3.893 \(\int F^{c (a+b x)} (f+i f \sinh (d+e x))^2 \, dx\)

Optimal. Leaf size=254 \[ \frac {b c f^2 \log (F) \sinh ^2(d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {2 i b c f^2 \log (F) \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 i e f^2 \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}-\frac {2 e f^2 \sinh (d+e x) \cosh (d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {f^2 F^{a c+b c x}}{b c \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.41, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6741, 12, 6742, 2194, 5474, 5476} \[ \frac {b c f^2 \log (F) \sinh ^2(d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {2 i b c f^2 \log (F) \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac {2 i e f^2 \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}-\frac {2 e f^2 \sinh (d+e x) \cosh (d+e x) F^{a c+b c x}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {f^2 F^{a c+b c x}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^2*F^(a*c + b*c*x))/(b*c*Log[F]) + ((2*I)*e*f^2*F^(a*c + b*c*x)*Cosh[d + e*x])/(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)) - ((2*I)*b*c*f^2*F^(a*c + b*c*x)*Log[F]*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*L
og[F]^2) + (b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sinh[d + e*x]^2)/(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 5474

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 5476

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]

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

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Mathematica [A]  time = 1.92, size = 196, normalized size = 0.77 \[ \frac {F^{c (a+b x)} (f+i f \sinh (d+e x))^2 \left (-\frac {2 e \sinh (2 (d+e x))}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh (2 (d+e x))}{b^2 c^2 \log ^2(F)-4 e^2}+\frac {4 i b c \log (F) \sinh (d+e x)}{(b c \log (F)-e) (b c \log (F)+e)}+\frac {4 i e \cosh (d+e x)}{(e-b c \log (F)) (b c \log (F)+e)}+\frac {3}{b c \log (F)}\right )}{2 \left (\cosh \left (\frac {1}{2} (d+e x)\right )+i \sinh \left (\frac {1}{2} (d+e x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 442, normalized size = 1.74 \[ \frac {{\left (24 \, e^{4} f^{2} e^{\left (2 \, e x + 2 \, d\right )} - {\left (b^{4} c^{4} f^{2} e^{\left (4 \, e x + 4 \, d\right )} - 4 i \, b^{4} c^{4} f^{2} e^{\left (3 \, e x + 3 \, d\right )} - 6 \, b^{4} c^{4} f^{2} e^{\left (2 \, e x + 2 \, d\right )} + 4 i \, b^{4} c^{4} f^{2} e^{\left (e x + d\right )} + b^{4} c^{4} f^{2}\right )} \log \relax (F)^{4} + {\left (2 \, b^{3} c^{3} e f^{2} e^{\left (4 \, e x + 4 \, d\right )} - 4 i \, b^{3} c^{3} e f^{2} e^{\left (3 \, e x + 3 \, d\right )} - 4 i \, b^{3} c^{3} e f^{2} e^{\left (e x + d\right )} - 2 \, b^{3} c^{3} e f^{2}\right )} \log \relax (F)^{3} + {\left (b^{2} c^{2} e^{2} f^{2} e^{\left (4 \, e x + 4 \, d\right )} - 16 i \, b^{2} c^{2} e^{2} f^{2} e^{\left (3 \, e x + 3 \, d\right )} - 30 \, b^{2} c^{2} e^{2} f^{2} e^{\left (2 \, e x + 2 \, d\right )} + 16 i \, b^{2} c^{2} e^{2} f^{2} e^{\left (e x + d\right )} + b^{2} c^{2} e^{2} f^{2}\right )} \log \relax (F)^{2} - {\left (2 \, b c e^{3} f^{2} e^{\left (4 \, e x + 4 \, d\right )} - 16 i \, b c e^{3} f^{2} e^{\left (3 \, e x + 3 \, d\right )} - 16 i \, b c e^{3} f^{2} e^{\left (e x + d\right )} - 2 \, b c e^{3} f^{2}\right )} \log \relax (F)\right )} F^{b c x + a c}}{4 \, {\left (b^{5} c^{5} e^{\left (2 \, e x + 2 \, d\right )} \log \relax (F)^{5} - 5 \, b^{3} c^{3} e^{2} e^{\left (2 \, e x + 2 \, d\right )} \log \relax (F)^{3} + 4 \, b c e^{4} e^{\left (2 \, e x + 2 \, d\right )} \log \relax (F)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(24*e^4*f^2*e^(2*e*x + 2*d) - (b^4*c^4*f^2*e^(4*e*x + 4*d) - 4*I*b^4*c^4*f^2*e^(3*e*x + 3*d) - 6*b^4*c^4*f
^2*e^(2*e*x + 2*d) + 4*I*b^4*c^4*f^2*e^(e*x + d) + b^4*c^4*f^2)*log(F)^4 + (2*b^3*c^3*e*f^2*e^(4*e*x + 4*d) -
4*I*b^3*c^3*e*f^2*e^(3*e*x + 3*d) - 4*I*b^3*c^3*e*f^2*e^(e*x + d) - 2*b^3*c^3*e*f^2)*log(F)^3 + (b^2*c^2*e^2*f
^2*e^(4*e*x + 4*d) - 16*I*b^2*c^2*e^2*f^2*e^(3*e*x + 3*d) - 30*b^2*c^2*e^2*f^2*e^(2*e*x + 2*d) + 16*I*b^2*c^2*
e^2*f^2*e^(e*x + d) + b^2*c^2*e^2*f^2)*log(F)^2 - (2*b*c*e^3*f^2*e^(4*e*x + 4*d) - 16*I*b*c*e^3*f^2*e^(3*e*x +
3*d) - 16*I*b*c*e^3*f^2*e^(e*x + d) - 2*b*c*e^3*f^2)*log(F))*F^(b*c*x + a*c)/(b^5*c^5*e^(2*e*x + 2*d)*log(F)^
5 - 5*b^3*c^3*e^2*e^(2*e*x + 2*d)*log(F)^3 + 4*b*c*e^4*e^(2*e*x + 2*d)*log(F))

