Integrand size = 18, antiderivative size = 199 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 b c e^2 F^{c (a+b x)} \cos (d+e x) \log (F)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 e^3 F^{c (a+b x)} \sin (d+e x)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)} \] Output:
b*c*F^(c*(b*x+a))*cos(e*x+d)^3*ln(F)/(9*e^2+b^2*c^2*ln(F)^2)+6*b*c*e^2*F^( c*(b*x+a))*cos(e*x+d)*ln(F)/(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))*cos(e*x+d)^2*sin(e*x+d)/(9*e^2+b^2*c^2*ln(F)^2)+6*e^3*F ^(c*(b*x+a))*sin(e*x+d)/(9*e^4+10*b^2*c^2*e^2*ln(F)^2+b^4*c^4*ln(F)^4)
Time = 0.41 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (b c \cos (3 (d+e x)) \log (F) \left (e^2+b^2 c^2 \log ^2(F)\right )+3 b c \cos (d+e x) \log (F) \left (9 e^2+b^2 c^2 \log ^2(F)\right )+6 e \left (5 e^2+b^2 c^2 \log ^2(F)+\cos (2 (d+e x)) \left (e^2+b^2 c^2 \log ^2(F)\right )\right ) \sin (d+e x)\right )}{4 \left (9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \] Input:
Integrate[F^(c*(a + b*x))*Cos[d + e*x]^3,x]
Output:
(F^(c*(a + b*x))*(b*c*Cos[3*(d + e*x)]*Log[F]*(e^2 + b^2*c^2*Log[F]^2) + 3 *b*c*Cos[d + e*x]*Log[F]*(9*e^2 + b^2*c^2*Log[F]^2) + 6*e*(5*e^2 + b^2*c^2 *Log[F]^2 + Cos[2*(d + e*x)]*(e^2 + b^2*c^2*Log[F]^2))*Sin[d + e*x]))/(4*( 9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4))
Time = 0.36 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4935, 4933}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^3(d+e x) F^{c (a+b x)} \, dx\) |
| \(\Big \downarrow \) 4935 |
| \(\displaystyle \frac {6 e^2 \int F^{c (a+b x)} \cos (d+e x)dx}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {b c \log (F) \cos ^3(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {3 e \sin (d+e x) \cos ^2(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}\) |
| \(\Big \downarrow \) 4933 |
| \(\displaystyle \frac {b c \log (F) \cos ^3(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {3 e \sin (d+e x) \cos ^2(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {6 e^2 \left (\frac {e \sin (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2}+\frac {b c \log (F) \cos (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2}\right )}{b^2 c^2 \log ^2(F)+9 e^2}\) |
Input:
Int[F^(c*(a + b*x))*Cos[d + e*x]^3,x]
Output:
(b*c*F^(c*(a + b*x))*Cos[d + e*x]^3*Log[F])/(9*e^2 + b^2*c^2*Log[F]^2) + ( 3*e*F^(c*(a + b*x))*Cos[d + e*x]^2*Sin[d + e*x])/(9*e^2 + b^2*c^2*Log[F]^2 ) + (6*e^2*((b*c*F^(c*(a + b*x))*Cos[d + e*x]*Log[F])/(e^2 + b^2*c^2*Log[F ]^2) + (e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)))/(9*e^2 + b^2*c^2*Log[F]^2)
Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Cos[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x ] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] /; F reeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbo l] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Lo g[F]^2)), x] + (Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1) /(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + Simp[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^ 2*Log[F]^2) Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x]) /; FreeQ[{F , a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]
Time = 1.92 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {\left (b c \ln \left (F \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \cos \left (3 e x +3 d \right )+\left (3 b^{2} c^{2} \ln \left (F \right )^{2} e +3 e^{3}\right ) \sin \left (3 e x +3 d \right )+3 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (\ln \left (F \right ) \cos \left (e x +d \right ) b c +e \sin \left (e x +d \right )\right )\right ) F^{c \left (b x +a \right )}}{4 b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+36 e^{4}}\) | \(144\) |
