Integrand size = 24, antiderivative size = 121 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=-\frac {F^{c (a+b x)} (2 e \cos (2 d+2 e x)-b c \log (F) \sin (2 d+2 e x))}{4 \left (4 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {F^{c (a+b x)} (4 e \cos (4 d+4 e x)-b c \log (F) \sin (4 d+4 e x))}{8 \left (16 e^2+b^2 c^2 \log ^2(F)\right )} \] Output:
-1/4*F^(c*(b*x+a))*(2*e*cos(2*e*x+2*d)-b*c*ln(F)*sin(2*e*x+2*d))/(4*e^2+b^ 2*c^2*ln(F)^2)+F^(c*(b*x+a))*(4*e*cos(4*e*x+4*d)-b*c*ln(F)*sin(4*e*x+4*d)) /(128*e^2+8*b^2*c^2*ln(F)^2)
Time = 1.35 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=\frac {1}{8} F^{c (a+b x)} \left (\frac {2 (-2 e \cos (2 (d+e x))+b c \log (F) \sin (2 (d+e x)))}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {4 e \cos (4 (d+e x))-b c \log (F) \sin (4 (d+e x))}{16 e^2+b^2 c^2 \log ^2(F)}\right ) \] Input:
Integrate[F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^3,x]
Output:
(F^(c*(a + b*x))*((2*(-2*e*Cos[2*(d + e*x)] + b*c*Log[F]*Sin[2*(d + e*x)]) )/(4*e^2 + b^2*c^2*Log[F]^2) + (4*e*Cos[4*(d + e*x)] - b*c*Log[F]*Sin[4*(d + e*x)])/(16*e^2 + b^2*c^2*Log[F]^2)))/8
Time = 0.32 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.45, 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.
| \(\displaystyle \int \sin ^3(d+e x) \cos (d+e x) F^{c (a+b x)} \, dx\) |
| \(\Big \downarrow \) 4972 |
| \(\displaystyle \int \left (\frac {1}{4} \sin (2 d+2 e x) F^{c (a+b x)}-\frac {1}{8} \sin (4 d+4 e x) F^{c (a+b x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \log (F) \sin (2 d+2 e x) F^{c (a+b x)}}{4 \left (b^2 c^2 \log ^2(F)+4 e^2\right )}-\frac {b c \log (F) \sin (4 d+4 e x) F^{c (a+b x)}}{8 \left (b^2 c^2 \log ^2(F)+16 e^2\right )}-\frac {e \cos (2 d+2 e x) F^{c (a+b x)}}{2 \left (b^2 c^2 \log ^2(F)+4 e^2\right )}+\frac {e \cos (4 d+4 e x) F^{c (a+b x)}}{2 \left (b^2 c^2 \log ^2(F)+16 e^2\right )}\) |
Input:
Int[F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^3,x]
Output:
-1/2*(e*F^(c*(a + b*x))*Cos[2*d + 2*e*x])/(4*e^2 + b^2*c^2*Log[F]^2) + (e* F^(c*(a + b*x))*Cos[4*d + 4*e*x])/(2*(16*e^2 + b^2*c^2*Log[F]^2)) + (b*c*F ^(c*(a + b*x))*Log[F]*Sin[2*d + 2*e*x])/(4*(4*e^2 + b^2*c^2*Log[F]^2)) - ( b*c*F^(c*(a + b*x))*Log[F]*Sin[4*d + 4*e*x])/(8*(16*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 = 1.45 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.27
| method | result | size |
| parallelrisch | \(\frac {4 \left (\left (-\frac {\ln \left (F \right )^{3} b^{3} c^{3}}{4}-\ln \left (F \right ) b c \,e^{2}\right ) \sin \left (4 e x +4 d \right )+\left (b^{2} c^{2} \ln \left (F \right )^{2} e +4 e^{3}\right ) \cos \left (4 e x +4 d \right )+\frac {\left (16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (b c \ln \left (F \right ) \sin \left (2 e x +2 d \right )-2 e \cos \left (2 e x +2 d \right )\right )}{2}\right ) F^{c \left (b x +a \right )}}{8 b^{4} c^{4} \ln \left (F \right )^{4}+160 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+512 e^{4}}\) | \(154\) |
