\(\int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [263]

Optimal result

Integrand size = 28, antiderivative size = 219 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d}-\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {(e+f x)^3 \sin (c+d x)}{a d}+\frac {3 f^3 \cos (c+d x) \sin (c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d} \]

[Out]

-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*cos(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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4619, 3377, 2718, 4489, 3392, 32, 2715, 8} \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f^3 \sin (c+d x) \cos (c+d x)}{8 a d^4}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 f (e+f x)^2 \sin (c+d x) \cos (c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d} \]

[In]

Int[((e + f*x)^3*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(-3*f^3*x)/(8*a*d^3) + (e + f*x)^3/(4*a*d) - (6*f^3*Cos[c + d*x])/(a*d^4) + (3*f*(e + f*x)^2*Cos[c + d*x])/(a*
d^2) - (6*f^2*(e + f*x)*Sin[c + d*x])/(a*d^3) + ((e + f*x)^3*Sin[c + d*x])/(a*d) + (3*f^3*Cos[c + d*x]*Sin[c +
d*x])/(8*a*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(4*a*d^2) + (3*f^2*(e + f*x)*Sin[c + d*x]^2)/(4
*a*d^3) - ((e + f*x)^3*Sin[c + d*x]^2)/(2*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 4489

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] - Dist[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]

Rule 4619

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)*S
in[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 1] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \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} \\ & = \frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d}+\frac {(3 f) \int (e+f x)^2 \sin ^2(c+d x) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d} \\ & = \frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}+\frac {(e+f x)^3 \sin (c+d x)}{a d}-\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}+\frac {(3 f) \int (e+f x)^2 \, dx}{4 a d}-\frac {\left (6 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}-\frac {\left (3 f^3\right ) \int \sin ^2(c+d x) \, dx}{4 a d^3} \\ & = \frac {(e+f x)^3}{4 a d}+\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}-\frac {\left (3 f^3\right ) \int 1 \, dx}{8 a d^3}+\frac {\left (6 f^3\right ) \int \sin (c+d x) \, dx}{a d^3} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.60 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {96 f \left (-2 f^2+d^2 (e+f x)^2\right ) \cos (c+d x)+4 d (e+f x) \left (-3 f^2+2 d^2 (e+f x)^2\right ) \cos (2 (c+d x))+4 \left (8 d (e+f x) \left (-6 f^2+d^2 (e+f x)^2\right )-3 f \left (-f^2+2 d^2 (e+f x)^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{32 a d^4} \]

[In]

Integrate[((e + f*x)^3*Cos[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(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)

Maple [A] (verified)

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}\)	\(736\)

[In]

int((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} - 2 \, {\left (2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} + 2 \, d^{3} e^{3} - 3 \, d e f^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x - 24 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + d^{2} e^{2} f - 2 \, f^{3}\right )} \cos \left (d x + c\right ) - {\left (8 \, d^{3} f^{3} x^{3} + 24 \, d^{3} e f^{2} x^{2} + 8 \, d^{3} e^{3} - 48 \, d e f^{2} + 24 \, {\left (d^{3} e^{2} f - 2 \, d f^{3}\right )} x - 3 \, {\left (2 \, d^{2} f^{3} x^{2} + 4 \, d^{2} e f^{2} x + 2 \, d^{2} e^{2} f - f^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, a d^{4}} \]

[In]

integrate((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-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)

Sympy [B] (verification not implemented)

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 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)**3*cos(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

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 + 16*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*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*f**3*x**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) - 12*d**3*f**3*x**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*f**3*x**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) + 2*d**3*f**3*x**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) + 12*d*
*2*e**2*f*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) + 48*d
**2*e**2*f*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) - 12*
d**2*e**2*f*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) + 48*d*
*2*e**2*f/(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**2*e*f**2*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) + 24*d**2*e*f**2*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) - 24*d**2*e*f**2*x*t
an(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) + 48*d**2*e*f**2*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) - 24*d**2*f**3*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) + 12*d**2*f**3*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) - 12*d**2*f**3*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) + 24*d**2*f**3*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) - 96*d*e*f**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) + 24*d*e*f**2*tan(c/2 + d*x/2)**2/(8*a*d**4*ta
n(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 96*d*e*f**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) - 3*d*f**3*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) - 96*d*f**3*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) + 18*d*f**3*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) - 96*d*f**3*x*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) - 3*d*f**3*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*f**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) - 96*f**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) + 6*f**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) -
96*f**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), Ne(d, 0)), ((e**3*x + 3*e*
*2*f*x**2/2 + e*f**2*x**3 + f**3*x**4/4)*cos(c)**3/(a*sin(c) + a), True))

