\(\int F^{c (a+b x)} \sinh ^3(d+e x) \, dx\) [322]

Optimal result

Integrand size = 18, antiderivative size = 202 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {6 e^3 F^{c (a+b x)} \cosh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {6 b c e^2 F^{c (a+b x)} \log (F) \sinh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^3(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)} \]

[Out]

-6*e^3*F^(c*(b*x+a))*cosh(e*x+d)/(9*e^4-10*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln(F)^4)+6*b*c*e^2*F^(c*(b*x+a))*ln(F)*
sinh(e*x+d)/(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)*sinh(e*x+d)^2/(9*e^2-
b^2*c^2*ln(F)^2)-b*c*F^(c*(b*x+a))*ln(F)*sinh(e*x+d)^3/(9*e^2-b^2*c^2*ln(F)^2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 202, 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 = {5584, 5582} \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {b c \log (F) \sinh ^3(d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {3 e \sinh ^2(d+e x) \cosh (d+e x) F^{c (a+b x)}}{9 e^2-b^2 c^2 \log ^2(F)}+\frac {6 b c e^2 \log (F) \sinh (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)-10 b^2 c^2 e^2 \log ^2(F)+9 e^4}-\frac {6 e^3 \cosh (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)-10 b^2 c^2 e^2 \log ^2(F)+9 e^4} \]

[In]

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

[Out]

(-6*e^3*F^(c*(a + b*x))*Cosh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (6*b*c*e^2*F^(c*
(a + b*x))*Log[F]*Sinh[d + e*x])/(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))*C
osh[d + e*x]*Sinh[d + e*x]^2)/(9*e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^3)/(9*e^2
- b^2*c^2*Log[F]^2)

Rule 5582

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x)
)*(Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Cosh[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 5584

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Sinh[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))*Sinh[d + e*x]^(n - 2), x], x] + Simp[e*n*F^(c*(a + b*x))*Cosh[d + e*x]*(Sinh[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*} \text {integral}& = \frac {3 e F^{c (a+b x)} \cosh (d+e x) \sinh ^2(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^3(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)} \sinh (d+e x) \, dx}{9 e^2-b^2 c^2 \log ^2(F)} \\ & = -\frac {6 e^3 F^{c (a+b x)} \cosh (d+e x)}{9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {6 b c e^2 F^{c (a+b x)} \log (F) \sinh (d+e x)}{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 (d+e x) \sinh ^2(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^3(d+e x)}{9 e^2-b^2 c^2 \log ^2(F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (3 \cosh (3 (d+e x)) \left (e^3-b^2 c^2 e \log ^2(F)\right )+3 \cosh (d+e x) \left (-9 e^3+b^2 c^2 e \log ^2(F)\right )+2 b c \log (F) \left (13 e^2-b^2 c^2 \log ^2(F)+\cosh (2 (d+e x)) \left (-e^2+b^2 c^2 \log ^2(F)\right )\right ) \sinh (d+e x)\right )}{4 \left (9 e^4-10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \]

[In]

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

[Out]

(F^(c*(a + b*x))*(3*Cosh[3*(d + e*x)]*(e^3 - b^2*c^2*e*Log[F]^2) + 3*Cosh[d + e*x]*(-9*e^3 + b^2*c^2*e*Log[F]^
2) + 2*b*c*Log[F]*(13*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))

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73

method	result	size
parallelrisch	\(-\frac {3 \left (\left (\ln \left (F \right )^{2} b^{2} c^{2} e -e^{3}\right ) \cosh \left (3 e x +3 d \right )+\frac {\left (-\ln \left (F \right )^{3} b^{3} c^{3}+\ln \left (F \right ) b c \,e^{2}\right ) \sinh \left (3 e x +3 d \right )}{3}+\left (b c \ln \left (F \right )-3 e \right ) \left (b c \ln \left (F \right )+3 e \right ) \left (\sinh \left (e x +d \right ) \ln \left (F \right ) b c -e \cosh \left (e x +d \right )\right )\right ) F^{c \left (b x +a \right )}}{4 \left (9 e^{4}-10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}\right )}\)	\(147\)
risch	\(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{6 e x +6 d}-3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{4 e x +4 d}-3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{6 e x +6 d}+3 \ln \left (F \right )^{3} b^{3} c^{3} {\mathrm e}^{2 e x +2 d}+3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{4 e x +4 d}-\ln \left (F \right ) b c \,e^{2} {\mathrm e}^{6 e x +6 d}-\ln \left (F \right )^{3} b^{3} c^{3}+3 \ln \left (F \right )^{2} b^{2} c^{2} e \,{\mathrm e}^{2 e x +2 d}+27 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{4 e x +4 d}+3 e^{3} {\mathrm e}^{6 e x +6 d}-3 \ln \left (F \right )^{2} b^{2} c^{2} e -27 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{2 e x +2 d}-27 e^{3} {\mathrm e}^{4 e x +4 d}+\ln \left (F \right ) b c \,e^{2}-27 e^{3} {\mathrm e}^{2 e x +2 d}+3 e^{3}\right ) {\mathrm e}^{-3 e x -3 d} F^{c \left (b x +a \right )}}{8 \left (b c \ln \left (F \right )-e \right ) \left (b c \ln \left (F \right )-3 e \right ) \left (e +b c \ln \left (F \right )\right ) \left (b c \ln \left (F \right )+3 e \right )}\)	\(326\)

