Integrand size = 28, antiderivative size = 219 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d}-\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {(e+f x)^3 \sin (c+d x)}{a d}+\frac {3 f^3 \cos (c+d x) \sin (c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d} \]
-3/8*f^3*x/a/d^3+1/4*(f*x+e)^3/a/d-6*f^3*cos(d*x+c)/a/d^4+3*f*(f*x+e)^2*co s(d*x+c)/a/d^2-6*f^2*(f*x+e)*sin(d*x+c)/a/d^3+(f*x+e)^3*sin(d*x+c)/a/d+3/8 *f^3*cos(d*x+c)*sin(d*x+c)/a/d^4-3/4*f*(f*x+e)^2*cos(d*x+c)*sin(d*x+c)/a/d ^2+3/4*f^2*(f*x+e)*sin(d*x+c)^2/a/d^3-1/2*(f*x+e)^3*sin(d*x+c)^2/a/d
Time = 1.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.60 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {96 f \left (-2 f^2+d^2 (e+f x)^2\right ) \cos (c+d x)+4 d (e+f x) \left (-3 f^2+2 d^2 (e+f x)^2\right ) \cos (2 (c+d x))+4 \left (8 d (e+f x) \left (-6 f^2+d^2 (e+f x)^2\right )-3 f \left (-f^2+2 d^2 (e+f x)^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{32 a d^4} \]
(96*f*(-2*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] + 4*d*(e + f*x)*(-3*f^2 + 2* d^2*(e + f*x)^2)*Cos[2*(c + d*x)] + 4*(8*d*(e + f*x)*(-6*f^2 + d^2*(e + f* x)^2) - 3*f*(-f^2 + 2*d^2*(e + f*x)^2)*Cos[c + d*x])*Sin[c + d*x])/(32*a*d ^4)
Time = 1.08 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5034, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118, 4904, 3042, 3792, 17, 3042, 3115, 24}
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.
\(\displaystyle \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 5034 |
\(\displaystyle \frac {\int (e+f x)^3 \cos (c+d x)dx}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (e+f x)^3 \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {3 f \int -(e+f x)^2 \sin (c+d x)dx}{d}+\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sin (c+d x)dx}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sin (c+d x)dx}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \int (e+f x) \cos (c+d x)dx}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4904 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \sin ^2(c+d x)dx}{2 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \sin (c+d x)^2dx}{2 d}}{a}\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \int \sin ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \int \sin ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \int \sin (c+d x)^2dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\) |
(((e + f*x)^3*Sin[c + d*x])/d - (3*f*(-(((e + f*x)^2*Cos[c + d*x])/d) + (2 *f*((f*Cos[c + d*x])/d^2 + ((e + f*x)*Sin[c + d*x])/d))/d))/d)/a - (((e + f*x)^3*Sin[c + d*x]^2)/(2*d) - (3*f*((e + f*x)^3/(6*f) - ((e + f*x)^2*Cos[ c + d*x]*Sin[c + d*x])/(2*d) + (f*(e + f*x)*Sin[c + d*x]^2)/(2*d^2) - (f^2 *(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/(2*d^2)))/(2*d))/a
3.3.63.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x _)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) , x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
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]
Time = 0.36 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.73
method | result | size |
parallelrisch | \(\frac {2 \left (f x +e \right ) d \left (\left (f x +e \right )^{2} d^{2}-\frac {3 f^{2}}{2}\right ) \cos \left (2 d x +2 c \right )-3 \left (\left (f x +e \right )^{2} d^{2}-\frac {f^{2}}{2}\right ) f \sin \left (2 d x +2 c \right )+8 \left (f x +e \right ) d \left (\left (f x +e \right )^{2} d^{2}-6 f^{2}\right ) \sin \left (d x +c \right )+24 f \left (\left (f x +e \right )^{2} d^{2}-2 f^{2}\right ) \cos \left (d x +c \right )-2 d^{3} e^{3}+24 d^{2} e^{2} f +3 d e \,f^{2}-48 f^{3}}{8 a \,d^{4}}\) | \(159\) |
risch | \(\frac {3 f \left (d^{2} x^{2} f^{2}+2 f e x \,d^{2}+d^{2} e^{2}-2 f^{2}\right ) \cos \left (d x +c \right )}{a \,d^{4}}+\frac {\left (d^{2} x^{3} f^{3}+3 d^{2} e \,f^{2} x^{2}+3 d^{2} e^{2} f x +d^{2} e^{3}-6 f^{3} x -6 e \,f^{2}\right ) \sin \left (d x +c \right )}{d^{3} a}+\frac {\left (2 d^{2} x^{3} f^{3}+6 d^{2} e \,f^{2} x^{2}+6 d^{2} e^{2} f x +2 d^{2} e^{3}-3 f^{3} x -3 e \,f^{2}\right ) \cos \left (2 d x +2 c \right )}{8 d^{3} a}-\frac {3 f \left (2 d^{2} x^{2} f^{2}+4 f e x \,d^{2}+2 d^{2} e^{2}-f^{2}\right ) \sin \left (2 d x +2 c \right )}{16 a \,d^{4}}\) | \(235\) |