| risch | \(\frac {e \,F^{c \left (b x +a \right )} \cos \left (4 e x +4 d \right )}{2 b^{2} c^{2} \ln \left (F \right )^{2}+32 e^{2}}-\frac {c b \ln \left (F \right ) F^{c \left (b x +a \right )} \sin \left (4 e x +4 d \right )}{8 \left (16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}-\frac {e \,F^{c \left (b x +a \right )} \cos \left (2 e x +2 d \right )}{2 \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {c b \ln \left (F \right ) F^{c \left (b x +a \right )} \sin \left (2 e x +2 d \right )}{4 b^{2} c^{2} \ln \left (F \right )^{2}+16 e^{2}}\) | \(168\) |
| default | \(-\frac {F^{a c} \left (\frac {\frac {4 e \,{\mathrm e}^{b c x \ln \left (F \right )}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{2}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan \left (e x +d \right )^{2}}+\frac {-\frac {4 e \,{\mathrm e}^{b c x \ln \left (F \right )}}{16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {24 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{2}}{16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{4}}{16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )}{16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )^{3}}{16 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{\left (1+\tan \left (e x +d \right )^{2}\right )^{2}}\right )}{8}\) | \(322\) |
| norman | \(\frac {-\frac {6 e^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{64 e^{4}+20 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {6 e^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{8}}{64 e^{4}+20 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}+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 )^{2}}{64 e^{4}+20 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}+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 )^{6}}{64 e^{4}+20 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {20 e \left (2 b^{2} c^{2} \ln \left (F \right )^{2}+11 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}}{64 e^{4}+20 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {12 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{64 e^{4}+20 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {12 \ln \left (F \right ) b c \,e^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{7}}{64 e^{4}+20 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 (2 b^{2} c^{2} \ln \left (F \right )^{2}+11 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}}{64 e^{4}+20 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 (2 b^{2} c^{2} \ln \left (F \right )^{2}+11 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}}{64 e^{4}+20 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 )^{4}}\) | \(637\) |
| orering | \(\frac {4 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+10 e^{2}\right ) F^{c \left (b x +a \right )} \cos \left (e x +d \right ) \sin \left (e x +d \right )^{3}}{64 e^{4}+20 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}+10 e^{2}\right ) \left (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 )^{3}-F^{c \left (b x +a \right )} e \sin \left (e x +d \right )^{4}+3 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )^{2} e \right )}{64 e^{4}+20 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 ) \sin \left (e x +d \right )^{3}-2 F^{c \left (b x +a \right )} b c \ln \left (F \right ) e \sin \left (e