Integrand size = 24, antiderivative size = 112 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=-\frac {F^{c (a+b x)} (e \cos (d+e x)-b c \log (F) \sin (d+e x))}{4 \left (e^2+b^2 c^2 \log ^2(F)\right )}-\frac {F^{c (a+b x)} (3 e \cos (3 d+3 e x)-b c \log (F) \sin (3 d+3 e x))}{4 \left (9 e^2+b^2 c^2 \log ^2(F)\right )} \] Output:
-1/4*F^(c*(b*x+a))*(e*cos(e*x+d)-b*c*ln(F)*sin(e*x+d))/(e^2+b^2*c^2*ln(F)^ 2)-F^(c*(b*x+a))*(3*e*cos(3*e*x+3*d)-b*c*ln(F)*sin(3*e*x+3*d))/(36*e^2+4*b ^2*c^2*ln(F)^2)
Time = 0.84 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=\frac {1}{4} F^{c (a+b x)} \left (\frac {-e \cos (d+e x)+b c \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {-3 e \cos (3 (d+e x))+b c \log (F) \sin (3 (d+e x))}{9 e^2+b^2 c^2 \log ^2(F)}\right ) \] Input:
Integrate[F^(c*(a + b*x))*Cos[d + e*x]^2*Sin[d + e*x],x]
Output:
(F^(c*(a + b*x))*((-(e*Cos[d + e*x]) + b*c*Log[F]*Sin[d + e*x])/(e^2 + b^2 *c^2*Log[F]^2) + (-3*e*Cos[3*(d + e*x)] + b*c*Log[F]*Sin[3*(d + e*x)])/(9* e^2 + b^2*c^2*Log[F]^2)))/4
Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4972, 2009}
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.
| \(\displaystyle \int \sin (d+e x) \cos ^2(d+e x) F^{c (a+b x)} \, dx\) |
| \(\Big \downarrow \) 4972 |
| \(\displaystyle \int \left (\frac {1}{4} \sin (d+e x) F^{c (a+b x)}+\frac {1}{4} \sin (3 d+3 e x) F^{c (a+b x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \log (F) \sin (d+e x) F^{c (a+b x)}}{4 \left (b^2 c^2 \log ^2(F)+e^2\right )}+\frac {b c \log (F) \sin (3 d+3 e x) F^{c (a+b x)}}{4 \left (b^2 c^2 \log ^2(F)+9 e^2\right )}-\frac {e \cos (d+e x) F^{c (a+b x)}}{4 \left (b^2 c^2 \log ^2(F)+e^2\right )}-\frac {3 e \cos (3 d+3 e x) F^{c (a+b x)}}{4 \left (b^2 c^2 \log ^2(F)+9 e^2\right )}\) |
Input:
Int[F^(c*(a + b*x))*Cos[d + e*x]^2*Sin[d + e*x],x]
Output:
-1/4*(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) - (3*e*F^(c *(a + b*x))*Cos[3*d + 3*e*x])/(4*(9*e^2 + b^2*c^2*Log[F]^2)) + (b*c*F^(c*( a + b*x))*Log[F]*Sin[d + e*x])/(4*(e^2 + b^2*c^2*Log[F]^2)) + (b*c*F^(c*(a + b*x))*Log[F]*Sin[3*d + 3*e*x])/(4*(9*e^2 + b^2*c^2*Log[F]^2))
Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_ .) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 0.88 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.29
| method | result | size |
| parallelrisch | \(-\frac {3 F^{c \left (b x +a \right )} \left (\left (b^{2} c^{2} \ln \left (F \right )^{2} e +e^{3}\right ) \cos \left (3 e x +3 d \right )-\frac {b c \ln \left (F \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \sin \left (3 e x +3 d \right )}{3}-\frac {\left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (b c \ln \left (F \right ) \sin \left (e x +d \right )-e \cos \left (e x +d \right )\right )}{3}\right )}{4 b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+36 e^{4}}\) | \(144\) |
| risch | \(-\frac {e \,F^{c \left (b x +a \right )} \cos \left (e x +d \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {c b \ln \left (F \right ) F^{c \left (b x +a \right )} \sin \left (e x +d \right )}{4 e^{2}+4 b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {3 e \,F^{c \left (b x +a \right )} \cos \left (3 e x +3 d \right )}{4 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {c b \ln \left (F \right ) F^{c \left (b x +a \right )} \sin \left (3 e x +3 d \right )}{36 e^{2}+4 b^{2} c^{2} \ln \left (F \right )^{2}}\) | \(158\) |
| norman | \(\frac {\frac {e \left (11 b^{2} c^{2} \ln \left (F \right )^{2}+9 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{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 {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 {e x}{2}+\frac {d}{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 {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 )}}{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 {e \left (11 b^{2} c^{2} \ln \left (F \right )^{2}+9 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{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}}+\frac {2 b c \ln \left (F \right ) \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 {e x}{2}+\frac {d}{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 {2 b c \ln \left (F \right ) \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 {e x}{2}+\frac {d}{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 {4 \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 {e x}{2}+\frac {d}{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}}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(539\) |
