Integrand size = 28, antiderivative size = 175 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {\cos (c+d x)}{a f (e+f x)}-\frac {d \cos \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2}-\frac {d \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right ) \sin \left (c-\frac {d e}{f}\right )}{a f^2}+\frac {\sin (2 c+2 d x)}{2 a f (e+f x)}-\frac {d \cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{a f^2}+\frac {d \sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{a f^2} \]
-d*Ci(2*d*e/f+2*d*x)*cos(2*c-2*d*e/f)/a/f^2-cos(d*x+c)/a/f/(f*x+e)-d*cos(c -d*e/f)*Si(d*e/f+d*x)/a/f^2+d*Si(2*d*e/f+2*d*x)*sin(2*c-2*d*e/f)/a/f^2-d*C i(d*e/f+d*x)*sin(c-d*e/f)/a/f^2+1/2*sin(2*d*x+2*c)/a/f/(f*x+e)
Time = 0.50 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {-2 f \cos (c+d x)-2 d (e+f x) \cos \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d (e+f x)}{f}\right )-2 d (e+f x) \operatorname {CosIntegral}\left (d \left (\frac {e}{f}+x\right )\right ) \sin \left (c-\frac {d e}{f}\right )+f \sin (2 (c+d x))-2 d e \cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )-2 d f x \cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (d \left (\frac {e}{f}+x\right )\right )+2 d e \sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )+2 d f x \sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d (e+f x)}{f}\right )}{2 a f^2 (e+f x)} \]
(-2*f*Cos[c + d*x] - 2*d*(e + f*x)*Cos[2*c - (2*d*e)/f]*CosIntegral[(2*d*( e + f*x))/f] - 2*d*(e + f*x)*CosIntegral[d*(e/f + x)]*Sin[c - (d*e)/f] + f *Sin[2*(c + d*x)] - 2*d*e*Cos[c - (d*e)/f]*SinIntegral[d*(e/f + x)] - 2*d* f*x*Cos[c - (d*e)/f]*SinIntegral[d*(e/f + x)] + 2*d*e*Sin[2*c - (2*d*e)/f] *SinIntegral[(2*d*(e + f*x))/f] + 2*d*f*x*Sin[2*c - (2*d*e)/f]*SinIntegral [(2*d*(e + f*x))/f])/(2*a*f^2*(e + f*x))
Time = 1.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5034, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783, 4906, 27, 3042, 3778, 3042, 3784, 3042, 3780, 3783}
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.
\(\displaystyle \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)} \, dx\) |
\(\Big \downarrow \) 5034 |
\(\displaystyle \frac {\int \frac {\cos (c+d x)}{(e+f x)^2}dx}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{(e+f x)^2}dx}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {\frac {d \int -\frac {\sin (c+d x)}{e+f x}dx}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {d \int \frac {\sin (c+d x)}{e+f x}dx}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d \int \frac {\sin (c+d x)}{e+f x}dx}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {-\frac {d \left (\sin \left (c-\frac {d e}{f}\right ) \int \frac {\cos \left (\frac {d e}{f}+d x\right )}{e+f x}dx+\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x}dx\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d \left (\sin \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x+\frac {\pi }{2}\right )}{e+f x}dx+\cos \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x\right )}{e+f x}dx\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {-\frac {d \left (\sin \left (c-\frac {d e}{f}\right ) \int \frac {\sin \left (\frac {d e}{f}+d x+\frac {\pi }{2}\right )}{e+f x}dx+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\cos (c+d x) \sin (c+d x)}{(e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\sin (2 c+2 d x)}{2 (e+f x)^2}dx}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\sin (2 c+2 d x)}{(e+f x)^2}dx}{2 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\int \frac {\sin (2 c+2 d x)}{(e+f x)^2}dx}{2 a}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\frac {2 d \int \frac {\cos (2 c+2 d x)}{e+f x}dx}{f}-\frac {\sin (2 c+2 d x)}{f (e+f x)}}{2 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\frac {2 d \int \frac {\sin \left (2 c+2 d x+\frac {\pi }{2}\right )}{e+f x}dx}{f}-\frac {\sin (2 c+2 d x)}{f (e+f x)}}{2 