3.9.93 \(\int F^{c (a+b x)} (f+i f \sinh (d+e x))^2 \, dx\) [893]

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

[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 = 0.30, 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 = {6873, 12, 6874, 2225, 5582, 5584} \begin {gather*} \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)} \end {gather*}

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 2225

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 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]

Rule 6873

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

Rule 6874

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 = 0.88, size = 196, normalized size = 0.77 \begin {gather*} \frac {F^{c (a+b x)} (f+i f \sinh (d+e x))^2 \left (\frac {3}{b c \log (F)}+\frac {4 i e \cosh (d+e x)}{(e-b c \log (F)) (e+b c \log (F))}-\frac {b c \cosh (2 (d+e x)) \log (F)}{-4 e^2+b^2 c^2 \log ^2(F)}+\frac {4 i b c \log (F) \sinh (d+e x)}{(-e+b c \log (F)) (e+b c \log (F))}-\frac {2 e \sinh (2 (d+e x))}{4 e^2-b^2 c^2 \log ^2(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} \end {gather*}

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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Maple [A]
time = 4.36, size = 434, normalized size = 1.71

method	result	size
risch	\(\frac {f^{2} \left (16 i \ln \left (F \right ) b c \,e^{3} {\mathrm e}^{e x +d}-\ln \left (F \right )^{4} b^{4} c^{4} {\mathrm e}^{4 e x +4 d}+16 i \ln \left (F \right ) b c \,e^{3} {\mathrm e}^{3 e x +3 d}+6 \ln \left (F \right )^{4} b^{4} c^{4} {\mathrm e}^{2 e x +2 d}+4 i \ln \left (F \right )^{4} b^{4} c^{4} {\mathrm e}^{3 e x +3 d}+2 \ln \left (F \right )^{3} b^{3} c^{3} e \,{\mathrm e}^{4 e x +4 d}-b^{4} c^{4} \ln \left (F \right )^{4}+16 i \ln \left (F \right )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{e x +d}-4 i \ln \left (F \right )^{3} b^{3} c^{3} e \,{\mathrm e}^{3 e x +3 d}+\ln \left (F \right )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{4 e x +4 d}-2 \ln \left (F \right )^{3} b^{3} c^{3} e -4 i \ln \left (F \right )^{3} b^{3} c^{3} e \,{\mathrm e}^{e x +d}-30 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{2 e x +2 d}-4 i \ln \left (F \right )^{4} b^{4} c^{4} {\mathrm e}^{e x +d}-2 \ln \left (F \right ) b c \,e^{3} {\mathrm e}^{4 e x +4 d}+b^{2} c^{2} e^{2} \ln \left (F \right )^{2}-16 i \ln \left (F \right )^{2} b^{2} c^{2} e^{2} {\mathrm e}^{3 e x +3 d}+2 \ln \left (F \right ) b c \,e^{3}+24 e^{4} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 b c \ln \left (F \right ) \left (e -b c \ln \left (F \right )\right ) \left (2 e -b c \ln \left (F \right )\right ) \left (e +b c \ln \left (F \right )\right ) \left (b c \ln \left (F \right )+2 e \right )}\)	\(434\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 197, normalized size = 0.78 \begin {gather*} -\frac {1}{4} \, f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, x e + 2 \, d\right )}}{b c \log \left (F\right ) + 2 \, e} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, x e\right )}}{b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e^{\left (2 \, d + 1\right )}} - \frac {2 \, F^{b c x + a c}}{b c \log \left (F\right )}\right )} + i \, f^{2} {\left (\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + x e + d\right )}}{b c \log \left (F\right ) + e} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - x e\right )}}{b c e^{d} \log \left (F\right ) - e^{\left (d + 1\right )}}\right )} + \frac {F^{b c x + a c} f^{2}}{b c \log \left (F\right )} \end {gather*}