| 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 )^{2} \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}}-\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 )^{2} \sin \left (e x +d \right )-2 F^{c \left (b x +a \right )} \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2} e +F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3} e \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 )^{2} \sin \left (e x +d \right )-4 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2} e +2 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{3} e +2 F^{c \left (b x +a \right )} e^{2} \sin \left (e x +d \right )^{3}-7 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{2} \sin \left (e x +d \right ) 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 )^{2} \sin \left (e x +d \right )-6 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2} e +3 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cos \left (e x +d \right )^{3} e +6 F^{c \left (b x +a \right )} b c \ln \left (F \right ) e^{2} \sin \left (e x +d \right )^{3}-21 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cos \left (e x +d \right )^{2} \sin \left (e x +d \right ) e^{2}+20 F^{c \left (b x +a \right )} e^{3} \sin \left (e x +d \right )^{2} \cos \left (e x +d \right )-7 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3} e^{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}}\) | \(634\) |
Input:
int(F^(c*(b*x+a))*cos(e*x+d)^2*sin(e*x+d),x,method=_RETURNVERBOSE)
Output:
-3*F^(c*(b*x+a))*((b^2*c^2*ln(F)^2*e+e^3)*cos(3*e*x+3*d)-1/3*b*c*ln(F)*(e^ 2+b^2*c^2*ln(F)^2)*sin(3*e*x+3*d)-1/3*(9*e^2+b^2*c^2*ln(F)^2)*(b*c*ln(F)*s in(e*x+d)-e*cos(e*x+d)))/(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 = 151, normalized size of antiderivative = 1.35 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=-\frac {{\left (3 \, e^{3} \cos \left (e x + d\right )^{3} + {\left (3 \, b^{2} c^{2} e \cos \left (e x + d\right )^{3} - 2 \, b^{2} c^{2} e \cos \left (e x + d\right )\right )} \log \left (F\right )^{2} - {\left (b^{3} c^{3} \cos \left (e x + d\right )^{2} \log \left (F\right )^{3} + {\left (b c e^{2} \cos \left (e x + d\right )^{2} + 2 \, b c e^{2}\right )} \log \left (F\right )\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 10 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 9 \, e^{4}} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)^2*sin(e*x+d),x, algorithm="fricas")
Output:
-(3*e^3*cos(e*x + d)^3 + (3*b^2*c^2*e*cos(e*x + d)^3 - 2*b^2*c^2*e*cos(e*x + d))*log(F)^2 - (b^3*c^3*cos(e*x + d)^2*log(F)^3 + (b*c*e^2*cos(e*x + d) ^2 + 2*b*c*e^2)*log(F))*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 = 3.67 (sec) , antiderivative size = 1571, normalized size of antiderivative = 14.03 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*cos(e*x+d)**2*sin(e*x+d),x)
Output:
Piecewise((x*sin(d)*cos(d)**2, Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sin(d)*co s(d)**2, Eq(b, 0) & Eq(e, 0)), (x*sin(d)*cos(d)**2, Eq(c, 0) & Eq(e, 0)), (-F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F) - d)**3/8 + I*F**(a*c + b*c*x)*x*s in(I*b*c*x*log(F) - d)**2*cos(I*b*c*x*log(F) - d)/8 - F**(a*c + b*c*x)*x*s in(I*b*c*x*log(F) - d)*cos(I*b*c*x*log(F) - d)**2/8 + I*F**(a*c + b*c*x)*x *cos(I*b*c*x*log(F) - d)**3/8 + F**(a*c + b*c*x)*sin(I*b*c*x*log(F) - d)** 3/(8*b*c*log(F)) + F**(a*c + b*c*x)*sin(I*b*c*x*log(F) - d)*cos(I*b*c*x*lo g(F) - d)**2/(4*b*c*log(F)) - 3*I*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))), (F**(a*c + b*c*x)*x*sin(I*b*c*x *log(F)/3 - d)**3/8 - 3*I*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*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 + I*F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F )/3 - d)**3/8 - F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/3 - d)**3/(8*b*c*log(F )) - 3*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)) + I*F**(a*c + b*c*x)*cos(I*b*c*x*log(F)/3 - d)**3/(8*b *c*log(F)), Eq(e, -I*b*c*log(F)/3)), (-F**(a*c + b*c*x)*x*sin(I*b*c*x*log( F)/3 + d)**3/8 + 3*I*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*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/3 + d)*co s(I*b*c*x*log(F)/3 + d)**2/8 - I*F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/3 + d)**3/8 + 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. 