a}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\frac {2 d \left (\cos \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\cos \left (\frac {2 d e}{f}+2 d x\right )}{e+f x}dx-\sin \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x\right )}{e+f x}dx\right )}{f}-\frac {\sin (2 c+2 d x)}{f (e+f x)}}{2 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\frac {2 d \left (\cos \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x+\frac {\pi }{2}\right )}{e+f x}dx-\sin \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x\right )}{e+f x}dx\right )}{f}-\frac {\sin (2 c+2 d x)}{f (e+f x)}}{2 a}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\frac {2 d \left (\cos \left (2 c-\frac {2 d e}{f}\right ) \int \frac {\sin \left (\frac {2 d e}{f}+2 d x+\frac {\pi }{2}\right )}{e+f x}dx-\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{f}\right )}{f}-\frac {\sin (2 c+2 d x)}{f (e+f x)}}{2 a}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {-\frac {d \left (\frac {\sin \left (c-\frac {d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {d e}{f}+d x\right )}{f}+\frac {\cos \left (c-\frac {d e}{f}\right ) \text {Si}\left (\frac {d e}{f}+d x\right )}{f}\right )}{f}-\frac {\cos (c+d x)}{f (e+f x)}}{a}-\frac {\frac {2 d \left (\frac {\cos \left (2 c-\frac {2 d e}{f}\right ) \operatorname {CosIntegral}\left (\frac {2 d e}{f}+2 d x\right )}{f}-\frac {\sin \left (2 c-\frac {2 d e}{f}\right ) \text {Si}\left (\frac {2 d e}{f}+2 d x\right )}{f}\right )}{f}-\frac {\sin (2 c+2 d x)}{f (e+f x)}}{2 a}\) |
(-(Cos[c + d*x]/(f*(e + f*x))) - (d*((CosIntegral[(d*e)/f + d*x]*Sin[c - ( d*e)/f])/f + (Cos[c - (d*e)/f]*SinIntegral[(d*e)/f + d*x])/f))/f)/a - (-(S in[2*c + 2*d*x]/(f*(e + f*x))) + (2*d*((Cos[2*c - (2*d*e)/f]*CosIntegral[( 2*d*e)/f + 2*d*x])/f - (Sin[2*c - (2*d*e)/f]*SinIntegral[(2*d*e)/f + 2*d*x ])/f))/f)/(2*a)
3.3.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.) *Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] - Simp[1/b Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*Sin[ c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^ 2 - b^2, 0]
Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.70
method | result | size |
risch | \(\frac {i d \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (i d x +i c -\frac {i \left (c f -d e \right )}{f}\right )}{2 a \,f^{2}}-\frac {i d \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (-i d x -i c -\frac {-i c f +i d e}{f}\right )}{2 a \,f^{2}}+\frac {d \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (-2 i d x -2 i c -\frac {2 \left (-i c f +i d e \right )}{f}\right )}{2 a \,f^{2}}+\frac {d \,{\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (2 i d x +2 i c -\frac {2 i \left (c f -d e \right )}{f}\right )}{2 a \,f^{2}}-\frac {\left (-2 d x f -2 d e \right ) \cos \left (d x +c \right )}{2 f a \left (f x +e \right ) \left (-d x f -d e \right )}+\frac {\left (-2 d x f -2 d e \right ) \sin \left (2 d x +2 c \right )}{4 f a \left (f x +e \right ) \left (-d x f -d e \right )}\) | \(297\) |
1/2*I/a*d/f^2*exp(-I*(c*f-d*e)/f)*Ei(1,I*d*x+I*c-I*(c*f-d*e)/f)-1/2*I*d/a/ f^2*exp(I*(c*f-d*e)/f)*Ei(1,-I*d*x-I*c-(-I*c*f+I*d*e)/f)+1/2*d/a/f^2*exp(2 *I*(c*f-d*e)/f)*Ei(1,-2*I*d*x-2*I*c-2*(-I*c*f+I*d*e)/f)+1/2/a*d/f^2*exp(-2 *I*(c*f-d*e)/f)*Ei(1,2*I*d*x+2*I*c-2*I*(c*f-d*e)/f)-1/2/f*(-2*d*f*x-2*d*e) /a/(f*x+e)/(-d*f*x-d*e)*cos(d*x+c)+1/4/f*(-2*d*f*x-2*d*e)/a/(f*x+e)/(-d*f* x-d*e)*sin(2*d*x+2*c)
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {{\left (d f x + d e\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) - f \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (d f x + d e\right )} \operatorname {Ci}\left (\frac {d f x + d e}{f}\right ) \sin \left (-\frac {d e - c f}{f}\right ) - {\left (d f x + d e\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) + {\left (d f x + d e\right )} \cos \left (-\frac {d e - c f}{f}\right ) \operatorname {Si}\left (\frac {d f x + d e}{f}\right ) + f \cos \left (d x + c\right )}{a f^{3} x + a e f^{2}} \]
