3.285 \(\int F^{c (a+b x)} \cosh ^3(d+e x) \, dx\)
Optimal. Leaf size=202 \[ -\frac {b c \log (F) \cosh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e \sinh (d+e x) \cosh ^2(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {6 b c e^2 \log (F) \cosh (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)-10 b^2 c^2 e^2 \log ^2(F)+9 e^4}+\frac {6 e^3 \sinh (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)-10 b^2 c^2 e^2 \log ^2(F)+9 e^4} \]
[Out]
-b*c*F^(c*(b*x+a))*cosh(e*x+d)^3*ln(F)/(9*e^2-b^2*c^2*ln(F)^2)-6*b*c*e^2*F^(c*(b*x+a))*cosh(e*x+d)*ln(F)/(9*e^
4-10*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln(F)^4)+3*e*F^(c*(b*x+a))*cosh(e*x+d)^2*sinh(e*x+d)/(9*e^2-b^2*c^2*ln(F)^2)+
6*e^3*F^(c*(b*x+a))*sinh(e*x+d)/(9*e^4-10*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln(F)^4)
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Rubi [A] time = 0.08, antiderivative size = 202, normalized size of antiderivative = 1.00,
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, 5475} \[ \frac {6 e^3 \sinh (d+e x) F^{c (a+b x)}}{-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4}-\frac {b c \log (F) \cosh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {6 b c e^2 \log (F) \cosh (d+e x) F^{c (a+b x)}}{-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)+9 e^4}+\frac {3 e \sinh (d+e x) \cosh ^2(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(c*(a + b*x))*Cosh[d + e*x]^3,x]
[Out]
-((b*c*F^(c*(a + b*x))*Cosh[d + e*x]^3*Log[F])/(9*e^2 - b^2*c^2*Log[F]^2)) - (6*b*c*e^2*F^(c*(a + b*x))*Cosh[d
+ e*x]*Log[F])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (3*e*F^(c*(a + b*x))*Cosh[d + e*x]^2*Si
nh[d + e*x])/(9*e^2 - b^2*c^2*Log[F]^2) + (6*e^3*F^(c*(a + b*x))*Sinh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]
^2 + b^4*c^4*Log[F]^4)
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]
Rubi steps
\begin {align*} \int F^{c (a+b x)} \cosh ^3(d+e x) \, dx &=-\frac {b c F^{c (a+b x)} \cosh ^3(d+e x) \log (F)}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {\left (6 e^2\right ) \int F^{c (a+b x)} \cosh (d+e x) \, dx}{9 e^2-b^2 c^2 \log ^2(F)}\\ &=-\frac {b c F^{c (a+b x)} \cosh ^3(d+e x) \log (F)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {6 b c e^2 F^{c (a+b x)} \cosh (d+e x) \log (F)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cosh ^2(d+e x) \sinh (d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {6 e^3 F^{c (a+b x)} \sinh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 159, normalized size = 0.79 \[ \frac {F^{c (a+b x)} \left (3 \cosh (d+e x) \left (b^3 c^3 \log ^3(F)-9 b c e^2 \log (F)\right )+\cosh (3 (d+e x)) \left (b^3 c^3 \log ^3(F)-b c e^2 \log (F)\right )+6 e \sinh (d+e x) \left (\cosh (2 (d+e x)) \left (e^2-b^2 c^2 \log ^2(F)\right )-b^2 c^2 \log ^2(F)+5 e^2\right )\right )}{4 \left (b^4 c^4 \log ^4(F)-10 b^2 c^2 e^2 \log ^2(F)+9 e^4\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^3,x]
[Out]
(F^(c*(a + b*x))*(3*Cosh[d + e*x]*(-9*b*c*e^2*Log[F] + b^3*c^3*Log[F]^3) + Cosh[3*(d + e*x)]*(-(b*c*e^2*Log[F]
) + b^3*c^3*Log[F]^3) + 6*e*(5*e^2 - b^2*c^2*Log[F]^2 + Cosh[2*(d + e*x)]*(e^2 - b^2*c^2*Log[F]^2))*Sinh[d + e
*x]))/(4*(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
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fricas [B] time = 0.65, size = 2218, normalized size = 10.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*cosh(e*x+d)^3,x, algorithm="fricas")