| risch | \(\frac {3 b c \,F^{c \left (b x +a \right )} \cos \left (e x +d \right ) \ln \left (F \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {3 e \,F^{c \left (b x +a \right )} \sin \left (e x +d \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (3 e x +3 d \right )}{4 b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}}+\frac {3 e \,F^{c \left (b x +a \right )} \sin \left (3 e x +3 d \right )}{4 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}\) | \(158\) |
| orering | \(\frac {4 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+5 e^{2}\right ) F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {2 \left (3 b^{2} c^{2} \ln \left (F \right )^{2}+5 e^{2}\right ) \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{3}-3 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{2} e \sin \left (e x +d \right )\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {4 b c \ln \left (F \right ) \left (F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right )^{3}-6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{2} e \sin \left (e x +d \right )+6 F^{c \left (b x +a \right )} \cos \left (e x +d \right ) e^{2} \sin \left (e x +d \right )^{2}-3 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3} e^{2}\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {F^{c \left (b x +a \right )} b^{3} c^{3} \ln \left (F \right )^{3} \cos \left (e x +d \right )^{3}-9 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right )^{2} e \sin \left (e x +d \right )+18 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right ) e^{2} \sin \left (e x +d \right )^{2}-9 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{3} e^{2}-6 F^{c \left (b x +a \right )} e^{3} \sin \left (e x +d \right )^{3}+21 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{2} e^{3} \sin \left (e x +d \right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}\) | \(537\) |
| norman | \(\frac {\frac {\ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+7 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {12 e \left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{3}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {6 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+3 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {6 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+3 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{5}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {\ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+7 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{6}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {3 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{2}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {3 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{4}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}}{\left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{2}\right )^{3}}\) | \(541\) |
Input:
int(F^(c*(b*x+a))*cos(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
(b*c*ln(F)*(e^2+b^2*c^2*ln(F)^2)*cos(3*e*x+3*d)+(3*b^2*c^2*ln(F)^2*e+3*e^3 )*sin(3*e*x+3*d)+3*(9*e^2+b^2*c^2*ln(F)^2)*(ln(F)*cos(e*x+d)*b*c+e*sin(e*x +d)))*F^(c*(b*x+a))/(4*b^4*c^4*ln(F)^4+40*b^2*c^2*e^2*ln(F)^2+36*e^4)
Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {{\left (b^{3} c^{3} \cos \left (e x + d\right )^{3} \log \left (F\right )^{3} + {\left (b c e^{2} \cos \left (e x + d\right )^{3} + 6 \, b c e^{2} \cos \left (e x + d\right )\right )} \log \left (F\right ) + 3 \, {\left (b^{2} c^{2} e \cos \left (e x + d\right )^{2} \log \left (F\right )^{2} + e^{3} \cos \left (e x + d\right )^{2} + 2 \, e^{3}\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 10 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 9 \, e^{4}} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)^3,x, algorithm="fricas")
Output:
(b^3*c^3*cos(e*x + d)^3*log(F)^3 + (b*c*e^2*cos(e*x + d)^3 + 6*b*c*e^2*cos (e*x + d))*log(F) + 3*(b^2*c^2*e*cos(e*x + d)^2*log(F)^2 + e^3*cos(e*x + d )^2 + 2*e^3)*sin(e*x + d))*F^(b*c*x + a*c)/(b^4*c^4*log(F)^4 + 10*b^2*c^2* e^2*log(F)^2 + 9*e^4)
Result contains complex when optimal does not.