derivativedivides | \(\frac {-\frac {c^{3} f^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {3 c^{2} d e \,f^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-3 c^{2} f^{3} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\frac {3 c \,d^{2} e^{2} f \left (\cos ^{2}\left (d x +c \right )\right )}{2}+6 c d e \,f^{2} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+3 c \,f^{3} \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )+\frac {d^{3} e^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-3 d^{2} e^{2} f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-3 d e \,f^{2} \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )-f^{3} \left (-\frac {\left (d x +c \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {3 \left (d x +c \right )^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{4}-\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}-\frac {\left (d x +c \right )^{3}}{2}\right )-\sin \left (d x +c \right ) c^{3} f^{3}+3 \sin \left (d x +c \right ) c^{2} d e \,f^{2}+3 c^{2} f^{3} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 \sin \left (d x +c \right ) c \,d^{2} e^{2} f -6 c d e \,f^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 c \,f^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+\sin \left (d x +c \right ) d^{3} e^{3}+3 d^{2} e^{2} f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+f^{3} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4} a}\) | \(736\) |
default | \(\frac {-\frac {c^{3} f^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {3 c^{2} d e \,f^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-3 c^{2} f^{3} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\frac {3 c \,d^{2} e^{2} f \left (\cos ^{2}\left (d x +c \right )\right )}{2}+6 c d e \,f^{2} \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+3 c \,f^{3} \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )+\frac {d^{3} e^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}-3 d^{2} e^{2} f \left (-\frac {\left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-3 d e \,f^{2} \left (-\frac {\left (d x +c \right )^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}\right )-f^{3} \left (-\frac {\left (d x +c \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+\frac {3 \left (d x +c \right )^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{4}-\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}-\frac {\left (d x +c \right )^{3}}{2}\right )-\sin \left (d x +c \right ) c^{3} f^{3}+3 \sin \left (d x +c \right ) c^{2} d e \,f^{2}+3 c^{2} f^{3} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 \sin \left (d x +c \right ) c \,d^{2} e^{2} f -6 c d e \,f^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 c \,f^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+\sin \left (d x +c \right ) d^{3} e^{3}+3 d^{2} e^{2} f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+f^{3} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4} a}\) | \(736\) |
1/8*(2*(f*x+e)*d*((f*x+e)^2*d^2-3/2*f^2)*cos(2*d*x+2*c)-3*((f*x+e)^2*d^2-1 /2*f^2)*f*sin(2*d*x+2*c)+8*(f*x+e)*d*((f*x+e)^2*d^2-6*f^2)*sin(d*x+c)+24*f *((f*x+e)^2*d^2-2*f^2)*cos(d*x+c)-2*d^3*e^3+24*d^2*e^2*f+3*d*e*f^2-48*f^3) /a/d^4
Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} - 2 \, {\left (2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} + 2 \, d^{3} e^{3} - 3 \, d e f^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x - 24 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + d^{2} e^{2} f - 2 \, f^{3}\right )} \cos \left (d x + c\right ) - {\left (8 \, d^{3} f^{3} x^{3} + 24 \, d^{3} e f^{2} x^{2} + 8 \, d^{3} e^{3} - 48 \, d e f^{2} + 24 \, {\left (d^{3} e^{2} f - 2 \, d f^{3}\right )} x - 3 \, {\left (2 \, d^{2} f^{3} x^{2} + 4 \, d^{2} e f^{2} x + 2 \, d^{2} e^{2} f - f^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, a d^{4}} \]
-1/8*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 - 2*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 + 2*d^3*e^3 - 3*d*e*f^2 + 3*(2*d^3*e^2*f - d*f^3)*x)*cos(d*x + c)^2 + 3*( 2*d^3*e^2*f - d*f^3)*x - 24*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f - 2*f ^3)*cos(d*x + c) - (8*d^3*f^3*x^3 + 24*d^3*e*f^2*x^2 + 8*d^3*e^3 - 48*d*e* f^2 + 24*(d^3*e^2*f - 2*d*f^3)*x - 3*(2*d^2*f^3*x^2 + 4*d^2*e*f^2*x + 2*d^ 2*e^2*f - f^3)*cos(d*x + c))*sin(d*x + c))/(a*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 2725 vs. \(2 (204) = 408\).