[Out]
1/8*((3*e^3*cosh(e*x + d)^6 + 27*e^3*cosh(e*x + d)^4 + (b^3*c^3*log(F)^3 - 3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(
F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c^3*cosh(e*x + d)*log(F)^3 - 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2
*cosh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 - 27*e^3*cosh(e*x + d)^2 + 3*(15*e^3*cosh(e*x + d
)^2 + (5*b^3*c^3*cosh(e*x + d)^2 + b^3*c^3)*log(F)^3 + 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^2 + b^2*c^2*e)*log(
F)^2 - (5*b*c*e^2*cosh(e*x + d)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^4 + (b^3*c^3*cosh(e*x + d)^6 + 3*b^3*c^3*
cosh(e*x + d)^4 + 3*b^3*c^3*cosh(e*x + d)^2 + b^3*c^3)*log(F)^3 + 4*(15*e^3*cosh(e*x + d)^3 + 27*e^3*cosh(e*x
+ d) + (5*b^3*c^3*cosh(e*x + d)^3 + 3*b^3*c^3*cosh(e*x + d))*log(F)^3 - 3*(5*b^2*c^2*e*cosh(e*x + d)^3 + b^2*c
^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^3 + 27*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3
- 3*e^3 - 3*(b^2*c^2*e*cosh(e*x + d)^6 + b^2*c^2*e*cosh(e*x + d)^4 - b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*lo
g(F)^2 + 3*(15*e^3*cosh(e*x + d)^4 + 54*e^3*cosh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^4 + 6*b^3*c^3*cosh(e*x
+ d)^2 + b^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d)^4 + 6*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*
log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^4 + 54*b*c*e^2*cosh(e*x + d)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c
*e^2*cosh(e*x + d)^6 + 27*b*c*e^2*cosh(e*x + d)^4 + 27*b*c*e^2*cosh(e*x + d)^2 + b*c*e^2)*log(F) + 6*(3*e^3*co
sh(e*x + d)^5 + 18*e^3*cosh(e*x + d)^3 - 9*e^3*cosh(e*x + d) + (b^3*c^3*cosh(e*x + d)^5 + 2*b^3*c^3*cosh(e*x +
d)^3 + b^3*c^3*cosh(e*x + d))*log(F)^3 - (3*b^2*c^2*e*cosh(e*x + d)^5 + 2*b^2*c^2*e*cosh(e*x + d)^3 - b^2*c^2
*e*cosh(e*x + d))*log(F)^2 - (b*c*e^2*cosh(e*x + d)^5 + 18*b*c*e^2*cosh(e*x + d)^3 + 9*b*c*e^2*cosh(e*x + d))*
log(F))*sinh(e*x + d))*cosh((b*c*x + a*c)*log(F)) + (3*e^3*cosh(e*x + d)^6 + 27*e^3*cosh(e*x + d)^4 + (b^3*c^3
*log(F)^3 - 3*b^2*c^2*e*log(F)^2 - b*c*e^2*log(F) + 3*e^3)*sinh(e*x + d)^6 + 6*(b^3*c^3*cosh(e*x + d)*log(F)^3
- 3*b^2*c^2*e*cosh(e*x + d)*log(F)^2 - b*c*e^2*cosh(e*x + d)*log(F) + 3*e^3*cosh(e*x + d))*sinh(e*x + d)^5 -
27*e^3*cosh(e*x + d)^2 + 3*(15*e^3*cosh(e*x + d)^2 + (5*b^3*c^3*cosh(e*x + d)^2 + b^3*c^3)*log(F)^3 + 9*e^3 -
(15*b^2*c^2*e*cosh(e*x + d)^2 + b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^2 + 9*b*c*e^2)*log(F))*sinh(e*x
+ d)^4 + (b^3*c^3*cosh(e*x + d)^6 + 3*b^3*c^3*cosh(e*x + d)^4 + 3*b^3*c^3*cosh(e*x + d)^2 + b^3*c^3)*log(F)^3
+ 4*(15*e^3*cosh(e*x + d)^3 + 27*e^3*cosh(e*x + d) + (5*b^3*c^3*cosh(e*x + d)^3 + 3*b^3*c^3*cosh(e*x + d))*lo
g(F)^3 - 3*(5*b^2*c^2*e*cosh(e*x + d)^3 + b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^3 + 27*
b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d)^3 - 3*e^3 - 3*(b^2*c^2*e*cosh(e*x + d)^6 + b^2*c^2*e*cosh(e*x + d
)^4 - b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 + 3*(15*e^3*cosh(e*x + d)^4 + 54*e^3*cosh(e*x + d)^2 + (
5*b^3*c^3*cosh(e*x + d)^4 + 6*b^3*c^3*cosh(e*x + d)^2 + b^3*c^3)*log(F)^3 - 9*e^3 - (15*b^2*c^2*e*cosh(e*x + d
)^4 + 6*b^2*c^2*e*cosh(e*x + d)^2 - b^2*c^2*e)*log(F)^2 - (5*b*c*e^2*cosh(e*x + d)^4 + 54*b*c*e^2*cosh(e*x + d