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

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Fricas [A]
time = 0.35, size = 438, normalized size = 1.72 \begin {gather*} -\frac {{\left ({\left (b^{4} c^{4} f^{2} e^{\left (4 \, x e + 4 \, d\right )} - 4 i \, b^{4} c^{4} f^{2} e^{\left (3 \, x e + 3 \, d\right )} - 6 \, b^{4} c^{4} f^{2} e^{\left (2 \, x e + 2 \, d\right )} + 4 i \, b^{4} c^{4} f^{2} e^{\left (x e + d\right )} + b^{4} c^{4} f^{2}\right )} \log \left (F\right )^{4} + 2 \, {\left (b^{3} c^{3} f^{2} e - b^{3} c^{3} f^{2} e^{\left (4 \, x e + 4 \, d + 1\right )} + 2 i \, b^{3} c^{3} f^{2} e^{\left (3 \, x e + 3 \, d + 1\right )} + 2 i \, b^{3} c^{3} f^{2} e^{\left (x e + d + 1\right )}\right )} \log \left (F\right )^{3} - 24 \, f^{2} e^{\left (2 \, x e + 2 \, d + 4\right )} - {\left (b^{2} c^{2} f^{2} e^{2} + b^{2} c^{2} f^{2} e^{\left (4 \, x e + 4 \, d + 2\right )} - 16 i \, b^{2} c^{2} f^{2} e^{\left (3 \, x e + 3 \, d + 2\right )} - 30 \, b^{2} c^{2} f^{2} e^{\left (2 \, x e + 2 \, d + 2\right )} + 16 i \, b^{2} c^{2} f^{2} e^{\left (x e + d + 2\right )}\right )} \log \left (F\right )^{2} - 2 \, {\left (b c f^{2} e^{3} - b c f^{2} e^{\left (4 \, x e + 4 \, d + 3\right )} + 8 i \, b c f^{2} e^{\left (3 \, x e + 3 \, d + 3\right )} + 8 i \, b c f^{2} e^{\left (x e + d + 3\right )}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{4 \, {\left (b^{5} c^{5} e^{\left (2 \, x e + 2 \, d\right )} \log \left (F\right )^{5} - 5 \, b^{3} c^{3} e^{\left (2 \, x e + 2 \, d + 2\right )} \log \left (F\right )^{3} + 4 \, b c e^{\left (2 \, x e + 2 \, d + 4\right )} \log \left (F\right )\right )}} \end {gather*}

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*((b^4*c^4*f^2*e^(4*x*e + 4*d) - 4*I*b^4*c^4*f^2*e^(3*x*e + 3*d) - 6*b^4*c^4*f^2*e^(2*x*e + 2*d) + 4*I*b^4
*c^4*f^2*e^(x*e + d) + b^4*c^4*f^2)*log(F)^4 + 2*(b^3*c^3*f^2*e - b^3*c^3*f^2*e^(4*x*e + 4*d + 1) + 2*I*b^3*c^
3*f^2*e^(3*x*e + 3*d + 1) + 2*I*b^3*c^3*f^2*e^(x*e + d + 1))*log(F)^3 - 24*f^2*e^(2*x*e + 2*d + 4) - (b^2*c^2*
f^2*e^2 + b^2*c^2*f^2*e^(4*x*e + 4*d + 2) - 16*I*b^2*c^2*f^2*e^(3*x*e + 3*d + 2) - 30*b^2*c^2*f^2*e^(2*x*e + 2
*d + 2) + 16*I*b^2*c^2*f^2*e^(x*e + d + 2))*log(F)^2 - 2*(b*c*f^2*e^3 - b*c*f^2*e^(4*x*e + 4*d + 3) + 8*I*b*c*
f^2*e^(3*x*e + 3*d + 3) + 8*I*b*c*f^2*e^(x*e + d + 3))*log(F))*F^(b*c*x + a*c)/(b^5*c^5*e^(2*x*e + 2*d)*log(F)
^5 - 5*b^3*c^3*e^(2*x*e + 2*d + 2)*log(F)^3 + 4*b*c*e^(2*x*e + 2*d + 4)*log(F))

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 7759 vs. \(2 (241) = 482\).
time = 78.75, size = 7759, normalized size = 30.55 \begin {gather*} \text {Too large to display} \end {gather*}

