3.135 \(\int F^{c (a+b x)} (f+f \sin (d+e x))^2 \, dx\)

Optimal. Leaf size=245 \[ \frac{b c f^2 \log (F) \sin ^2(d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac{2 b c f^2 \log (F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac{2 e f^2 \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac{2 e f^2 \sin (d+e x) \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac{2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (b^2 c^2 \log ^2(F)+4 e^2\right )}+\frac{f^2 F^{a c+b c x}}{b c \log (F)} \]

[Out]

(f^2*F^(a*c + b*c*x))/(b*c*Log[F]) - (2*e*f^2*F^(a*c + b*c*x)*Cos[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*b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x])
/(e^2 + b^2*c^2*Log[F]^2) - (2*e*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2) + (
b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x]^2)/(4*e^2 + b^2*c^2*Log[F]^2)

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Rubi [A]  time = 0.358488, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6741, 12, 6742, 2194, 4432, 4434} \[ \frac{b c f^2 \log (F) \sin ^2(d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac{2 b c f^2 \log (F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac{2 e f^2 \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac{2 e f^2 \sin (d+e x) \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac{2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (b^2 c^2 \log ^2(F)+4 e^2\right )}+\frac{f^2 F^{a c+b c x}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^2*F^(a*c + b*c*x))/(b*c*Log[F]) - (2*e*f^2*F^(a*c + b*c*x)*Cos[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*b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x])
/(e^2 + b^2*c^2*Log[F]^2) - (2*e*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2) + (
b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x]^2)/(4*e^2 + b^2*c^2*Log[F]^2)

Rule 6741

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

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

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 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[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 4434

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*
x))*Sin[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))*Sin[d + e*x]^(n - 2), x], x] - Simp[(e*n*F^(c*(a + b*x))*Cos[d + e*x]*Sin[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)} (f+f \sin (d+e x))^2 \, dx &=\int f^2 F^{a c+b c x} (1+\sin (d+e x))^2 \, dx\\ &=f^2 \int F^{a c+b c x} (1+\sin (d+e x))^2 \, dx\\ &=f^2 \int \left (F^{a c+b c x}+2 F^{a c+b c x} \sin (d+e x)+F^{a c+b c x} \sin ^2(d+e x)\right ) \, dx\\ &=f^2 \int F^{a c+b c x} \, dx+f^2 \int F^{a c+b c x} \sin ^2(d+e x) \, dx+\left (2 f^2\right ) \int F^{a c+b c x} \sin (d+e x) \, dx\\ &=\frac{f^2 F^{a c+b c x}}{b c \log (F)}-\frac{2 e f^2 F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac{2 b c f^2 F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac{2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (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) \sin ^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 e f^2 F^{a c+b c x} \cos (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 b c f^2 F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}-\frac{2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (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) \sin ^2(d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}\\ \end{align*}

Mathematica [A]  time = 1.60363, size = 180, normalized size = 0.73 \[ \frac{f^2 (\sin (d+e x)+1)^2 F^{c (a+b x)} \left (\frac{4 b c \log (F) \sin (d+e x)}{b^2 c^2 \log ^2(F)+e^2}-\frac{2 e \sin (2 (d+e x))}{b^2 c^2 \log ^2(F)+4 e^2}-\frac{4 e \cos (d+e x)}{b^2 c^2 \log ^2(F)+e^2}-\frac{b c \log (F) \cos (2 (d+e x))}{b^2 c^2 \log ^2(F)+4 e^2}+\frac{3}{b c \log (F)}\right )}{2 \left (\sin \left (\frac{1}{2} (d+e x)\right )+\cos \left (\frac{1}{2} (d+e x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^2*F^(c*(a + b*x))*(1 + Sin[d + e*x])^2*(3/(b*c*Log[F]) - (4*e*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) - (b*c
*Cos[2*(d + e*x)]*Log[F])/(4*e^2 + b^2*c^2*Log[F]^2) + (4*b*c*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) -
(2*e*Sin[2*(d + e*x)])/(4*e^2 + b^2*c^2*Log[F]^2)))/(2*(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^4)