Maxima [B] (verification not implemented)

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} \]

[In]

integrate((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-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*(si
n(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

Giac [B] (verification not implemented)

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 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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*t
an(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*tan(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*d*x)*tan(1/2*c)^4 - 36*d^3*e^2*f*x*tan(1/2*d*x)^2*tan(1/2*c)^4
- 16*d^3*e^3*tan(1/2*d*x)^3*tan(1/2*c)^4 + 24*d^2*e*f^2*x*tan(1/2*d*x)^3*tan(1/2*c)^4 + 24*d^2*e^2*f*tan(1/2*d
*x)^4*tan(1/2*c)^4 - 3*d*f^3*x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 16*d^3*f^3*x^3*tan(1/2*d*x)^3 + 6*d^3*e*f^2*x^2*t
an(1/2*d*x)^4 + 96*d^3*e*f^2*x^2*tan(1/2*d*x)^3*tan(1/2*c) - 48*d^3*e^2*f*x*tan(1/2*d*x)^4*tan(1/2*c) - 12*d^2
*f^3*x^2*tan(1/2*d*x)^4*tan(1/2*c) + 216*d^3*e*f^2*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 72*d^2*f^3*x^2*tan(1/2*d*
x)^3*tan(1/2*c)^2 - 12*d^3*e^3*tan(1/2*d*x)^4*tan(1/2*c)^2 + 16*d^3*f^3*x^3*tan(1/2*c)^3 + 96*d^3*e*f^2*x^2*ta
n(1/2*d*x)*tan(1/2*c)^3 - 72*d^2*f^3*x^2*tan(1/2*d*x)^2*tan(1/2*c)^3 - 32*d^3*e^3*tan(1/2*d*x)^3*tan(1/2*c)^3
- 192*d^2*e*f^2*x*tan(1/2*d*x)^3*tan(1/2*c)^3 + 12*d^2*e^2*f*tan(1/2*d*x)^4*tan(1/2*c)^3 + 96*d*f^3*x*tan(1/2*
d*x)^4*tan(1/2*c)^3 + 6*d^3*e*f^2*x^2*tan(1/2*c)^4 - 48*d^3*e^2*f*x*tan(1/2*d*x)*tan(1/2*c)^4 - 12*d^2*f^3*x^2
*tan(1/2*d*x)*tan(1/2*c)^4 - 12*d^3*e^3*tan(1/2*d*x)^2*tan(1/2*c)^4 + 12*d^2*e^2*f*tan(1/2*d*x)^3*tan(1/2*c)^4
+ 96*d*f^3*x*tan(1/2*d*x)^3*tan(1/2*c)^4 - 3*d*e*f^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 12*d^3*f^3*x^3*tan(1/2*d*x
)^2 + 48*d^3*e*f^2*x^2*tan(1/2*d*x)^3 + 6*d^3*e^2*f*x*tan(1/2*d*x)^4 - 24*d^2*f^3*x^2*tan(1/2*d*x)^4 - 32*d^3*
f^3*x^3*tan(1/2*d*x)*tan(1/2*c) + 96*d^3*e^2*f*x*tan(1/2*d*x)^3*tan(1/2*c) - 96*d^2*f^3*x^2*tan(1/2*d*x)^3*tan
(1/2*c) - 16*d^3*e^3*tan(1/2*d*x)^4*tan(1/2*c) - 24*d^2*e*f^2*x*tan(1/2*d*x)^4*tan(1/2*c) - 12*d^3*f^3*x^3*tan
(1/2*c)^2 + 216*d^3*e^2*f*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 144*d^2*e*f^2*x*tan(1/2*d*x)^3*tan(1/2*c)^2 + 18*d*f