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)), (-b**4*c**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*
log(exp(-2*e/(b*c)))**4*sinh(d + e*x)**2/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b
*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) + 2*I*b**4*c**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*
x)*log(exp(-2*e/(b*c)))**4*sinh(d + e*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b
*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) + b**4*c**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*l
og(exp(-2*e/(b*c)))**4/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b*c)))**3 + 4*b*c*e
**4*log(exp(-2*e/(b*c)))) + 2*b**3*c**3*e*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*log(exp(-2*e/(b
*c)))**3*sinh(d + e*x)*cosh(d + e*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b*c))
)**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) - 2*I*b**3*c**3*e*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)
*log(exp(-2*e/(b*c)))**3*cosh(d + e*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b*c
)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) + 3*b**2*c**2*e**2*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c
*x)*log(exp(-2*e/(b*c)))**2*sinh(d + e*x)**2/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*
e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) - 8*I*b**2*c**2*e**2*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c
))**(b*c*x)*log(exp(-2*e/(b*c)))**2*sinh(d + e*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(ex
p(-2*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) - 2*b**2*c**2*e**2*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(
b*c))**(b*c*x)*log(exp(-2*e/(b*c)))**2*cosh(d + e*x)**2/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*
log(exp(-2*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) - 5*b**2*c**2*e**2*f**2*exp(-2*e/(b*c))**(a*c)*exp(
-2*e/(b*c))**(b*c*x)*log(exp(-2*e/(b*c)))**2/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*
e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) - 2*b*c*e**3*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c
*x)*log(exp(-2*e/(b*c)))*sinh(d + e*x)*cosh(d + e*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log
(exp(-2*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) + 8*I*b*c*e**3*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b
*c))**(b*c*x)*log(exp(-2*e/(b*c)))*cosh(d + e*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp
(-2*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) - 2*e**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c
*x)*sinh(d + e*x)**2/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b*c)))**3 + 4*b*c*e**
4*log(exp(-2*e/(b*c)))) + 2*e**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*cosh(d + e*x)**2/(b**5*c
**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-2*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))) + 4*
e**4*f**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)/(b**5*c**5*log(exp(-2*e/(b*c)))**5 - 5*b**3*c**3*e**
2*log(exp(-2*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*e/(b*c)))), Eq(F, exp(-2*e/(b*c)))), (-b**4*c**4*f**2*exp(-e
/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))**4*sinh(d + e*x)**2/(b**5*c**5*log(exp(-e/(b*c)))**5
- 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3 + 4*b*c*e**4*log(exp(-e/(b*c)))) + 2*I*b**4*c**4*f**2*exp(-e/(b*c))**
(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))**4*sinh(d + e*x)/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**
3*e**2*log(exp(-e/(b*c)))**3 + 4*b*c*e**4*log(exp(-e/(b*c)))) + b**4*c**4*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*
c))**(b*c*x)*log(exp(-e/(b*c)))**4/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3 +
4*b*c*e**4*log(exp(-e/(b*c)))) + 2*b**3*c**3*e*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b
*c)))**3*sinh(d + e*x)*cosh(d + e*x)/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3
+ 4*b*c*e**4*log(exp(-e/(b*c)))) - 2*I*b**3*c**3*e*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-
e/(b*c)))**3*cosh(d + e*x)/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3 + 4*b*c*e
**4*log(exp(-e/(b*c)))) + 3*b**2*c**2*e**2*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))
**2*sinh(d + e*x)**2/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3 + 4*b*c*e**4*lo
g(exp(-e/(b*c)))) - 8*I*b**2*c**2*e**2*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))**2*
sinh(d + e*x)/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3 + 4*b*c*e**4*log(exp(-
e/(b*c)))) - 2*b**2*c**2*e**2*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))**2*cosh(d +
e*x)**2/(b**5*c**5*log(exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e/(b*c)))**3 + 4*b*c*e**4*log(exp(-e/(b*c
)))) - 5*b**2*c**2*e**2*f**2*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))**2/(b**5*c**5*log(
exp(-e/(b*c)))**5 - 5*b**3*c**3*e**2*log(exp(-e...

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (250) = 500\).
time = 0.45, size = 1546, normalized size = 6.09 \begin {gather*} \text {Too large to display} \end {gather*}

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))) + 3*I*(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))) - 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(ab
s(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) + I*(-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) + 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*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) - (-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) - 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*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) - (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) + 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*p
i*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) + I*(-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) + 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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Mupad [B]
time = 2.78, size = 252, normalized size = 0.99 \begin {gather*} \frac {F^{c\,\left (a+b\,x\right )}\,f^2\,\left (12\,e^4+3\,b^4\,c^4\,{\ln \left (F\right )}^4+b^4\,c^4\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \left (F\right )}^4\,4{}\mathrm {i}-b^4\,c^4\,{\ln \left (F\right )}^4\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )-15\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+2\,b^3\,c^3\,e\,{\ln \left (F\right )}^3\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )-b^2\,c^2\,e^2\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \left (F\right )}^2\,16{}\mathrm {i}-2\,b\,c\,e^3\,\ln \left (F\right )\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )+b^2\,c^2\,e^2\,{\ln \left (F\right )}^2\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )-b^3\,c^3\,e\,\mathrm {cosh}\left (d+e\,x\right )\,{\ln \left (F\right )}^3\,4{}\mathrm {i}+b\,c\,e^3\,\mathrm {cosh}\left (d+e\,x\right )\,\ln \left (F\right )\,16{}\mathrm {i}\right )}{2\,b\,c\,\ln \left (F\right )\,\left (b^4\,c^4\,{\ln \left (F\right )}^4-5\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+4\,e^4\right )} \end {gather*}

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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