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Maple [A]  time = 0.099, size = 368, normalized size = 1.5 \begin{align*}{\frac{3\,{F}^{ac}{f}^{2}{F}^{bcx}}{2\,bc\ln \left ( F \right ) }}-{\frac{{F}^{ac}{f}^{2}\ln \left ( F \right ) bc{{\rm e}^{bcx\ln \left ( F \right ) }}}{ \left ( 2+2\, \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) \left ( 4\,{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ) }}-2\,{\frac{{F}^{ac}{f}^{2}e{{\rm e}^{bcx\ln \left ( F \right ) }}\tan \left ( ex+d \right ) }{ \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) \left ( 4\,{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ) }}+{\frac{{F}^{ac}{f}^{2}\ln \left ( F \right ) bc{{\rm e}^{bcx\ln \left ( F \right ) }} \left ( \tan \left ( ex+d \right ) \right ) ^{2}}{ \left ( 2+2\, \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) \left ( 4\,{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ) }}-2\,{\frac{{F}^{ac}{f}^{2}e{{\rm e}^{bcx\ln \left ( F \right ) }}}{ \left ( 1+ \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2} \right ) \left ({e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ) }}+2\,{\frac{{F}^{ac}{f}^{2}e{{\rm e}^{bcx\ln \left ( F \right ) }} \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}}{ \left ( 1+ \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2} \right ) \left ({e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ) }}+4\,{\frac{{F}^{ac}{f}^{2}\ln \left ( F \right ) bc{{\rm e}^{bcx\ln \left ( F \right ) }}\tan \left ( d/2+1/2\,ex \right ) }{ \left ( 1+ \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2} \right ) \left ({e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.1844, size = 784, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(f+f*sin(e*x+d))^2,x, algorithm="maxima")

[Out]

-1/4*((F^(a*c)*b^2*c^2*cos(2*d)*log(F)^2 + 2*F^(a*c)*b*c*e*log(F)*sin(2*d))*F^(b*c*x)*cos(2*e*x) + (F^(a*c)*b^
2*c^2*cos(2*d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(2*d))*F^(b*c*x)*cos(2*e*x + 4*d) - (F^(a*c)*b^2*c^2*log(F
)^2*sin(2*d) - 2*F^(a*c)*b*c*e*cos(2*d)*log(F))*F^(b*c*x)*sin(2*e*x) + (F^(a*c)*b^2*c^2*log(F)^2*sin(2*d) + 2*
F^(a*c)*b*c*e*cos(2*d)*log(F))*F^(b*c*x)*sin(2*e*x + 4*d) - 2*(F^(a*c)*b^2*c^2*cos(2*d)^2*log(F)^2 + F^(a*c)*b
^2*c^2*log(F)^2*sin(2*d)^2 + 4*(F^(a*c)*cos(2*d)^2 + F^(a*c)*sin(2*d)^2)*e^2)*F^(b*c*x))*f^2/(b^3*c^3*cos(2*d)
^2*log(F)^3 + b^3*c^3*log(F)^3*sin(2*d)^2 + 4*(b*c*cos(2*d)^2*log(F) + b*c*log(F)*sin(2*d)^2)*e^2) - ((F^(a*c)
*b*c*log(F)*sin(d) + F^(a*c)*e*cos(d))*F^(b*c*x)*cos(e*x + 2*d) - (F^(a*c)*b*c*log(F)*sin(d) - F^(a*c)*e*cos(d
))*F^(b*c*x)*cos(e*x) - (F^(a*c)*b*c*cos(d)*log(F) - F^(a*c)*e*sin(d))*F^(b*c*x)*sin(e*x + 2*d) - (F^(a*c)*b*c
*cos(d)*log(F) + F^(a*c)*e*sin(d))*F^(b*c*x)*sin(e*x))*f^2/(b^2*c^2*cos(d)^2*log(F)^2 + b^2*c^2*log(F)^2*sin(d
)^2 + (cos(d)^2 + sin(d)^2)*e^2) + F^(b*c*x + a*c)*f^2/(b*c*log(F))

