3.9.98 \(\int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx\) [898]

Optimal. Leaf size=101 \[ \frac {f F^{a c+b c x}}{b c \log (F)}-\frac {b c f F^{a c+b c x} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}+\frac {e f F^{a c+b c x} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6873, 12, 6874, 2225, 5583} \begin {gather*} \frac {e f \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c f \log (F) \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac {f F^{a c+b c x}}{b c \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f*F^(a*c + b*c*x))/(b*c*Log[F]) - (b*c*f*F^(a*c + b*c*x)*Cosh[d + e*x]*Log[F])/(e^2 - b^2*c^2*Log[F]^2) + (e*
f*F^(a*c + b*c*x)*Sinh[d + e*x])/(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 5583

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

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Mathematica [A]
time = 0.13, size = 88, normalized size = 0.87 \begin {gather*} \frac {f F^{c (a+b x)} \left (-e^2+b^2 c^2 \log ^2(F)+b^2 c^2 \cosh (d+e x) \log ^2(F)-b c e \log (F) \sinh (d+e x)\right )}{b c \log (F) (-e+b c \log (F)) (e+b c \log (F))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.15, size = 135, normalized size = 1.34

method	result	size
risch	\(\frac {f \left (\ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}+2 \ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{e x +d}+b^{2} c^{2} \ln \left (F \right )^{2}-\ln \left (F \right ) b c e \,{\mathrm e}^{2 e x +2 d}+\ln \left (F \right ) b c e -2 e^{2} {\mathrm e}^{e x +d}\right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 b c \ln \left (F \right ) \left (b c \ln \left (F \right )-e \right ) \left (e +b c \ln \left (F \right )\right )}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 91, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, f {\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}{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+f*cosh(e*x+d)),x, algorithm="maxima")

[Out]

