2 + 6*(4*e^4*f^2 + (b^4*c^4*f^2*cosh(e*x + d)^2 + 2*b^4*c^4*f^2*cos
h(e*x + d) + b^4*c^4*f^2)*log(F)^4 - 2*(b^3*c^3*e*f^2*cosh(e*x + d)^2 + b^3*c^3*e*f^2*cosh(e*x + d))*log(F)^3
- (b^2*c^2*e^2*f^2*cosh(e*x + d)^2 + 8*b^2*c^2*e^2*f^2*cosh(e*x + d) + 5*b^2*c^2*e^2*f^2)*log(F)^2 + 2*(b*c*e^
3*f^2*cosh(e*x + d)^2 + 4*b*c*e^3*f^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^2 + 2*(b*c*e^3*f^2*cosh(e*x + d)^4
+ 8*b*c*e^3*f^2*cosh(e*x + d)^3 - 8*b*c*e^3*f^2*cosh(e*x + d) - b*c*e^3*f^2)*log(F) + 4*(12*e^4*f^2*cosh(e*x +
d) + (b^4*c^4*f^2*cosh(e*x + d)^3 + 3*b^4*c^4*f^2*cosh(e*x + d)^2 + 3*b^4*c^4*f^2*cosh(e*x + d) + b^4*c^4*f^2
)*log(F)^4 - (2*b^3*c^3*e*f^2*cosh(e*x + d)^3 + 3*b^3*c^3*e*f^2*cosh(e*x + d)^2 - b^3*c^3*e*f^2)*log(F)^3 - (b
^2*c^2*e^2*f^2*cosh(e*x + d)^3 + 12*b^2*c^2*e^2*f^2*cosh(e*x + d)^2 + 15*b^2*c^2*e^2*f^2*cosh(e*x + d) + 4*b^2
*c^2*e^2*f^2)*log(F)^2 + 2*(b*c*e^3*f^2*cosh(e*x + d)^3 + 6*b*c*e^3*f^2*cosh(e*x + d)^2 - 2*b*c*e^3*f^2)*log(F
))*sinh(e*x + d))*sinh((b*c*x + a*c)*log(F)))/(b^5*c^5*cosh(e*x + d)^2*log(F)^5 - 5*b^3*c^3*e^2*cosh(e*x + d)^
2*log(F)^3 + 4*b*c*e^4*cosh(e*x + d)^2*log(F) + (b^5*c^5*log(F)^5 - 5*b^3*c^3*e^2*log(F)^3 + 4*b*c*e^4*log(F))
*sinh(e*x + d)^2 + 2*(b^5*c^5*cosh(e*x + d)*log(F)^5 - 5*b^3*c^3*e^2*cosh(e*x + d)*log(F)^3 + 4*b*c*e^4*cosh(e
*x + d)*log(F))*sinh(e*x + d))

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giac [C]  time = 0.29, size = 1576, normalized size = 6.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(2*b*c*f^2*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(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*f^2*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(ab
s(F)) + a*c*log(abs(F))) - 1/2*I*(-6*I*f^2*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))) + 6*I*f^2*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))))*e^(b*c*x
*log(abs(F)) + a*c*log(abs(F))) + 1/2*(2*(b*c*log(abs(F)) + 2*e)*f^2*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)*f^2*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*f^2*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*f^2*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) + 2*(2*(b*c*log(abs(F)) + e)*f^2*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)) + e)^2) - (pi*b*c*sgn(F) - pi*b*c)*f^2*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)) + e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) - 1/2*I*(-2*I*f^2*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) + 2*e) + 2*
I*f^2*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)/(-I*pi*b*c*sgn(F) + I*p
i*b*c + 2*b*c*log(abs(F)) + 2*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 2*(2*(b*c*log(abs(F)) -
e)*f^2*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)) - e)^2) - (pi*b*c*sgn(F) - pi*b*c)*f^2*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)) - e)^2))*e^(a*c*log(abs(F)) + (b*c*log
(abs(F)) - e)*x - d) - 1/2*I*(-2*I*f^2*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) - 2*e) + 2*I*f^2*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)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F)) - 2*e))*e^(a*c*l
og(abs(F)) + (b*c*log(abs(F)) - e)*x - d) + 1/2*(2*(b*c*log(abs(F)) - 2*e)*f^2*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)*f^2*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*f^2*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*f^2*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*lo
g(abs(F)) - 2*e)*x - 2*d)