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Fricas [A]  time = 0.511159, size = 581, normalized size = 2.37 \begin{align*} -\frac{{\left (2 \, b^{3} c^{3} e f^{2} \cos \left (e x + d\right ) \log \left (F\right )^{3} + 8 \, b c e^{3} f^{2} \cos \left (e x + d\right ) \log \left (F\right ) - 6 \, e^{4} f^{2} +{\left (b^{4} c^{4} f^{2} \cos \left (e x + d\right )^{2} - 2 \, b^{4} c^{4} f^{2}\right )} \log \left (F\right )^{4} +{\left (b^{2} c^{2} e^{2} f^{2} \cos \left (e x + d\right )^{2} - 8 \, b^{2} c^{2} e^{2} f^{2}\right )} \log \left (F\right )^{2} - 2 \,{\left (b^{4} c^{4} f^{2} \log \left (F\right )^{4} - b^{3} c^{3} e f^{2} \cos \left (e x + d\right ) \log \left (F\right )^{3} + 4 \, b^{2} c^{2} e^{2} f^{2} \log \left (F\right )^{2} - b c e^{3} f^{2} \cos \left (e x + d\right ) \log \left (F\right )\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5} + 5 \, b^{3} c^{3} e^{2} \log \left (F\right )^{3} + 4 \, b c e^{4} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^3*c^3*e*f^2*cos(e*x + d)*log(F)^3 + 8*b*c*e^3*f^2*cos(e*x + d)*log(F) - 6*e^4*f^2 + (b^4*c^4*f^2*cos(e*x
+ d)^2 - 2*b^4*c^4*f^2)*log(F)^4 + (b^2*c^2*e^2*f^2*cos(e*x + d)^2 - 8*b^2*c^2*e^2*f^2)*log(F)^2 - 2*(b^4*c^4
*f^2*log(F)^4 - b^3*c^3*e*f^2*cos(e*x + d)*log(F)^3 + 4*b^2*c^2*e^2*f^2*log(F)^2 - b*c*e^3*f^2*cos(e*x + d)*lo
g(F))*sin(e*x + d))*F^(b*c*x + a*c)/(b^5*c^5*log(F)^5 + 5*b^3*c^3*e^2*log(F)^3 + 4*b*c*e^4*log(F))

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Sympy [A]  time = 162.303, size = 1760, normalized size = 7.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*(f+f*sin(e*x+d))**2,x)

[Out]