)^2 + 9*b*c*e^2)*log(F))*sinh(e*x + d)^2 - (b*c*e^2*cosh(e*x + d)^6 + 27*b*c*e^2*cosh(e*x + d)^4 + 27*b*c*e^2*
cosh(e*x + d)^2 + b*c*e^2)*log(F) + 6*(3*e^3*cosh(e*x + d)^5 + 18*e^3*cosh(e*x + d)^3 - 9*e^3*cosh(e*x + d) +
(b^3*c^3*cosh(e*x + d)^5 + 2*b^3*c^3*cosh(e*x + d)^3 + b^3*c^3*cosh(e*x + d))*log(F)^3 - (3*b^2*c^2*e*cosh(e*x
+ d)^5 + 2*b^2*c^2*e*cosh(e*x + d)^3 - b^2*c^2*e*cosh(e*x + d))*log(F)^2 - (b*c*e^2*cosh(e*x + d)^5 + 18*b*c*
e^2*cosh(e*x + d)^3 + 9*b*c*e^2*cosh(e*x + d))*log(F))*sinh(e*x + d))*sinh((b*c*x + a*c)*log(F)))/(b^4*c^4*cos
h(e*x + d)^3*log(F)^4 - 10*b^2*c^2*e^2*cosh(e*x + d)^3*log(F)^2 + 9*e^4*cosh(e*x + d)^3 + (b^4*c^4*log(F)^4 -
10*b^2*c^2*e^2*log(F)^2 + 9*e^4)*sinh(e*x + d)^3 + 3*(b^4*c^4*cosh(e*x + d)*log(F)^4 - 10*b^2*c^2*e^2*cosh(e*x
+ d)*log(F)^2 + 9*e^4*cosh(e*x + d))*sinh(e*x + d)^2 + 3*(b^4*c^4*cosh(e*x + d)^2*log(F)^4 - 10*b^2*c^2*e^2*c
osh(e*x + d)^2*log(F)^2 + 9*e^4*cosh(e*x + d)^2)*sinh(e*x + d))
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giac [C] time = 0.27, size = 1239, normalized size = 6.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*cosh(e*x+d)^3,x, algorithm="giac")
[Out]
1/4*(2*(b*c*log(abs(F)) + 3*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)) + 3*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)) + 3*e)^2))*e^(a*
c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 48*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b
*c*log(abs(F)) + 48*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 3*e)*x + 3*d) + 3/4*(2*(b*c*log(abs(F)) + e)*c
os(-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)*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*(-6*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)/(8*I*pi
*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e) + 6*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) + 16*e))*e^(a*c*log(abs(F
)) + (b*c*log(abs(F)) + e)*x + d) + 3/4*(2*(b*c*log(abs(F)) - 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)) - 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)) - e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - e)*x - d) - 1/2*I*(-6*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)
) - 16*e) + 6*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)/(-8*I*pi*b*c*
sgn(F) + 8*I*pi*b*c + 16*b*c*log(abs(F)) - 16*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - e)*x - d) + 1/4*(2*(
b*c*log(abs(F)) - 3*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)) - 3*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)) - 3*e)^2))*e^(a*c*log(ab
s(F)) + (b*c*log(abs(F)) - 3*e)*x - 3*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)/(8*I*pi*b*c*sgn(F) - 8*I*pi*b*c + 16*b*c*log(abs(F)) - 48*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)/(-8*I*pi*b*c*sgn(F) + 8*I*pi*b*c + 16*b*c*log(a
bs(F)) - 48*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - 3*e)*x - 3*d)