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giac [B]  time = 0.30, size = 1574, normalized size = 6.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(f+I*f*sinh(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*((pi*b*c*sgn(F) - pi*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + e)^2) + 2*(b*c*log(abs(F)) + e)*f^2*sin(-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))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 1/2*(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*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + 2*((pi*b*c*sgn(F) - pi*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)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*
(b*c*log(abs(F)) - e)^2) + 2*(b*c*log(abs(F)) - e)*f^2*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sg
n(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*(-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*log(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) - p
i*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)

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maple [A]  time = 0.65, size = 434, normalized size = 1.71 \[ \frac {f^{2} \left (-4 i \ln \relax (F )^{3} b^{3} c^{3} e \,{\mathrm e}^{3 e x +3 d}-\ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{4 e x +4 d}+16 i \ln \relax (F ) b c \,e^{3} {\mathrm e}^{3 e x +3 d}+6 \ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{2 e x +2 d}+16 i \ln \relax (F ) b c \,e^{3} {\mathrm e}^{e x +d}+2 \ln \relax (F )^{3} b^{3} c^{3} e \,{\mathrm e}^{4 e x +4 d}-b^{4} c^{4} \ln \relax (F )^{4}+16 i \ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{e x +d}-4 i \ln \relax (F )^{3} b^{3} c^{3} e \,{\mathrm e}^{e x +d}+\ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{4 e x +4 d}-2 \ln \relax (F )^{3} b^{3} c^{3} e +4 i \ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{3 e x +3 d}-30 \ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{2 e x +2 d}-16 i \ln \relax (F )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{3 e x +3 d}-2 \ln \relax (F ) b c \,e^{3} {\mathrm e}^{4 e x +4 d}+b^{2} c^{2} e^{2} \ln \relax (F )^{2}-4 i \ln \relax (F )^{4} b^{4} c^{4} {\mathrm e}^{e x +d}+2 \ln \relax (F ) b c \,e^{3}+24 e^{4} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 b c \ln \relax (F ) \left (e -b c \ln \relax (F )\right ) \left (2 e -b c \ln \relax (F )\right ) \left (e +b c \ln \relax (F )\right ) \left (b c \ln \relax (F )+2 e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 189, normalized size = 0.74 \[ -\frac {1}{4} \, f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \relax (F) + 2 \, e x + 2 \, d\right )}}{b c \log \relax (F) + 2 \, e} + \frac {F^{a c} e^{\left (b c x \log \relax (F) - 2 \, e x\right )}}{b c e^{\left (2 \, d\right )} \log \relax (F) - 2 \, e e^{\left (2 \, d\right )}} - \frac {2 \, F^{b c x + a c}}{b c \log \relax (F)}\right )} + i \, f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \relax (F) + e x + d\right )}}{b c \log \relax (F) + e} - \frac {F^{a c} e^{\left (b c x \log \relax (F) - e x\right )}}{b c e^{d} \log \relax (F) - e e^{d}}\right )} + \frac {F^{b c x + a c} f^{2}}{b c \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(f+I*f*sinh(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))) + I*f^2*(F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c