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maple [A]  time = 0.60, size = 426, normalized size = 1.70 \[ \frac {f^{2} \left (\ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{4 e x +4 d}+4 \ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{3 e x +3 d}+6 \ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{2 e x +2 d}-2 \ln \relax (F )^{3} b^{3} c^{3} e \,{\mathrm e}^{4 e x +4 d}+4 \ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{e x +d}-4 \ln \relax (F )^{3} b^{3} c^{3} e \,{\mathrm e}^{3 e x +3 d}+b^{4} c^{4} \ln \relax (F )^{4}-\ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{4 e x +4 d}+4 \ln \relax (F )^{3} b^{3} c^{3} e \,{\mathrm e}^{e x +d}-16 \ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{3 e x +3 d}+2 \ln \relax (F )^{3} b^{3} c^{3} e -30 \ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{2 e x +2 d}+2 \ln \relax (F ) b c \,e^{3} {\mathrm e}^{4 e x +4 d}-16 \ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{e x +d}+16 \ln \relax (F ) b c \,e^{3} {\mathrm e}^{3 e x +3 d}-b^{2} c^{2} e^{2} \ln \relax (F )^{2}-16 \ln \relax (F ) b c \,e^{3} {\mathrm e}^{e x +d}-2 \ln \relax (F ) b c \,e^{3}+24 e^{4} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 b c \ln \relax (F ) \left (b c \ln \relax (F )-e \right ) \left (b c \ln \relax (F )-2 e \right ) \left (e +b c \ln \relax (F )\right ) \left (b c \ln \relax (F )+2 e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*f^2*(ln(F)^4*b^4*c^4*exp(4*e*x+4*d)+4*ln(F)^4*b^4*c^4*exp(3*e*x+3*d)+6*ln(F)^4*b^4*c^4*exp(2*e*x+2*d)-2*ln
(F)^3*b^3*c^3*e*exp(4*e*x+4*d)+4*ln(F)^4*b^4*c^4*exp(e*x+d)-4*ln(F)^3*b^3*c^3*e*exp(3*e*x+3*d)+b^4*c^4*ln(F)^4
-ln(F)^2*b^2*c^2*e^2*exp(4*e*x+4*d)+4*ln(F)^3*b^3*c^3*e*exp(e*x+d)-16*ln(F)^2*b^2*c^2*e^2*exp(3*e*x+3*d)+2*ln(
F)^3*b^3*c^3*e-30*ln(F)^2*b^2*c^2*e^2*exp(2*e*x+2*d)+2*ln(F)*b*c*e^3*exp(4*e*x+4*d)-16*ln(F)^2*b^2*c^2*e^2*exp
(e*x+d)+16*ln(F)*b*c*e^3*exp(3*e*x+3*d)-b^2*c^2*e^2*ln(F)^2-16*ln(F)*b*c*e^3*exp(e*x+d)-2*ln(F)*b*c*e^3+24*e^4
*exp(2*e*x+2*d))/b/c/ln(F)/(b*c*ln(F)-e)*exp(-2*e*x-2*d)/(b*c*ln(F)-2*e)/(e+b*c*ln(F))/(b*c*ln(F)+2*e)*F^(c*(b
*x+a))

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maxima [A]  time = 0.37, size = 187, normalized size = 0.75 \[ \frac {1}{4} \, f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \relax (F) + 2 \, e x + 2 \, d\right )}}{b c \log \relax (F) + 2 \, e} + \frac {F^{a c} e^{\left (b c x \log \relax (F) - 2 \, e x\right )}}{b c e^{\left (2 \, d\right )} \log \relax (F) - 2 \, e e^{\left (2 \, d\right )}} + \frac {2 \, F^{b c x + a c}}{b c \log \relax (F)}\right )} + f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \relax (F) + e x + d\right )}}{b c \log \relax (F) + e} + \frac {F^{a c} e^{\left (b c x \log \relax (F) - e x\right )}}{b c e^{d} \log \relax (F) - e e^{d}}\right )} + \frac {F^{b c x + a c} f^{2}}{b c \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.99, size = 288, normalized size = 1.15 \[ \frac {2\,F^{b\,c\,x}\,F^{a\,c}\,e\,f^2\,\mathrm {sinh}\left (d+e\,x\right )}{e^2-b^2\,c^2\,{\ln \relax (F)}^2}+\frac {F^{b\,c\,x}\,F^{a\,c}\,f^2}{b\,c\,\ln \relax (F)}+\frac {2\,F^{b\,c\,x}\,F^{a\,c}\,e\,f^2\,\mathrm {cosh}\left (d+e\,x\right )\,\mathrm {sinh}\left (d+e\,x\right )}{4\,e^2-b^2\,c^2\,{\ln \relax (F)}^2}-\frac {2\,F^{b\,c\,x}\,F^{a\,c}\,b\,c\,f^2\,\mathrm {cosh}\left (d+e\,x\right )\,\ln \relax (F)}{e^2-b^2\,c^2\,{\ln \relax (F)}^2}-\frac {2\,F^{b\,c\,x}\,F^{a\,c}\,e^2\,f^2\,{\mathrm {sinh}\left (d+e\,x\right )}^2}{b\,c\,\ln \relax (F)\,\left (4\,e^2-b^2\,c^2\,{\ln \relax (F)}^2\right )}+\frac {F^{b\,c\,x}\,F^{a\,c}\,f^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\left (2\,e^2-b^2\,c^2\,{\ln \relax (F)}^2\right )}{b\,c\,\ln \relax (F)\,\left (4\,e^2-b^2\,c^2\,{\ln \relax (F)}^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 61.57, size = 1719, normalized size = 6.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-f**2*x*sinh(d + e*x)**2/2 + f**2*x*cosh(d + e*x)**2/2 + f**2*x + f**2*sinh(d + e*x)*cosh(d + e*x)/