[In]

int(F^(c*(b*x+a))*sinh(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

-3/4*((ln(F)^2*b^2*c^2*e-e^3)*cosh(3*e*x+3*d)+1/3*(-ln(F)^3*b^3*c^3+ln(F)*b*c*e^2)*sinh(3*e*x+3*d)+(b*c*ln(F)-
3*e)*(b*c*ln(F)+3*e)*(sinh(e*x+d)*ln(F)*b*c-e*cosh(e*x+d)))*F^(c*(b*x+a))/(9*e^4-10*b^2*c^2*e^2*ln(F)^2+b^4*c^
4*ln(F)^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2228 vs. \(2 (199) = 398\).

Time = 0.36 (sec) , antiderivative size = 2228, normalized size of antiderivative = 11.03 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \]

[In]

integrate(F^(c*(b*x+a))*sinh(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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1525 vs. \(2 (199) = 398\).

Time = 7.77 (sec) , antiderivative size = 1525, normalized size of antiderivative = 7.55 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((x*sinh(d)**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sinh(d)**3, Eq(b, 0) & Eq(e, 0)), (x*sinh(d)**3, Eq
(c, 0) & Eq(e, 0)), (-3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F
) - d)**2*cosh(b*c*x*log(F) - d)/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) - d)*cosh(b*c*x*log(F) - d)**2/8 -
3*F**(a*c + b*c*x)*x*cosh(b*c*x*log(F) - d)**3/8 + F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)**3/(8*b*c*log(F))
- 3*F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)**2*cosh(b*c*x*log(F) - d)/(4*b*c*log(F)) + 3*F**(a*c + b*c*x)*cosh
(b*c*x*log(F) - d)**3/(8*b*c*log(F)), Eq(e, -b*c*log(F))), (-F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)**3/8
+ 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 - d)**2*cosh(b*c*x*log(F)/3 - d)/8 - 3*F**(a*c + b*c*x)*x*sinh(b*c*
x*log(F)/3 - d)*cosh(b*c*x*log(F)/3 - d)**2/8 + F**(a*c + b*c*x)*x*cosh(b*c*x*log(F)/3 - d)**3/8 - 9*F**(a*c +
b*c*x)*sinh(b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)) + 3*F**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 - d)**2*cosh(b*c*x
*log(F)/3 - d)/(4*b*c*log(F)) - F**(a*c + b*c*x)*cosh(b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)), Eq(e, -b*c*log(F)
/3)), (F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 + d)**3/8 - 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 + d)**2*cos
h(b*c*x*log(F)/3 + d)/8 + 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/3 + d)*cosh(b*c*x*log(F)/3 + d)**2/8 - F**(a*
c + b*c*x)*x*cosh(b*c*x*log(F)/3 + d)**3/8 - F**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 + d)**3/(8*b*c*log(F)) + 3*F
**(a*c + b*c*x)*sinh(b*c*x*log(F)/3 + d)**2*cosh(b*c*x*log(F)/3 + d)/(b*c*log(F)) - 15*F**(a*c + b*c*x)*sinh(b
*c*x*log(F)/3 + d)*cosh(b*c*x*log(F)/3 + d)**2/(4*b*c*log(F)) + 11*F**(a*c + b*c*x)*cosh(b*c*x*log(F)/3 + d)**
3/(8*b*c*log(F)), Eq(e, b*c*log(F)/3)), (3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) + d)**3/8 - 3*F**(a*c + b*c*x)
*x*sinh(b*c*x*log(F) + d)**2*cosh(b*c*x*log(F) + d)/8 - 3*F**(a*c + b*c*x)*x*sinh(b*c*x*log(F) + d)*cosh(b*c*x
*log(F) + d)**2/8 + 3*F**(a*c + b*c*x)*x*cosh(b*c*x*log(F) + d)**3/8 - F**(a*c + b*c*x)*sinh(b*c*x*log(F) + d)
**3/(8*b*c*log(F)) + 3*F**(a*c + b*c*x)*sinh(b*c*x*log(F) + d)**2*cosh(b*c*x*log(F) + d)/(4*b*c*log(F)) - 3*F*
*(a*c + b*c*x)*cosh(b*c*x*log(F) + d)**3/(8*b*c*log(F)), Eq(e, b*c*log(F))), (F**(a*c + b*c*x)*b**3*c**3*log(F
)**3*sinh(d + e*x)**3/(b**4*c**4*log(F)**4 - 10*b**2*c**2*e**2*log(F)**2 + 9*e**4) - 3*F**(a*c + b*c*x)*b**2*c
**2*e*log(F)**2*sinh(d + e*x)**2*cosh(d + e*x)/(b**4*c**4*log(F)**4 - 10*b**2*c**2*e**2*log(F)**2 + 9*e**4) -
7*F**(a*c + b*c*x)*b*c*e**2*log(F)*sinh(d + e*x)**3/(b**4*c**4*log(F)**4 - 10*b**2*c**2*e**2*log(F)**2 + 9*e**
4) + 6*F**(a*c + b*c*x)*b*c*e**2*log(F)*sinh(d + e*x)*cosh(d + e*x)**2/(b**4*c**4*log(F)**4 - 10*b**2*c**2*e**
2*log(F)**2 + 9*e**4) + 9*F**(a*c + b*c*x)*e**3*sinh(d + e*x)**2*cosh(d + e*x)/(b**4*c**4*log(F)**4 - 10*b**2*
c**2*e**2*log(F)**2 + 9*e**4) - 6*F**(a*c + b*c*x)*e**3*cosh(d + e*x)**3/(b**4*c**4*log(F)**4 - 10*b**2*c**2*e
**2*log(F)**2 + 9*e**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.66 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 3 \, e x + 3 \, d\right )}}{8 \, {\left (b c \log \left (F\right ) + 3 \, e\right )}} - \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{8 \, {\left (b c \log \left (F\right ) + e\right )}} + \frac {3 \, F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{8 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 3 \, e x\right )}}{8 \, {\left (b c e^{\left (3 \, d\right )} \log \left (F\right ) - 3 \, e e^{\left (3 \, d\right )}\right )}} \]