^3*x*tan(1/2*d*x)^4*tan(1/2*c)^2 + 48*d^3*e*f^2*x^2*tan(1/2*c)^3 + 96*d^3*e^2*f*x*tan(1/2*d*x)*tan(1/2*c)^3 -
96*d^2*f^3*x^2*tan(1/2*d*x)*tan(1/2*c)^3 - 144*d^2*e*f^2*x*tan(1/2*d*x)^2*tan(1/2*c)^3 - 96*d^2*e^2*f*tan(1/2*
d*x)^3*tan(1/2*c)^3 + 48*d*f^3*x*tan(1/2*d*x)^3*tan(1/2*c)^3 + 96*d*e*f^2*tan(1/2*d*x)^4*tan(1/2*c)^3 + 6*d^3*
e^2*f*x*tan(1/2*c)^4 - 24*d^2*f^3*x^2*tan(1/2*c)^4 - 16*d^3*e^3*tan(1/2*d*x)*tan(1/2*c)^4 - 24*d^2*e*f^2*x*tan
(1/2*d*x)*tan(1/2*c)^4 + 18*d*f^3*x*tan(1/2*d*x)^2*tan(1/2*c)^4 + 96*d*e*f^2*tan(1/2*d*x)^3*tan(1/2*c)^4 - 48*
f^3*tan(1/2*d*x)^4*tan(1/2*c)^4 + 16*d^3*f^3*x^3*tan(1/2*d*x) - 36*d^3*e*f^2*x^2*tan(1/2*d*x)^2 + 48*d^3*e^2*f
*x*tan(1/2*d*x)^3 + 12*d^2*f^3*x^2*tan(1/2*d*x)^3 + 2*d^3*e^3*tan(1/2*d*x)^4 - 48*d^2*e*f^2*x*tan(1/2*d*x)^4 +
16*d^3*f^3*x^3*tan(1/2*c) - 96*d^3*e*f^2*x^2*tan(1/2*d*x)*tan(1/2*c) + 72*d^2*f^3*x^2*tan(1/2*d*x)^2*tan(1/2*
c) + 32*d^3*e^3*tan(1/2*d*x)^3*tan(1/2*c) - 192*d^2*e*f^2*x*tan(1/2*d*x)^3*tan(1/2*c) - 12*d^2*e^2*f*tan(1/2*d
*x)^4*tan(1/2*c) + 96*d*f^3*x*tan(1/2*d*x)^4*tan(1/2*c) - 36*d^3*e*f^2*x^2*tan(1/2*c)^2 + 72*d^2*f^3*x^2*tan(1
/2*d*x)*tan(1/2*c)^2 + 72*d^3*e^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - 72*d^2*e^2*f*tan(1/2*d*x)^3*tan(1/2*c)^2 + 18*
d*e*f^2*tan(1/2*d*x)^4*tan(1/2*c)^2 + 48*d^3*e^2*f*x*tan(1/2*c)^3 + 12*d^2*f^3*x^2*tan(1/2*c)^3 + 32*d^3*e^3*t
an(1/2*d*x)*tan(1/2*c)^3 - 192*d^2*e*f^2*x*tan(1/2*d*x)*tan(1/2*c)^3 - 72*d^2*e^2*f*tan(1/2*d*x)^2*tan(1/2*c)^
3 + 48*d*e*f^2*tan(1/2*d*x)^3*tan(1/2*c)^3 - 6*f^3*tan(1/2*d*x)^4*tan(1/2*c)^3 + 2*d^3*e^3*tan(1/2*c)^4 - 48*d
^2*e*f^2*x*tan(1/2*c)^4 - 12*d^2*e^2*f*tan(1/2*d*x)*tan(1/2*c)^4 + 96*d*f^3*x*tan(1/2*d*x)*tan(1/2*c)^4 + 18*d
*e*f^2*tan(1/2*d*x)^2*tan(1/2*c)^4 - 6*f^3*tan(1/2*d*x)^3*tan(1/2*c)^4 + 2*d^3*f^3*x^3 + 48*d^3*e*f^2*x^2*tan(
1/2*d*x) - 36*d^3*e^2*f*x*tan(1/2*d*x)^2 + 16*d^3*e^3*tan(1/2*d*x)^3 + 24*d^2*e*f^2*x*tan(1/2*d*x)^3 - 24*d^2*
e^2*f*tan(1/2*d*x)^4 - 3*d*f^3*x*tan(1/2*d*x)^4 + 48*d^3*e*f^2*x^2*tan(1/2*c) - 96*d^3*e^2*f*x*tan(1/2*d*x)*ta
n(1/2*c) - 96*d^2*f^3*x^2*tan(1/2*d*x)*tan(1/2*c) + 144*d^2*e*f^2*x*tan(1/2*d*x)^2*tan(1/2*c) - 96*d^2*e^2*f*t
an(1/2*d*x)^3*tan(1/2*c) - 48*d*f^3*x*tan(1/2*d*x)^3*tan(1/2*c) + 96*d*e*f^2*tan(1/2*d*x)^4*tan(1/2*c) - 36*d^
3*e^2*f*x*tan(1/2*c)^2 + 144*d^2*e*f^2*x*tan(1/2*d*x)*tan(1/2*c)^2 - 108*d*f^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2 +