Piecewise((f**2*x*sin(d + e*x)**2/2 + f**2*x*cos(d + e*x)**2/2 + f**2*x - f**2*sin(d + e*x)*cos(d + e*x)/(2*e)
- 2*f**2*cos(d + e*x)/e, Eq(F, 1)), (zoo*e**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*sin(d
+ e*x)**2 + zoo*e**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*sin(d + e*x)*cos(d + e*x) + zoo*
e**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*cos(d + e*x)**2, Eq(F, exp(-2*I*e/(b*c)))), (zoo
*e**4*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*sin(d + e*x) + zoo*e**4*f**2*exp(-I*e/(b*c))**(a*c)
*exp(-I*e/(b*c))**(b*c*x)*cos(d + e*x), Eq(F, exp(-I*e/(b*c)))), (zoo*e**4*f**2*exp(I*e/(b*c))**(a*c)*exp(I*e/
(b*c))**(b*c*x)*sin(d + e*x) + zoo*e**4*f**2*exp(I*e/(b*c))**(a*c)*exp(I*e/(b*c))**(b*c*x)*cos(d + e*x), Eq(F,
exp(I*e/(b*c)))), (zoo*e**4*f**2*exp(2*I*e/(b*c))**(a*c)*exp(2*I*e/(b*c))**(b*c*x)*sin(d + e*x)**2 + zoo*e**4
*f**2*exp(2*I*e/(b*c))**(a*c)*exp(2*I*e/(b*c))**(b*c*x)*sin(d + e*x)*cos(d + e*x) + zoo*e**4*f**2*exp(2*I*e/(b
*c))**(a*c)*exp(2*I*e/(b*c))**(b*c*x)*cos(d + e*x)**2, Eq(F, exp(2*I*e/(b*c)))), (F**(a*c)*(f**2*x*sin(d + e*x
)**2/2 + f**2*x*cos(d + e*x)**2/2 + f**2*x - f**2*sin(d + e*x)*cos(d + e*x)/(2*e) - 2*f**2*cos(d + e*x)/e), Eq
(b, 0)), (f**2*x*sin(d + e*x)**2/2 + f**2*x*cos(d + e*x)**2/2 + f**2*x - f**2*sin(d + e*x)*cos(d + e*x)/(2*e)
- 2*f**2*cos(d + e*x)/e, Eq(c, 0)), (F**(a*c)*F**(b*c*x)*b**4*c**4*f**2*log(F)**4*sin(d + e*x)**2/(b**5*c**5*l
og(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**4*c**4*f**2*log(F)**4*si
n(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*sin(d + e*x)*cos(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**3*c**3*e*f**2*log(F)**3*cos(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*sin(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*F**(a*c)*F**(b*c*x)*b**2*c**2*e*
*2*f**2*log(F)**2*sin(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*cos(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)*sin(d + e*x)*cos(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*F**(a*c)*F**(b*c*x)*b*c*e**3*f**2*lo
g(F)*cos(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*sin(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*cos(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*log(F)), Tru
e))

________________________________________________________________________________________

Giac [C]  time = 1.45405, size = 2395, normalized size = 9.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(f+f*sin(e*x+d))^2,x, algorithm="giac")

[Out]

-1/2*(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 + 2*x*e + 2*d)*log(abs
(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 4*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 4*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 + 2*x*e + 2*d)/(4*b^2*c^2*log(abs(F))^2 + (pi
*b*c*sgn(F) - pi*b*c + 4*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 1/2*(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 - 2*x*e - 2*d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b
*c*sgn(F) - pi*b*c - 4*e)^2) + (pi*b*c*sgn(F) - pi*b*c - 4*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 - 2*x*e - 2*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 4*e)^2))*e^(b*
c*x*log(abs(F)) + a*c*log(abs(F))) + 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(abs(F)) + a*c*log(abs(F))) + 2*(2*b*c*f^2*log(abs(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 + x*e + d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b
*c + 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c + 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 + x*e + d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 2*e)^2))*e^(b*c*x*log(abs(F)) + a*
c*log(abs(F))) - 2*(2*b*c*f^2*log(abs(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 - x*e - d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c - 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 - x*e - d)/(4*b^2*c^2*log(abs(F))^2
+ (pi*b*c*sgn(F) - pi*b*c - 2*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 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 + 2*I*x*e + 2*I*d)/(4*I*pi*b*c*sgn(F) - 4*I*pi
*b*c + 8*b*c*log(abs(F)) + 16*I*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 - 2*I*x*e - 2*I*d)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*I*e))*e^(b*c*x*log
(abs(F)) + a*c*log(abs(F))) + 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*x*e + I*d)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) + 2*I*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*x*e - I*d)/(-I*pi*b*c*sgn(F) + I*pi*b*c +
2*b*c*log(abs(F)) - 2*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 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*x*e - I*d)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(ab
s(F)) - 2*I*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*x
*e + I*d)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F)) + 2*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) -
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 - 2*I*x*e - 2*I
*d)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*I*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 + 2*I*x*e + 2*I*d)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(
abs(F)) + 16*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 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)/(2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F))) + 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)/(-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*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F))) + 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))))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F)))