Time = 4.37 (sec) , antiderivative size = 2725, normalized size of antiderivative = 12.44 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
Piecewise((16*d**3*e**3*tan(c/2 + d*x/2)**3/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 16*d**3*e**3*tan(c/2 + d*x/2 )**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d **4) + 16*d**3*e**3*tan(c/2 + d*x/2)/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a* d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e**2*f*x*tan(c/2 + d*x/2)**4 /(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 48*d**3*e**2*f*x*tan(c/2 + d*x/2)**3/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16 *a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 36*d**3*e**2*f*x*tan(c/2 + d*x/2 )**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d **4) + 48*d**3*e**2*f*x*tan(c/2 + d*x/2)/(8*a*d**4*tan(c/2 + d*x/2)**4 + 1 6*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e**2*f*x/(8*a*d**4*tan(c /2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e*f**2 *x**2*tan(c/2 + d*x/2)**4/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/ 2 + d*x/2)**2 + 8*a*d**4) + 48*d**3*e*f**2*x**2*tan(c/2 + d*x/2)**3/(8*a*d **4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 36*d **3*e*f**2*x**2*tan(c/2 + d*x/2)**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d **4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 48*d**3*e*f**2*x**2*tan(c/2 + d*x/2) /(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e*f**2*x**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 2*d**3*f**3*x**3*tan(c/2 + d*x/2)**4/(8*a*d**4*...
Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (207) = 414\).
Time = 0.24 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.61 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {8 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} e^{3}}{a} - \frac {24 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c e^{2} f}{a d} + \frac {24 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c^{2} e f^{2}}{a d^{2}} - \frac {8 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )} c^{3} f^{3}}{a d^{3}} - \frac {6 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} e^{2} f}{a d} + \frac {12 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} c e f^{2}}{a d^{2}} - \frac {6 \, {\left (2 \, {\left (d x + c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 8 \, \cos \left (d x + c\right ) - \sin \left (2 \, d x + 2 \, c\right )\right )} c^{2} f^{3}}{a d^{3}} - \frac {6 \, {\left ({\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \cos \left (2 \, d x + 2 \, c\right ) + 16 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (2 \, d x + 2 \, c\right ) + 8 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} e f^{2}}{a d^{2}} + \frac {6 \, {\left ({\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \cos \left (2 \, d x + 2 \, c\right ) + 16 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (2 \, d x + 2 \, c\right ) + 8 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} c f^{3}}{a d^{3}} - \frac {{\left (2 \, {\left (2 \, {\left (d x + c\right )}^{3} - 3 \, d x - 3 \, c\right )} \cos \left (2 \, d x + 2 \, c\right ) + 48 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, {\left (d x + c\right )}^{2} - 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + 16 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} f^{3}}{a d^{3}}}{16 \, d} \]