(2*e) + 2*f**2*sinh(d + e*x)/e, Eq(F, 1)), (zoo*e**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*sinh
(d + e*x)**2 + zoo*e**4*f**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**4*f**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**
4*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*sinh(d + e*x) + zoo*e**4*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(
b*c))**(b*c*x)*cosh(d + e*x), Eq(F, exp(-e/(b*c)))), (zoo*e**4*f**2*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*
sinh(d + e*x) + zoo*e**4*f**2*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*cosh(d + e*x), Eq(F, exp(e/(b*c)))), (
zoo*e**4*f**2*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*sinh(d + e*x)**2 + zoo*e**4*f**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**4*f**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)*(-f**2*x*sinh(d + e*x)**2/2 + f**2*x*cosh(d + e*x
)**2/2 + f**2*x + f**2*sinh(d + e*x)*cosh(d + e*x)/(2*e) + 2*f**2*sinh(d + e*x)/e), Eq(b, 0)), (-f**2*x*sinh(d
+ e*x)**2/2 + f**2*x*cosh(d + e*x)**2/2 + f**2*x + f**2*sinh(d + e*x)*cosh(d + e*x)/(2*e) + 2*f**2*sinh(d + e
*x)/e, Eq(c, 0)), (F**(a*c)*F**(b*c*x)*b**4*c**4*f**2*log(F)**4*cosh(d + e*x)**2/(b**5*c**5*log(F)**5 - 5*b**3
*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c)*F**(b*c*x)*b**4*c**4*f**2*log(F)**4*cosh(d + e*x)/(b**5
*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + F**(a*c)*F**(b*c*x)*b**4*c**4*f**2*log(F)*
*4/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) - 2*F**(a*c)*F**(b*c*x)*b**3*c**3*e*
f**2*log(F)**3*sinh(d + e*x)*cosh(d + e*x)/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(
F)) - 2*F**(a*c)*F**(b*c*x)*b**3*c**3*e*f**2*log(F)**3*sinh(d + e*x)/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*l
og(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c)*F**(b*c*x)*b**2*c**2*e**2*f**2*log(F)**2*sinh(d + e*x)**2/(b**5*c**
5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) - 3*F**(a*c)*F**(b*c*x)*b**2*c**2*e**2*f**2*log(
F)**2*cosh(d + e*x)**2/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) - 8*F**(a*c)*F**
(b*c*x)*b**2*c**2*e**2*f**2*log(F)**2*cosh(d + e*x)/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*
e**4*log(F)) - 5*F**(a*c)*F**(b*c*x)*b**2*c**2*e**2*f**2*log(F)**2/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log
(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c)*F**(b*c*x)*b*c*e**3*f**2*log(F)*sinh(d + e*x)*cosh(d + e*x)/(b**5*c**
5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 8*F**(a*c)*F**(b*c*x)*b*c*e**3*f**2*log(F)*sin
h(d + e*x)/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) - 2*F**(a*c)*F**(b*c*x)*e**4
*f**2*sinh(d + e*x)**2/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 2*F**(a*c)*F**
(b*c*x)*e**4*f**2*cosh(d + e*x)**2/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)) + 4*
F**(a*c)*F**(b*c*x)*e**4*f**2/(b**5*c**5*log(F)**5 - 5*b**3*c**3*e**2*log(F)**3 + 4*b*c*e**4*log(F)), True))

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