1/2*f*(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/(b*c*log(F))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (104) = 208\).
time = 0.36, size = 685, normalized size = 6.78 \begin {gather*} \frac {{\left ({\left (b^{2} c^{2} f \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + 2 \, b^{2} c^{2} f \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2} + {\left (b^{2} c^{2} f \log \left (F\right )^{2} - {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right )\right )} \log \left (F\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - 2 \, {\left (f \cosh \left (1\right )^{2} + 2 \, f \cosh \left (1\right ) \sinh \left (1\right ) + f \sinh \left (1\right )^{2}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right ) - {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2}\right )} \log \left (F\right ) - 2 \, {\left (f \cosh \left (1\right )^{2} + {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right ) - {\left (b^{2} c^{2} f \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2} + 2 \, f \cosh \left (1\right ) \sinh \left (1\right ) + f \sinh \left (1\right )^{2}\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left ({\left (b^{2} c^{2} f \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + 2 \, b^{2} c^{2} f \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2} + {\left (b^{2} c^{2} f \log \left (F\right )^{2} - {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right )\right )} \log \left (F\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - 2 \, {\left (f \cosh \left (1\right )^{2} + 2 \, f \cosh \left (1\right ) \sinh \left (1\right ) + f \sinh \left (1\right )^{2}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right ) - {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2}\right )} \log \left (F\right ) - 2 \, {\left (f \cosh \left (1\right )^{2} + {\left (b c f \cosh \left (1\right ) + b c f \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right ) - {\left (b^{2} c^{2} f \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2} + 2 \, f \cosh \left (1\right ) \sinh \left (1\right ) + f \sinh \left (1\right )^{2}\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \, {\left (b^{3} c^{3} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right )^{3} - {\left (b c \cosh \left (1\right )^{2} + 2 \, b c \cosh \left (1\right ) \sinh \left (1\right ) + b c \sinh \left (1\right )^{2}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right ) + {\left (b^{3} c^{3} \log \left (F\right )^{3} - {\left (b c \cosh \left (1\right )^{2} + 2 \, b c \cosh \left (1\right ) \sinh \left (1\right ) + b c \sinh \left (1\right )^{2}\right )} \log \left (F\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(((b^2*c^2*f*cosh(x*cosh(1) + x*sinh(1) + d)^2 + 2*b^2*c^2*f*cosh(x*cosh(1) + x*sinh(1) + d) + b^2*c^2*f)*
log(F)^2 + (b^2*c^2*f*log(F)^2 - (b*c*f*cosh(1) + b*c*f*sinh(1))*log(F))*sinh(x*cosh(1) + x*sinh(1) + d)^2 - 2
*(f*cosh(1)^2 + 2*f*cosh(1)*sinh(1) + f*sinh(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d) + (b*c*f*cosh(1) + b*c*f*si
nh(1) - (b*c*f*cosh(1) + b*c*f*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2)*log(F) - 2*(f*cosh(1)^2 + (b*c*f*co
sh(1) + b*c*f*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)*log(F) - (b^2*c^2*f*cosh(x*cosh(1) + x*sinh(1) + d) + b
^2*c^2*f)*log(F)^2 + 2*f*cosh(1)*sinh(1) + f*sinh(1)^2)*sinh(x*cosh(1) + x*sinh(1) + d))*cosh((b*c*x + a*c)*lo
g(F)) + ((b^2*c^2*f*cosh(x*cosh(1) + x*sinh(1) + d)^2 + 2*b^2*c^2*f*cosh(x*cosh(1) + x*sinh(1) + d) + b^2*c^2*
f)*log(F)^2 + (b^2*c^2*f*log(F)^2 - (b*c*f*cosh(1) + b*c*f*sinh(1))*log(F))*sinh(x*cosh(1) + x*sinh(1) + d)^2
- 2*(f*cosh(1)^2 + 2*f*cosh(1)*sinh(1) + f*sinh(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d) + (b*c*f*cosh(1) + b*c*f
*sinh(1) - (b*c*f*cosh(1) + b*c*f*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2)*log(F) - 2*(f*cosh(1)^2 + (b*c*f
*cosh(1) + b*c*f*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)*log(F) - (b^2*c^2*f*cosh(x*cosh(1) + x*sinh(1) + d)
+ b^2*c^2*f)*log(F)^2 + 2*f*cosh(1)*sinh(1) + f*sinh(1)^2)*sinh(x*cosh(1) + x*sinh(1) + d))*sinh((b*c*x + a*c)
*log(F)))/(b^3*c^3*cosh(x*cosh(1) + x*sinh(1) + d)*log(F)^3 - (b*c*cosh(1)^2 + 2*b*c*cosh(1)*sinh(1) + b*c*sin
h(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d)*log(F) + (b^3*c^3*log(F)^3 - (b*c*cosh(1)^2 + 2*b*c*cosh(1)*sinh(1) +
b*c*sinh(1)^2)*log(F))*sinh(x*cosh(1) + x*sinh(1) + d))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (94) = 188\).
time = 2.58, size = 852, normalized size = 8.44 \begin {gather*} \begin {cases} f x + \frac {f \sinh {\left (d + e x \right )}}{e} & \text {for}\: F = 1 \\\frac {b^{2} c^{2} f \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {e}{b c}} \right )}^{2} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{- \frac {e}{b c}} \right )}} + \frac {b^{2} c^{2} f \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {e}{b c}} \right )}^{2}}{b^{3} c^{3} \log {\left (e^{- \frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{- \frac {e}{b c}} \right )}} - \frac {b c e f \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {e}{b c}} \right )} \sinh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{- \frac {e}{b c}} \right )}} - \frac {e^{2} f \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x}}{b^{3} c^{3} \log {\left (e^{- \frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{- \frac {e}{b c}} \right )}} & \text {for}\: F = e^{- \frac {e}{b c}} \\\frac {b^{2} c^{2} f \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \log {\left (e^{\frac {e}{b c}} \right )}^{2} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{\frac {e}{b c}} \right )}} + \frac {b^{2} c^{2} f \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \log {\left (e^{\frac {e}{b c}} \right )}^{2}}{b^{3} c^{3} \log {\left (e^{\frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{\frac {e}{b c}} \right )}} - \frac {b c e f \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \log {\left (e^{\frac {e}{b c}} \right )} \sinh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{\frac {e}{b c}} \right )}} - \frac {e^{2} f \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x}}{b^{3} c^{3} \log {\left (e^{\frac {e}{b c}} \right )}^{3} - b c e^{2} \log {\left (e^{\frac {e}{b c}} \right )}} & \text {for}\: F = e^{\frac {e}{b c}} \\F^{a c} \left (f x + \frac {f \sinh {\left (d + e x \right )}}{e}\right ) & \text {for}\: b = 0 \\f x + \frac {f \sinh {\left (d + e x \right )}}{e} & \text {for}\: c = 0 \\\frac {F^{a c} F^{b c x} b^{2} c^{2} f \log {\left (F \right )}^{2} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} + \frac {F^{a c} F^{b c x} b^{2} c^{2} f \log {\left (F \right )}^{2}}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} - \frac {F^{a c} F^{b c x} b c e f \log {\left (F \right )} \sinh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} - \frac {F^{a c} F^{b c x} e^{2} f}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((f*x + f*sinh(d + e*x)/e, Eq(F, 1)), (b**2*c**2*f*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(ex