-1/16*(8*(sin(d*x + c)^2 - 2*sin(d*x + c))*e^3/a - 24*(sin(d*x + c)^2 - 2* sin(d*x + c))*c*e^2*f/(a*d) + 24*(sin(d*x + c)^2 - 2*sin(d*x + c))*c^2*e*f ^2/(a*d^2) - 8*(sin(d*x + c)^2 - 2*sin(d*x + c))*c^3*f^3/(a*d^3) - 6*(2*(d *x + c)*cos(2*d*x + 2*c) + 8*(d*x + c)*sin(d*x + c) + 8*cos(d*x + c) - sin (2*d*x + 2*c))*e^2*f/(a*d) + 12*(2*(d*x + c)*cos(2*d*x + 2*c) + 8*(d*x + c )*sin(d*x + c) + 8*cos(d*x + c) - sin(2*d*x + 2*c))*c*e*f^2/(a*d^2) - 6*(2 *(d*x + c)*cos(2*d*x + 2*c) + 8*(d*x + c)*sin(d*x + c) + 8*cos(d*x + c) - sin(2*d*x + 2*c))*c^2*f^3/(a*d^3) - 6*((2*(d*x + c)^2 - 1)*cos(2*d*x + 2*c ) + 16*(d*x + c)*cos(d*x + c) - 2*(d*x + c)*sin(2*d*x + 2*c) + 8*((d*x + c )^2 - 2)*sin(d*x + c))*e*f^2/(a*d^2) + 6*((2*(d*x + c)^2 - 1)*cos(2*d*x + 2*c) + 16*(d*x + c)*cos(d*x + c) - 2*(d*x + c)*sin(2*d*x + 2*c) + 8*((d*x + c)^2 - 2)*sin(d*x + c))*c*f^3/(a*d^3) - (2*(2*(d*x + c)^3 - 3*d*x - 3*c) *cos(2*d*x + 2*c) + 48*((d*x + c)^2 - 2)*cos(d*x + c) - 3*(2*(d*x + c)^2 - 1)*sin(2*d*x + 2*c) + 16*((d*x + c)^3 - 6*d*x - 6*c)*sin(d*x + c))*f^3/(a *d^3))/d
Leaf count of result is larger than twice the leaf count of optimal. 3893 vs. \(2 (207) = 414\).
Time = 0.48 (sec) , antiderivative size = 3893, normalized size of antiderivative = 17.78 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
1/8*(2*d^3*f^3*x^3*tan(1/2*d*x)^4*tan(1/2*c)^4 - 16*d^3*f^3*x^3*tan(1/2*d* x)^4*tan(1/2*c)^3 - 16*d^3*f^3*x^3*tan(1/2*d*x)^3*tan(1/2*c)^4 + 6*d^3*e*f ^2*x^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 12*d^3*f^3*x^3*tan(1/2*d*x)^4*tan(1/2 *c)^2 - 32*d^3*f^3*x^3*tan(1/2*d*x)^3*tan(1/2*c)^3 - 48*d^3*e*f^2*x^2*tan( 1/2*d*x)^4*tan(1/2*c)^3 - 12*d^3*f^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^4 - 48* d^3*e*f^2*x^2*tan(1/2*d*x)^3*tan(1/2*c)^4 + 6*d^3*e^2*f*x*tan(1/2*d*x)^4*t an(1/2*c)^4 + 24*d^2*f^3*x^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 16*d^3*f^3*x^3* tan(1/2*d*x)^4*tan(1/2*c) - 36*d^3*e*f^2*x^2*tan(1/2*d*x)^4*tan(1/2*c)^2 - 96*d^3*e*f^2*x^2*tan(1/2*d*x)^3*tan(1/2*c)^3 - 48*d^3*e^2*f*x*tan(1/2*d*x )^4*tan(1/2*c)^3 + 12*d^2*f^3*x^2*tan(1/2*d*x)^4*tan(1/2*c)^3 - 16*d^3*f^3 *x^3*tan(1/2*d*x)*tan(1/2*c)^4 - 36*d^3*e*f^2*x^2*tan(1/2*d*x)^2*tan(1/2*c )^4 - 48*d^3*e^2*f*x*tan(1/2*d*x)^3*tan(1/2*c)^4 + 12*d^2*f^3*x^2*tan(1/2* d*x)^3*tan(1/2*c)^4 + 2*d^3*e^3*tan(1/2*d*x)^4*tan(1/2*c)^4 + 48*d^2*e*f^2 *x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 2*d^3*f^3*x^3*tan(1/2*d*x)^4 + 32*d^3*f^3 *x^3*tan(1/2*d*x)^3*tan(1/2*c) - 48*d^3*e*f^2*x^2*tan(1/2*d*x)^4*tan(1/2*c ) + 72*d^3*f^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - 36*d^3*e^2*f*x*tan(1/2*d* x)^4*tan(1/2*c)^2 + 32*d^3*f^3*x^3*tan(1/2*d*x)*tan(1/2*c)^3 - 96*d^3*e^2* f*x*tan(1/2*d*x)^3*tan(1/2*c)^3 - 96*d^2*f^3*x^2*tan(1/2*d*x)^3*tan(1/2*c) ^3 - 16*d^3*e^3*tan(1/2*d*x)^4*tan(1/2*c)^3 + 24*d^2*e*f^2*x*tan(1/2*d*x)^ 4*tan(1/2*c)^3 + 2*d^3*f^3*x^3*tan(1/2*c)^4 - 48*d^3*e*f^2*x^2*tan(1/2*...