x +d \right )^{4}+6 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} e -10 F^{c \left (b x +a \right )} e^{2} \sin \left (e x +d \right )^{3} \cos \left (e x +d \right )+6 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{3} \sin \left (e x +d \right ) e^{2}\right )}{64 e^{4}+20 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 ) \sin \left (e x +d \right )^{3}-3 F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} e \sin \left (e x +d \right )^{4}+9 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 )^{2} e -30 F^{c \left (b x +a \right )} b c \ln \left (F \right ) e^{2} \sin \left (e x +d \right )^{3} \cos \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 )^{3} \sin \left (e x +d \right ) e^{2}-48 F^{c \left (b x +a \right )} e^{3} \sin \left (e x +d \right )^{2} \cos \left (e x +d \right )^{2}+10 F^{c \left (b x +a \right )} e^{3} \sin \left (e x +d \right )^{4}+6 F^{c \left (b x +a \right )} \cos \left (e x +d \right )^{4} e^{3}}{64 e^{4}+20 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}\) | \(677\) |
Input:
int(F^(c*(b*x+a))*cos(e*x+d)*sin(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
4*((-1/4*ln(F)^3*b^3*c^3-ln(F)*b*c*e^2)*sin(4*e*x+4*d)+(b^2*c^2*ln(F)^2*e+ 4*e^3)*cos(4*e*x+4*d)+1/2*(16*e^2+b^2*c^2*ln(F)^2)*(b*c*ln(F)*sin(2*e*x+2* d)-2*e*cos(2*e*x+2*d)))*F^(c*(b*x+a))/(8*b^4*c^4*ln(F)^4+160*b^2*c^2*e^2*l n(F)^2+512*e^4)
Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.67 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=\frac {{\left (16 \, e^{3} \cos \left (e x + d\right )^{4} - 32 \, e^{3} \cos \left (e x + d\right )^{2} + 10 \, e^{3} + {\left (4 \, b^{2} c^{2} e \cos \left (e x + d\right )^{4} - 5 \, b^{2} c^{2} e \cos \left (e x + d\right )^{2} + b^{2} c^{2} e\right )} \log \left (F\right )^{2} - {\left ({\left (b^{3} c^{3} \cos \left (e x + d\right )^{3} - b^{3} c^{3} \cos \left (e x + d\right )\right )} \log \left (F\right )^{3} + 2 \, {\left (2 \, b c e^{2} \cos \left (e x + d\right )^{3} - 5 \, b c e^{2} \cos \left (e x + d\right )\right )} \log \left (F\right )\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 20 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 64 \, e^{4}} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)*sin(e*x+d)^3,x, algorithm="fricas")
Output:
(16*e^3*cos(e*x + d)^4 - 32*e^3*cos(e*x + d)^2 + 10*e^3 + (4*b^2*c^2*e*cos (e*x + d)^4 - 5*b^2*c^2*e*cos(e*x + d)^2 + b^2*c^2*e)*log(F)^2 - ((b^3*c^3 *cos(e*x + d)^3 - b^3*c^3*cos(e*x + d))*log(F)^3 + 2*(2*b*c*e^2*cos(e*x + d)^3 - 5*b*c*e^2*cos(e*x + d))*log(F))*sin(e*x + d))*F^(b*c*x + a*c)/(b^4* c^4*log(F)^4 + 20*b^2*c^2*e^2*log(F)^2 + 64*e^4)
Result contains complex when optimal does not.
Time = 26.78 (sec) , antiderivative size = 2057, normalized size of antiderivative = 17.00 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*cos(e*x+d)*sin(e*x+d)**3,x)
Output:
Piecewise((x*sin(d)**3*cos(d), Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sin(d)**3 *cos(d), Eq(b, 0) & Eq(e, 0)), (x*sin(d)**3*cos(d), Eq(c, 0) & Eq(e, 0)), (-I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/2 - d)**4/8 - F**(a*c + b*c*x)*x *sin(I*b*c*x*log(F)/2 - d)**3*cos(I*b*c*x*log(F)/2 - d)/4 - F**(a*c + b*c* x)*x*sin(I*b*c*x*log(F)/2 - d)*cos(I*b*c*x*log(F)/2 - d)**3/4 + I*F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/2 - d)**4/8 + 7*I*F**(a*c + b*c*x)*sin(I*b*c *x*log(F)/2 - d)**4/(24*b*c*log(F)) + F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/ 2 - d)**3*cos(I*b*c*x*log(F)/2 - d)/(3*b*c*log(F)) - I*F**(a*c + b*c*x)*si n(I*b*c*x*log(F)/2 - d)**2*cos(I*b*c*x*log(F)/2 - d)**2/(2*b*c*log(F)) - I *F**(a*c + b*c*x)*cos(I*b*c*x*log(F)/2 - d)**4/(8*b*c*log(F)), Eq(e, -I*b* c*log(F)/2)), (-I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/4 - d)**4/16 - F** (a*c + b*c*x)*x*sin(I*b*c*x*log(F)/4 - d)**3*cos(I*b*c*x*log(F)/4 - d)/4 + 3*I*F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/4 - d)**2*cos(I*b*c*x*log(F)/4 - d)**2/8 + F**(a*c + b*c*x)*x*sin(I*b*c*x*log(F)/4 - d)*cos(I*b*c*x*log(F )/4 - d)**3/4 - I*F**(a*c + b*c*x)*x*cos(I*b*c*x*log(F)/4 - d)**4/16 - I*F **(a*c + b*c*x)*sin(I*b*c*x*log(F)/4 - d)**4/(6*b*c*log(F)) - 11*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/4 - d)**3*cos(I*b*c*x*log(F)/4 - d)/(12*b*c*log (F)) - 5*F**(a*c + b*c*x)*sin(I*b*c*x*log(F)/4 - d)*cos(I*b*c*x*log(F)/4 - d)**3/(12*b*c*log(F)) + I*F**(a*c + b*c*x)*cos(I*b*c*x*log(F)/4 - d)**4/( 6*b*c*log(F)), Eq(e, -I*b*c*log(F)/4)), (I*F**(a*c + b*c*x)*x*sin(I*b*c...
Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (115) = 230\).
Time = 0.07 (sec) , antiderivative size = 823, normalized size of antiderivative = 6.80 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx =\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)*sin(e*x+d)^3,x, algorithm="maxima")
Output:
-1/16*((F^(a*c)*b^3*c^3*log(F)^3*sin(4*d) - 4*F^(a*c)*b^2*c^2*e*cos(4*d)*l og(F)^2 + 4*F^(a*c)*b*c*e^2*log(F)*sin(4*d) - 16*F^(a*c)*e^3*cos(4*d))*F^( b*c*x)*cos(4*e*x) - (F^(a*c)*b^3*c^3*log(F)^3*sin(4*d) + 4*F^(a*c)*b^2*c^2 *e*cos(4*d)*log(F)^2 + 4*F^(a*c)*b*c*e^2*log(F)*sin(4*d) + 16*F^(a*c)*e^3* cos(4*d))*F^(b*c*x)*cos(4*e*x + 8*d) + 2*(F^(a*c)*b^3*c^3*log(F)^3*sin(4*d ) + 2*F^(a*c)*b^2*c^2*e*cos(4*d)*log(F)^2 + 16*F^(a*c)*b*c*e^2*log(F)*sin( 4*d) + 32*F^(a*c)*e^3*cos(4*d))*F^(b*c*x)*cos(2*e*x + 6*d) - 2*(F^(a*c)*b^ 3*c^3*log(F)^3*sin(4*d) - 2*F^(a*c)*b^2*c^2*e*cos(4*d)*log(F)^2 + 16*F^(a* c)*b*c*e^2*log(F)*sin(4*d) - 32*F^(a*c)*e^3*cos(4*d))*F^(b*c*x)*cos(2*e*x - 2*d) + (F^(a*c)*b^3*c^3*cos(4*d)*log(F)^3 + 4*F^(a*c)*b^2*c^2*e*log(F)^2 *sin(4*d) + 4*F^(a*c)*b*c*e^2*cos(4*d)*log(F) + 16*F^(a*c)*e^3*sin(4*d))*F ^(b*c*x)*sin(4*e*x) + (F^(a*c)*b^3*c^3*cos(4*d)*log(F)^3 - 4*F^(a*c)*b^2*c ^2*e*log(F)^2*sin(4*d) + 4*F^(a*c)*b*c*e^2*cos(4*d)*log(F) - 16*F^(a*c)*e^ 3*sin(4*d))*F^(b*c*x)*sin(4*e*x + 8*d) - 2*(F^(a*c)*b^3*c^3*cos(4*d)*log(F )^3 - 2*F^(a*c)*b^2*c^2*e*log(F)^2*sin(4*d) + 16*F^(a*c)*b*c*e^2*cos(4*d)* log(F) - 32*F^(a*c)*e^3*sin(4*d))*F^(b*c*x)*sin(2*e*x + 6*d) - 2*(F^(a*c)* b^3*c^3*cos(4*d)*log(F)^3 + 2*F^(a*c)*b^2*c^2*e*log(F)^2*sin(4*d) + 16*F^( a*c)*b*c*e^2*cos(4*d)*log(F) + 32*F^(a*c)*e^3*sin(4*d))*F^(b*c*x)*sin(2*e* x - 2*d))/(b^4*c^4*cos(4*d)^2*log(F)^4 + b^4*c^4*log(F)^4*sin(4*d)^2 + 64* (cos(4*d)^2 + sin(4*d)^2)*e^4 + 20*(b^2*c^2*cos(4*d)^2*log(F)^2 + b^2*c...