[In]

integrate(F^(c*(b*x+a))*sinh(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))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 1211, normalized size of antiderivative = 6.00 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=\text {Too large to display} \]

[In]

integrate(F^(c*(b*x+a))*sinh(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) + I*(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) - 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(ab
s(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)*cos(-1/2*p
i*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)
+ 3*I*(-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) + 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(a
bs(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) + 3*I*(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) - 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(abs(F)) + (b*c*log(abs(F
)) - 3*e)*x - 3*d) + I*(-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) + 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)

Mupad [B] (verification not implemented)

Time = 2.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.82 \[ \int F^{c (a+b x)} \sinh ^3(d+e x) \, dx=-\frac {F^{a\,c+b\,c\,x}\,\left (-b^3\,c^3\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,e\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\,{\ln \left (F\right )}^2-6\,b\,c\,e^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )+7\,b\,c\,e^2\,{\mathrm {sinh}\left (d+e\,x\right )}^3\,\ln \left (F\right )+6\,e^3\,{\mathrm {cosh}\left (d+e\,x\right )}^3-9\,e^3\,\mathrm {cosh}\left (d+e\,x\right )\,{\mathrm {sinh}\left (d+e\,x\right )}^2\right )}{b^4\,c^4\,{\ln \left (F\right )}^4-10\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+9\,e^4} \]

[In]

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

[Out]

-(F^(a*c + b*c*x)*(6*e^3*cosh(d + e*x)^3 - 9*e^3*cosh(d + e*x)*sinh(d + e*x)^2 - b^3*c^3*sinh(d + e*x)^3*log(F
)^3 + 7*b*c*e^2*sinh(d + e*x)^3*log(F) + 3*b^2*c^2*e*cosh(d + e*x)*sinh(d + e*x)^2*log(F)^2 - 6*b*c*e^2*cosh(d
+ e*x)^2*sinh(d + e*x)*log(F)))/(9*e^4 + b^4*c^4*log(F)^4 - 10*b^2*c^2*e^2*log(F)^2)