Time = 3.89 (sec) , antiderivative size = 1676, normalized size of antiderivative = 8.42 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*cos(e*x+d)**3,x)
Output:
Piecewise((x*cos(d)**3, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*cos(d)**3, Eq(b, 0) & Eq(e, 0)), (x*cos(d)**3, Eq(c, 0) & Eq(e, 0)), (3*I*F**(a*c + b*c*x) *x*sin(I*b*c*x*log(F) - d)**3/8 + 3*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d)/8 + 3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log (F) - d)*cos(I*b*c*x*log(F) - d)**2/8 + 3*F**(a*c + b*c*x)*x*cos(I*b*c*x*l og(F) - d)**3/8 - 3*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F) - d)**3/(8*b*c*l og(F)) - 3*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*log(F) - d)**2/(4*b*c*log(F)) - F**(a*c + b*c*x)*cos(I*b*c*x*log(F) - d)**3/(8*b*c *log(F)), Eq(e, -I*b*c*log(F))), (-I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F) /3 - d)**3/8 - 3*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 - d)**2*cos(I*b*c *x*log(F)/3 - d)/8 + 3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 - d)*cos( I*b*c*x*log(F)/3 - d)**2/8 + F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/3 - d)* *3/8 + I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 - d)**3/(8*b*c*log(F)) + 3* I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 - d)*cos(I*b*c*x*log(F)/3 - d)**2/ (4*b*c*log(F)) + 9*F**(a*c + b*c*x)*cos(I*b*c*x*log(F)/3 - d)**3/(8*b*c*lo g(F)), Eq(e, -I*b*c*log(F)/3)), (-I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/ 3 + d)**3/8 - 3*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 + d)**2*cos(I*b*c* x*log(F)/3 + d)/8 + 3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 + d)*cos(I *b*c*x*log(F)/3 + d)**2/8 + F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/3 + d)** 3/8 + 11*I*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 + d)**3/(8*b*c*log(F))...
Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (199) = 398\).
Time = 0.08 (sec) , antiderivative size = 813, normalized size of antiderivative = 4.09 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx =\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)^3,x, algorithm="maxima")
Output:
1/8*((F^(a*c)*b^3*c^3*cos(3*d)*log(F)^3 + 3*F^(a*c)*b^2*c^2*e*log(F)^2*sin (3*d) + F^(a*c)*b*c*e^2*cos(3*d)*log(F) + 3*F^(a*c)*e^3*sin(3*d))*F^(b*c*x )*cos(3*e*x) + (F^(a*c)*b^3*c^3*cos(3*d)*log(F)^3 - 3*F^(a*c)*b^2*c^2*e*lo g(F)^2*sin(3*d) + F^(a*c)*b*c*e^2*cos(3*d)*log(F) - 3*F^(a*c)*e^3*sin(3*d) )*F^(b*c*x)*cos(3*e*x + 6*d) + 3*(F^(a*c)*b^3*c^3*cos(3*d)*log(F)^3 - F^(a *c)*b^2*c^2*e*log(F)^2*sin(3*d) + 9*F^(a*c)*b*c*e^2*cos(3*d)*log(F) - 9*F^ (a*c)*e^3*sin(3*d))*F^(b*c*x)*cos(e*x + 4*d) + 3*(F^(a*c)*b^3*c^3*cos(3*d) *log(F)^3 + F^(a*c)*b^2*c^2*e*log(F)^2*sin(3*d) + 9*F^(a*c)*b*c*e^2*cos(3* d)*log(F) + 9*F^(a*c)*e^3*sin(3*d))*F^(b*c*x)*cos(e*x - 2*d) - (F^(a*c)*b^ 3*c^3*log(F)^3*sin(3*d) - 3*F^(a*c)*b^2*c^2*e*cos(3*d)*log(F)^2 + F^(a*c)* b*c*e^2*log(F)*sin(3*d) - 3*F^(a*c)*e^3*cos(3*d))*F^(b*c*x)*sin(3*e*x) + ( F^(a*c)*b^3*c^3*log(F)^3*sin(3*d) + 3*F^(a*c)*b^2*c^2*e*cos(3*d)*log(F)^2 + F^(a*c)*b*c*e^2*log(F)*sin(3*d) + 3*F^(a*c)*e^3*cos(3*d))*F^(b*c*x)*sin( 3*e*x + 6*d) + 3*(F^(a*c)*b^3*c^3*log(F)^3*sin(3*d) + F^(a*c)*b^2*c^2*e*co s(3*d)*log(F)^2 + 9*F^(a*c)*b*c*e^2*log(F)*sin(3*d) + 9*F^(a*c)*e^3*cos(3* d))*F^(b*c*x)*sin(e*x + 4*d) - 3*(F^(a*c)*b^3*c^3*log(F)^3*sin(3*d) - F^(a *c)*b^2*c^2*e*cos(3*d)*log(F)^2 + 9*F^(a*c)*b*c*e^2*log(F)*sin(3*d) - 9*F^ (a*c)*e^3*cos(3*d))*F^(b*c*x)*sin(e*x - 2*d))/(b^4*c^4*cos(3*d)^2*log(F)^4 + b^4*c^4*log(F)^4*sin(3*d)^2 + 9*(cos(3*d)^2 + sin(3*d)^2)*e^4 + 10*(b^2 *c^2*cos(3*d)^2*log(F)^2 + b^2*c^2*log(F)^2*sin(3*d)^2)*e^2)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.39 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)^3,x, algorithm="giac")