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 1281, normalized size of antiderivative = 10.59 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cos(e*x+d)*sin(e*x+d)^3,x, algorithm="giac")
Output:
-1/8*(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 + 4*e*x + 4*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sg n(F) - pi*b*c + 8*e)^2) - (pi*b*c*sgn(F) - pi*b*c + 8*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 + 4*e*x + 4*d)/(4*b ^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 8*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 + 2*e*x + 2*d)/(4*b^2*c^2*l og(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 4*e)^2) - (pi*b*c*sgn(F) - pi*b*c + 4*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*p i*a*c + 2*e*x + 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))) - 1/4*(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 - 2*e*x - 2*d)/(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)*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*e*x - 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))) + 1/8*(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 - 4*e*x - 4*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*s gn(F) - pi*b*c - 8*e)^2) - (pi*b*c*sgn(F) - pi*b*c - 8*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 - 4*e*x - 4*d)/...
Time = 16.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.67 \[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (2\,e\,x\right )+\sin \left (2\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,d\right )+\sin \left (2\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (b\,c\,\ln \left (F\right )+e\,2{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (2\,e\,x\right )-\sin \left (2\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,d\right )-\sin \left (2\,d\right )\,1{}\mathrm {i}\right )}{8\,\left (2\,e+b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}+\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (4\,e\,x\right )+\sin \left (4\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,d\right )+\sin \left (4\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,\left (b\,c\,\ln \left (F\right )+e\,4{}\mathrm {i}\right )}+\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (4\,e\,x\right )-\sin \left (4\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,d\right )-\sin \left (4\,d\right )\,1{}\mathrm {i}\right )}{16\,\left (4\,e+b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )} \] Input:
int(F^(c*(a + b*x))*cos(d + e*x)*sin(d + e*x)^3,x)
Output:
(F^(c*(a + b*x))*(cos(4*e*x) + sin(4*e*x)*1i)*(cos(4*d) + sin(4*d)*1i)*1i) /(16*(e*4i + b*c*log(F))) - (F^(c*(a + b*x))*(cos(2*e*x) - sin(2*e*x)*1i)* (cos(2*d) - sin(2*d)*1i))/(8*(2*e + b*c*log(F)*1i)) - (F^(c*(a + b*x))*(co s(2*e*x) + sin(2*e*x)*1i)*(cos(2*d) + sin(2*d)*1i)*1i)/(8*(e*2i + b*c*log( F))) + (F^(c*(a + b*x))*(cos(4*e*x) - sin(4*e*x)*1i)*(cos(4*d) - sin(4*d)* 1i))/(16*(4*e + b*c*log(F)*1i))
\[ \int F^{c (a+b x)} \cos (d+e x) \sin ^3(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cos \left (e x +d \right ) \sin \left (e x +d \right )^{3}d x \right ) \] Input:
int(F^(c*(b*x+a))*cos(e*x+d)*sin(e*x+d)^3,x)
Output:
f**(a*c)*int(f**(b*c*x)*cos(d + e*x)*sin(d + e*x)**3,x)