Time = 3.22 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {3\,f^3\,\sin \left (2\,c+2\,d\,x\right )}{2}-48\,f^3\,\cos \left (c+d\,x\right )+8\,d^3\,e^3\,\sin \left (c+d\,x\right )+2\,d^3\,e^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,e^2\,f\,\sin \left (2\,c+2\,d\,x\right )+24\,d^2\,f^3\,x^2\,\cos \left (c+d\,x\right )+8\,d^3\,f^3\,x^3\,\sin \left (c+d\,x\right )-48\,d\,e\,f^2\,\sin \left (c+d\,x\right )-48\,d\,f^3\,x\,\sin \left (c+d\,x\right )+2\,d^3\,f^3\,x^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,f^3\,x^2\,\sin \left (2\,c+2\,d\,x\right )-3\,d\,e\,f^2\,\cos \left (2\,c+2\,d\,x\right )+24\,d^2\,e^2\,f\,\cos \left (c+d\,x\right )-3\,d\,f^3\,x\,\cos \left (2\,c+2\,d\,x\right )+48\,d^2\,e\,f^2\,x\,\cos \left (c+d\,x\right )+24\,d^3\,e^2\,f\,x\,\sin \left (c+d\,x\right )+6\,d^3\,e^2\,f\,x\,\cos \left (2\,c+2\,d\,x\right )-6\,d^2\,e\,f^2\,x\,\sin \left (2\,c+2\,d\,x\right )+24\,d^3\,e\,f^2\,x^2\,\sin \left (c+d\,x\right )+6\,d^3\,e\,f^2\,x^2\,\cos \left (2\,c+2\,d\,x\right )}{8\,a\,d^4} \]
((3*f^3*sin(2*c + 2*d*x))/2 - 48*f^3*cos(c + d*x) + 8*d^3*e^3*sin(c + d*x) + 2*d^3*e^3*cos(2*c + 2*d*x) - 3*d^2*e^2*f*sin(2*c + 2*d*x) + 24*d^2*f^3* x^2*cos(c + d*x) + 8*d^3*f^3*x^3*sin(c + d*x) - 48*d*e*f^2*sin(c + d*x) - 48*d*f^3*x*sin(c + d*x) + 2*d^3*f^3*x^3*cos(2*c + 2*d*x) - 3*d^2*f^3*x^2*s in(2*c + 2*d*x) - 3*d*e*f^2*cos(2*c + 2*d*x) + 24*d^2*e^2*f*cos(c + d*x) - 3*d*f^3*x*cos(2*c + 2*d*x) + 48*d^2*e*f^2*x*cos(c + d*x) + 24*d^3*e^2*f*x *sin(c + d*x) + 6*d^3*e^2*f*x*cos(2*c + 2*d*x) - 6*d^2*e*f^2*x*sin(2*c + 2 *d*x) + 24*d^3*e*f^2*x^2*sin(c + d*x) + 6*d^3*e*f^2*x^2*cos(2*c + 2*d*x))/ (8*a*d^4)