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maple [A] time = 0.29, size = 326, normalized size = 1.61 \[ \frac {\left (\ln \relax (F )^{3} b^{3} c^{3} {\mathrm e}^{6 e x +6 d}+3 \ln \relax (F )^{3} b^{3} c^{3} {\mathrm e}^{4 e x +4 d}-3 \ln \relax (F )^{2} b^{2} c^{2} e \,{\mathrm e}^{6 e x +6 d}+3 \ln \relax (F )^{3} b^{3} c^{3} {\mathrm e}^{2 e x +2 d}-3 \ln \relax (F )^{2} b^{2} c^{2} e \,{\mathrm e}^{4 e x +4 d}-\ln \relax (F ) b c \,e^{2} {\mathrm e}^{6 e x +6 d}+\ln \relax (F )^{3} b^{3} c^{3}+3 \ln \relax (F )^{2} b^{2} c^{2} e \,{\mathrm e}^{2 e x +2 d}-27 \ln \relax (F ) b c \,e^{2} {\mathrm e}^{4 e x +4 d}+3 e^{3} {\mathrm e}^{6 e x +6 d}+3 \ln \relax (F )^{2} b^{2} c^{2} e -27 \ln \relax (F ) b c \,e^{2} {\mathrm e}^{2 e x +2 d}+27 e^{3} {\mathrm e}^{4 e x +4 d}-\ln \relax (F ) b c \,e^{2}-27 e^{3} {\mathrm e}^{2 e x +2 d}-3 e^{3}\right ) {\mathrm e}^{-3 e x -3 d} F^{c \left (b x +a \right )}}{8 \left (b c \ln \relax (F )-e \right ) \left (b c \ln \relax (F )-3 e \right ) \left (e +b c \ln \relax (F )\right ) \left (b c \ln \relax (F )+3 e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(b*x+a))*cosh(e*x+d)^3,x)
[Out]
1/8*(ln(F)^3*b^3*c^3*exp(6*e*x+6*d)+3*ln(F)^3*b^3*c^3*exp(4*e*x+4*d)-3*ln(F)^2*b^2*c^2*e*exp(6*e*x+6*d)+3*ln(F
)^3*b^3*c^3*exp(2*e*x+2*d)-3*ln(F)^2*b^2*c^2*e*exp(4*e*x+4*d)-ln(F)*b*c*e^2*exp(6*e*x+6*d)+ln(F)^3*b^3*c^3+3*l
n(F)^2*b^2*c^2*e*exp(2*e*x+2*d)-27*ln(F)*b*c*e^2*exp(4*e*x+4*d)+3*e^3*exp(6*e*x+6*d)+3*ln(F)^2*b^2*c^2*e-27*ln
(F)*b*c*e^2*exp(2*e*x+2*d)+27*e^3*exp(4*e*x+4*d)-ln(F)*b*c*e^2-27*e^3*exp(2*e*x+2*d)-3*e^3)/(b*c*ln(F)-e)*exp(
-3*e*x-3*d)/(b*c*ln(F)-3*e)/(e+b*c*ln(F))/(b*c*ln(F)+3*e)*F^(c*(b*x+a))
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maxima [A] time = 0.35, size = 134, normalized size = 0.66 \[ \frac {F^{a c} e^{\left (b c x \log \relax (F) + 3 \, e x + 3 \, d\right )}}{8 \, {\left (b c \log \relax (F) + 3 \, e\right )}} + \frac {3 \, F^{a c} e^{\left (b c x \log \relax (F) + e x + d\right )}}{8 \, {\left (b c \log \relax (F) + e\right )}} + \frac {3 \, F^{a c} e^{\left (b c x \log \relax (F) - e x\right )}}{8 \, {\left (b c e^{d} \log \relax (F) - e e^{d}\right )}} + \frac {F^{a c} e^{\left (b c x \log \relax (F) - 3 \, e x\right )}}{8 \, {\left (b c e^{\left (3 \, d\right )} \log \relax (F) - 3 \, e e^{\left (3 \, d\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*cosh(e*x+d)^3,x, algorithm="maxima")
[Out]
1/8*F^(a*c)*e^(b*c*x*log(F) + 3*e*x + 3*d)/(b*c*log(F) + 3*e) + 3/8*F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*lo
g(F) + e) + 3/8*F^(a*c)*e^(b*c*x*log(F) - e*x)/(b*c*e^d*log(F) - e*e^d) + 1/8*F^(a*c)*e^(b*c*x*log(F) - 3*e*x)
/(b*c*e^(3*d)*log(F) - 3*e*e^(3*d))
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mupad [B] time = 1.81, size = 154, normalized size = 0.76 \[ \frac {F^{a\,c+b\,c\,x}\,\left (6\,e^3\,\mathrm {sinh}\left (d+e\,x\right )+3\,e^3\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )+b^3\,c^3\,{\mathrm {cosh}\left (d+e\,x\right )}^3\,{\ln \relax (F)}^3-b\,c\,e^2\,{\mathrm {cosh}\left (d+e\,x\right )}^3\,\ln \relax (F)-6\,b\,c\,e^2\,\mathrm {cosh}\left (d+e\,x\right )\,\ln \relax (F)-3\,b^2\,c^2\,e\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \relax (F)}^2\right )}{b^4\,c^4\,{\ln \relax (F)}^4-10\,b^2\,c^2\,e^2\,{\ln \relax (F)}^2+9\,e^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(a + b*x))*cosh(d + e*x)^3,x)
[Out]
(F^(a*c + b*c*x)*(6*e^3*sinh(d + e*x) + 3*e^3*cosh(d + e*x)^2*sinh(d + e*x) + b^3*c^3*cosh(d + e*x)^3*log(F)^3
- b*c*e^2*cosh(d + e*x)^3*log(F) - 6*b*c*e^2*cosh(d + e*x)*log(F) - 3*b^2*c^2*e*cosh(d + e*x)^2*sinh(d + e*x)
*log(F)^2))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e^2*log(F)^2)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(c*(b*x+a))*cosh(e*x+d)**3,x)
[Out]
Timed out
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