*log(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 = 2.78, size = 252, normalized size = 0.99 \[ \frac {F^{c\,\left (a+b\,x\right )}\,f^2\,\left (12\,e^4+3\,b^4\,c^4\,{\ln \relax (F)}^4+b^4\,c^4\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \relax (F)}^4\,4{}\mathrm {i}-b^4\,c^4\,{\ln \relax (F)}^4\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )-15\,b^2\,c^2\,e^2\,{\ln \relax (F)}^2+2\,b^3\,c^3\,e\,{\ln \relax (F)}^3\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )-b^2\,c^2\,e^2\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \relax (F)}^2\,16{}\mathrm {i}-2\,b\,c\,e^3\,\ln \relax (F)\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )+b^2\,c^2\,e^2\,{\ln \relax (F)}^2\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )-b^3\,c^3\,e\,\mathrm {cosh}\left (d+e\,x\right )\,{\ln \relax (F)}^3\,4{}\mathrm {i}+b\,c\,e^3\,\mathrm {cosh}\left (d+e\,x\right )\,\ln \relax (F)\,16{}\mathrm {i}\right )}{2\,b\,c\,\ln \relax (F)\,\left (b^4\,c^4\,{\ln \relax (F)}^4-5\,b^2\,c^2\,e^2\,{\ln \relax (F)}^2+4\,e^4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))*(f + f*sinh(d + e*x)*1i)^2,x)

[Out]

(F^(c*(a + b*x))*f^2*(12*e^4 + 3*b^4*c^4*log(F)^4 + b^4*c^4*sinh(d + e*x)*log(F)^4*4i - b^4*c^4*log(F)^4*cosh(
2*d + 2*e*x) - 15*b^2*c^2*e^2*log(F)^2 + 2*b^3*c^3*e*log(F)^3*sinh(2*d + 2*e*x) - b^2*c^2*e^2*sinh(d + e*x)*lo
g(F)^2*16i - 2*b*c*e^3*log(F)*sinh(2*d + 2*e*x) + b^2*c^2*e^2*log(F)^2*cosh(2*d + 2*e*x) - b^3*c^3*e*cosh(d +
e*x)*log(F)^3*4i + b*c*e^3*cosh(d + e*x)*log(F)*16i))/(2*b*c*log(F)*(4*e^4 + b^4*c^4*log(F)^4 - 5*b^2*c^2*e^2*
log(F)^2))

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sympy [A]  time = 119.10, size = 1731, normalized size = 6.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*(f+I*f*sinh(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*I*f**2*cosh(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)*si
nh(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) + z
oo*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*I*f**2*cosh(d + e*x)/e), Eq(b, 0)), (-f**2*x*si
nh(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*I*f**2*cosh
(d + e*x)/e, Eq(c, 0)), (-F**(a*c)*F**(b*c*x)*b**4*c**4*f**2*log(F)**4*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*I*F**(a*c)*F**(b*c*x)*b**4*c**4*f**2*log(F)**4*sinh(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*I*F**(a*c)*F**(b*c*x)*b**3*c**3*e*f**2*log(F)**3*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)) + 3*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)) - 8*I*F**(a*c)*F**(b*c*x)*b**2*c**2*e
**2*f**2*log(F)**2*sinh(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**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)) - 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*I*F**(a*c)*F**(b*c*x)*b*c*e**3*
f**2*log(F)*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)*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*lo
g(F)), True))

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