-((d*f*x + d*e)*cos(-2*(d*e - c*f)/f)*cos_integral(2*(d*f*x + d*e)/f) - f* cos(d*x + c)*sin(d*x + c) + (d*f*x + d*e)*cos_integral((d*f*x + d*e)/f)*si n(-(d*e - c*f)/f) - (d*f*x + d*e)*sin(-2*(d*e - c*f)/f)*sin_integral(2*(d* f*x + d*e)/f) + (d*f*x + d*e)*cos(-(d*e - c*f)/f)*sin_integral((d*f*x + d* e)/f) + f*cos(d*x + c))/(a*f^3*x + a*e*f^2)
Timed out. \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.77 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=-\frac {2 \, d^{2} {\left (E_{2}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + E_{2}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \cos \left (-\frac {d e - c f}{f}\right ) - d^{2} {\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right ) + 2 \, d^{2} {\left (-i \, E_{2}\left (\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right ) + i \, E_{2}\left (-\frac {i \, d e + i \, {\left (d x + c\right )} f - i \, c f}{f}\right )\right )} \sin \left (-\frac {d e - c f}{f}\right ) - d^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, d e - i \, {\left (d x + c\right )} f + i \, c f\right )}}{f}\right )\right )} \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}{4 \, {\left (a d e f + {\left (d x + c\right )} a f^{2} - a c f^{2}\right )} d} \]
-1/4*(2*d^2*(exp_integral_e(2, (I*d*e + I*(d*x + c)*f - I*c*f)/f) + exp_in tegral_e(2, -(I*d*e + I*(d*x + c)*f - I*c*f)/f))*cos(-(d*e - c*f)/f) - d^2 *(-I*exp_integral_e(2, 2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f) + I*exp_integ ral_e(2, -2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f))*cos(-2*(d*e - c*f)/f) + 2 *d^2*(-I*exp_integral_e(2, (I*d*e + I*(d*x + c)*f - I*c*f)/f) + I*exp_inte gral_e(2, -(I*d*e + I*(d*x + c)*f - I*c*f)/f))*sin(-(d*e - c*f)/f) - d^2*( exp_integral_e(2, 2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f) + exp_integral_e(2 , -2*(-I*d*e - I*(d*x + c)*f + I*c*f)/f))*sin(-2*(d*e - c*f)/f))/((a*d*e*f + (d*x + c)*a*f^2 - a*c*f^2)*d)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.50 (sec) , antiderivative size = 46878, normalized size of antiderivative = 267.87 \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Too large to display} \]
-1/2*(d*f*x*imag_part(cos_integral(d*x + d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2 *tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - d*f*x*imag_part(cos_integral (-d*x - d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/ 2*d*e/f)^2 - d*f*x*real_part(cos_integral(2*d*x + 2*d*e/f))*tan(d*x)^2*tan (1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - d*f*x*real_part(c os_integral(-2*d*x - 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan( d*e/f)^2*tan(1/2*d*e/f)^2 + 2*d*f*x*sin_integral((d*f*x + d*e)/f)*tan(d*x) ^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f)^2 + 2*d*f*x*rea l_part(cos_integral(d*x + d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*t an(d*e/f)^2*tan(1/2*d*e/f) + 2*d*f*x*real_part(cos_integral(-d*x - d*e/f)) *tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)^2*tan(1/2*d*e/f) + 2*d* f*x*imag_part(cos_integral(2*d*x + 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan (1/2*c)^4*tan(d*e/f)*tan(1/2*d*e/f)^2 - 2*d*f*x*imag_part(cos_integral(-2* d*x - 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^4*tan(d*e/f)*tan(1/2* d*e/f)^2 + 4*d*f*x*sin_integral(2*(d*f*x + d*e)/f)*tan(d*x)^2*tan(1/2*d*x) ^2*tan(1/2*c)^4*tan(d*e/f)*tan(1/2*d*e/f)^2 - 4*d*f*x*imag_part(cos_integr al(2*d*x + 2*d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c)^3*tan(d*e/f)^2*t an(1/2*d*e/f)^2 + 4*d*f*x*imag_part(cos_integral(-2*d*x - 2*d*e/f))*tan(d* x)^2*tan(1/2*d*x)^2*tan(1/2*c)^3*tan(d*e/f)^2*tan(1/2*d*e/f)^2 - 2*d*f*x*r eal_part(cos_integral(d*x + d*e/f))*tan(d*x)^2*tan(1/2*d*x)^2*tan(1/2*c...
Timed out. \[ \int \frac {\cos ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]