p(-e/(b*c)))**2*cosh(d + e*x)/(b**3*c**3*log(exp(-e/(b*c)))**3 - b*c*e**2*log(exp(-e/(b*c)))) + b**2*c**2*f*ex
p(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))**2/(b**3*c**3*log(exp(-e/(b*c)))**3 - b*c*e**2*lo
g(exp(-e/(b*c)))) - b*c*e*f*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))*sinh(d + e*x)/(b**3
*c**3*log(exp(-e/(b*c)))**3 - b*c*e**2*log(exp(-e/(b*c)))) - e**2*f*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x
)/(b**3*c**3*log(exp(-e/(b*c)))**3 - b*c*e**2*log(exp(-e/(b*c)))), Eq(F, exp(-e/(b*c)))), (b**2*c**2*f*exp(e/(
b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*log(exp(e/(b*c)))**2*cosh(d + e*x)/(b**3*c**3*log(exp(e/(b*c)))**3 - b*c*e*
*2*log(exp(e/(b*c)))) + b**2*c**2*f*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*log(exp(e/(b*c)))**2/(b**3*c**3*
log(exp(e/(b*c)))**3 - b*c*e**2*log(exp(e/(b*c)))) - b*c*e*f*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*log(exp
(e/(b*c)))*sinh(d + e*x)/(b**3*c**3*log(exp(e/(b*c)))**3 - b*c*e**2*log(exp(e/(b*c)))) - e**2*f*exp(e/(b*c))**
(a*c)*exp(e/(b*c))**(b*c*x)/(b**3*c**3*log(exp(e/(b*c)))**3 - b*c*e**2*log(exp(e/(b*c)))), Eq(F, exp(e/(b*c)))
), (F**(a*c)*(f*x + f*sinh(d + e*x)/e), Eq(b, 0)), (f*x + f*sinh(d + e*x)/e, Eq(c, 0)), (F**(a*c)*F**(b*c*x)*b
**2*c**2*f*log(F)**2*cosh(d + e*x)/(b**3*c**3*log(F)**3 - b*c*e**2*log(F)) + F**(a*c)*F**(b*c*x)*b**2*c**2*f*l
og(F)**2/(b**3*c**3*log(F)**3 - b*c*e**2*log(F)) - F**(a*c)*F**(b*c*x)*b*c*e*f*log(F)*sinh(d + e*x)/(b**3*c**3
*log(F)**3 - b*c*e**2*log(F)) - F**(a*c)*F**(b*c*x)*e**2*f/(b**3*c**3*log(F)**3 - b*c*e**2*log(F)), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 886, normalized size = 8.77 \begin {gather*} 2 \, {\left (\frac {2 \, b c f \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right ) \log \left ({\left | F \right |}\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} f \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2}}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + i \, {\left (\frac {i \, f e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{i \, \pi b c \mathrm {sgn}\left (F\right ) - i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right )} - \frac {i \, f e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-i \, \pi b c \mathrm {sgn}\left (F\right ) + i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right )}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + {\left (\frac {2 \, {\left (b c \log \left ({\left | F \right |}\right ) + e\right )} f \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) + e\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} f \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) + e\right )}^{2}}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) + e\right )} x + d\right )} + i \, {\left (\frac {i \, f e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{2 i \, \pi b c \mathrm {sgn}\left (F\right ) - 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right ) + 4 \, e} - \frac {i \, f e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-2 i \, \pi b c \mathrm {sgn}\left (F\right ) + 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right ) + 4 \, e}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) + e\right )} x + d\right )} + {\left (\frac {2 \, {\left (b c \log \left ({\left | F \right |}\right ) - e\right )} f \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) - e\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} f \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) - e\right )}^{2}}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) - e\right )} x - d\right )} + i \, {\left (\frac {i \, f e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{2 i \, \pi b c \mathrm {sgn}\left (F\right ) - 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right ) - 4 \, e} - \frac {i \, f e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-2 i \, \pi b c \mathrm {sgn}\left (F\right ) + 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right ) - 4 \, e}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) - e\right )} x - d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*b*c*f*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*lo
g(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*f*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(abs(F)
) + a*c*log(abs(F))) + I*(I*f*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))) - I*f*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))))*e^(b*c*x*log(abs(F)) + a*c*log(ab
s(F))) + (2*(b*c*log(abs(F)) + e)*f*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*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*(I*f*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)) + 4*e) - I*f*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(ab
s(F)) + 4*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + (2*(b*c*log(abs(F)) - e)*f*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*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)/((p
i*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*(
I*f*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*p
i*b*c + 4*b*c*log(abs(F)) - 4*e) - I*f*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)) - 4*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) -
e)*x - d)

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Mupad [B]
time = 1.74, size = 134, normalized size = 1.33 \begin {gather*} -\frac {F^{b\,c\,x}\,F^{a\,c}\,f\,{\mathrm {e}}^{-d-e\,x}\,\left (b^2\,c^2\,{\ln \left (F\right )}^2-2\,e^2\,{\mathrm {e}}^{d+e\,x}+b\,c\,e\,\ln \left (F\right )+2\,b^2\,c^2\,{\mathrm {e}}^{d+e\,x}\,{\ln \left (F\right )}^2+b^2\,c^2\,{\mathrm {e}}^{2\,d+2\,e\,x}\,{\ln \left (F\right )}^2-b\,c\,e\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\ln \left (F\right )\right )}{2\,b\,c\,\ln \left (F\right )\,\left (e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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