Output:
1/4*(2*b*c*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 + 3*e*x + 3*d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn (F) - pi*b*c + 6*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 6*e)*sin(1/2*pi*b*c*x*s gn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + 3*e*x + 3*d)/(4*b^ 2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 6*e)^2))*e^(b*c*x*log(abs( F)) + a*c*log(abs(F))) + 3/4*(2*b*c*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 + e*x + d)*log(abs(F))/(4*b^2*c^2*log(ab s(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 2*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 2* e)*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 + e*x + 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))) + 3/4*(2*b*c*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 - e*x - d)*log(abs(F))/ (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)*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 - e*x - d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - p i*b*c - 2*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 1/4*(2*b*c*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 - 3*e* x - 3*d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 6*e)^2) + (pi*b*c*sgn(F) - pi*b*c - 6*e)*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 - 3*e*x - 3*d)/(4*b^2*c^2*log(ab...
Time = 20.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.96 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )+\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )+\sin \left (d\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,\left (e-b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )-\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )-\sin \left (3\,d\right )\,1{}\mathrm {i}\right )}{8\,\left (-b\,c\,\ln \left (F\right )+e\,3{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )+\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )+\sin \left (3\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (3\,e-b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {3\,F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )-\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )-\sin \left (d\right )\,1{}\mathrm {i}\right )}{8\,\left (-b\,c\,\ln \left (F\right )+e\,1{}\mathrm {i}\right )} \] Input:
int(F^(c*(a + b*x))*cos(d + e*x)^3,x)
Output:
- (F^(c*(a + b*x))*(cos(e*x) + sin(e*x)*1i)*(cos(d) + sin(d)*1i)*3i)/(8*(e - b*c*log(F)*1i)) - (F^(c*(a + b*x))*(cos(3*e*x) - sin(3*e*x)*1i)*(cos(3* d) - sin(3*d)*1i))/(8*(e*3i - b*c*log(F))) - (F^(c*(a + b*x))*(cos(3*e*x) + sin(3*e*x)*1i)*(cos(3*d) + sin(3*d)*1i)*1i)/(8*(3*e - b*c*log(F)*1i)) - (3*F^(c*(a + b*x))*(cos(e*x) - sin(e*x)*1i)*(cos(d) - sin(d)*1i))/(8*(e*1i - b*c*log(F)))
\[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cos \left (e x +d \right )^{3}d x \right ) \] Input:
int(F^(c*(b*x+a))*cos(e*x+d)^3,x)
Output:
f**(a*c)*int(f**(b*c*x)*cos(d + e*x)**3,x)