810 vs. \(2 (107) = 214\).
Time = 0.08 (sec) , antiderivative size = 810, normalized size of antiderivative = 7.23 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx =\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)^2*sin(e*x+d),x, algorithm="maxima")
Output:
1/8*((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 )*cos(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*co s(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)*cos(3*e*x + 6*d) - (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)*cos(e*x + 4*d) + (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)*cos(e*x - 2*d) + (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)*sin(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*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)*sin(3*e* x + 6*d) + (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)*sin(e*x + 4*d) + (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)*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 = 0.22 (sec) , antiderivative size = 1275, normalized size of antiderivative = 11.38 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)^2*sin(e*x+d),x, algorithm="giac")
Output:
1/4*(2*b*c*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 + 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) - (pi*b*c*sgn(F) - pi*b*c + 6*e)*cos(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))) + 1/4*(2*b*c*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 + e*x + d)/(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)*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)/(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))) - 1/4*(2*b*c*log(abs(F))*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 - e*x - 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)*cos(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*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*p i*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) - (pi*b*c*sgn(F) - pi*b*c - 6*e)*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)/(4*b^2*c^2*log(ab...
Time = 16.50 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.71 \[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )-\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )-\sin \left (d\right )\,1{}\mathrm {i}\right )}{8\,\left (e+b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )+\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )+\sin \left (3\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (b\,c\,\ln \left (F\right )+e\,3{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )-\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )-\sin \left (3\,d\right )\,1{}\mathrm {i}\right )}{8\,\left (3\,e+b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )+\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )+\sin \left (d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (b\,c\,\ln \left (F\right )+e\,1{}\mathrm {i}\right )} \] Input:
int(F^(c*(a + b*x))*cos(d + e*x)^2*sin(d + e*x),x)
Output:
- (F^(c*(a + b*x))*(cos(e*x) - sin(e*x)*1i)*(cos(d) - sin(d)*1i))/(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)*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))/(8*(3*e + b*c*log(F)*1i)) - (F^ (c*(a + b*x))*(cos(e*x) + sin(e*x)*1i)*(cos(d) + sin(d)*1i)*1i)/(8*(e*1i + b*c*log(F)))
\[ \int F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )d x \right ) \] Input:
int(F^(c*(b*x+a))*cos(e*x+d)^2*sin(e*x+d),x)
Output:
f**(a*c)*int(f**(b*c*x)*cos(d + e*x)**2*sin(d + e*x),x)