16*d^3*e^3*tan(1/2*c)^3 + 24*d^2*e*f^2*x*tan(1/2*c)^3 - 96*d^2*e^2*f*tan(1/2*d*x)*tan(1/2*c)^3 - 48*d*f^3*x*t
an(1/2*d*x)*tan(1/2*c)^3 + 192*f^3*tan(1/2*d*x)^3*tan(1/2*c)^3 - 24*d^2*e^2*f*tan(1/2*c)^4 - 3*d*f^3*x*tan(1/2
*c)^4 + 96*d*e*f^2*tan(1/2*d*x)*tan(1/2*c)^4 + 6*d^3*e*f^2*x^2 + 48*d^3*e^2*f*x*tan(1/2*d*x) - 12*d^2*f^3*x^2*
tan(1/2*d*x) - 12*d^3*e^3*tan(1/2*d*x)^2 + 12*d^2*e^2*f*tan(1/2*d*x)^3 - 96*d*f^3*x*tan(1/2*d*x)^3 - 3*d*e*f^2
*tan(1/2*d*x)^4 + 48*d^3*e^2*f*x*tan(1/2*c) - 12*d^2*f^3*x^2*tan(1/2*c) - 32*d^3*e^3*tan(1/2*d*x)*tan(1/2*c) -
192*d^2*e*f^2*x*tan(1/2*d*x)*tan(1/2*c) + 72*d^2*e^2*f*tan(1/2*d*x)^2*tan(1/2*c) - 48*d*e*f^2*tan(1/2*d*x)^3*
tan(1/2*c) + 6*f^3*tan(1/2*d*x)^4*tan(1/2*c) - 12*d^3*e^3*tan(1/2*c)^2 + 72*d^2*e^2*f*tan(1/2*d*x)*tan(1/2*c)^
2 - 108*d*e*f^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 36*f^3*tan(1/2*d*x)^3*tan(1/2*c)^2 + 12*d^2*e^2*f*tan(1/2*c)^3 -
96*d*f^3*x*tan(1/2*c)^3 - 48*d*e*f^2*tan(1/2*d*x)*tan(1/2*c)^3 + 36*f^3*tan(1/2*d*x)^2*tan(1/2*c)^3 - 3*d*e*f
^2*tan(1/2*c)^4 + 6*f^3*tan(1/2*d*x)*tan(1/2*c)^4 + 6*d^3*e^2*f*x + 24*d^2*f^3*x^2 + 16*d^3*e^3*tan(1/2*d*x) -
24*d^2*e*f^2*x*tan(1/2*d*x) + 18*d*f^3*x*tan(1/2*d*x)^2 - 96*d*e*f^2*tan(1/2*d*x)^3 + 48*f^3*tan(1/2*d*x)^4 +
16*d^3*e^3*tan(1/2*c) - 24*d^2*e*f^2*x*tan(1/2*c) - 96*d^2*e^2*f*tan(1/2*d*x)*tan(1/2*c) + 48*d*f^3*x*tan(1/2
*d*x)*tan(1/2*c) + 192*f^3*tan(1/2*d*x)^3*tan(1/2*c) + 18*d*f^3*x*tan(1/2*c)^2 - 96*d*e*f^2*tan(1/2*c)^3 + 192
*f^3*tan(1/2*d*x)*tan(1/2*c)^3 + 48*f^3*tan(1/2*c)^4 + 2*d^3*e^3 + 48*d^2*e*f^2*x - 12*d^2*e^2*f*tan(1/2*d*x)
- 96*d*f^3*x*tan(1/2*d*x) + 18*d*e*f^2*tan(1/2*d*x)^2 - 6*f^3*tan(1/2*d*x)^3 - 12*d^2*e^2*f*tan(1/2*c) - 96*d*
f^3*x*tan(1/2*c) + 48*d*e*f^2*tan(1/2*d*x)*tan(1/2*c) - 36*f^3*tan(1/2*d*x)^2*tan(1/2*c) + 18*d*e*f^2*tan(1/2*
c)^2 - 36*f^3*tan(1/2*d*x)*tan(1/2*c)^2 - 6*f^3*tan(1/2*c)^3 + 24*d^2*e^2*f - 3*d*f^3*x - 96*d*e*f^2*tan(1/2*d
*x) - 96*d*e*f^2*tan(1/2*c) + 192*f^3*tan(1/2*d*x)*tan(1/2*c) - 3*d*e*f^2 + 6*f^3*tan(1/2*d*x) + 6*f^3*tan(1/2
*c) - 48*f^3)/(a*d^4*tan(1/2*d*x)^4*tan(1/2*c)^4 + 2*a*d^4*tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*a*d^4*tan(1/2*d*x)^
2*tan(1/2*c)^4 + a*d^4*tan(1/2*d*x)^4 + 4*a*d^4*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*d^4*tan(1/2*c)^4 + 2*a*d^4*tan
(1/2*d*x)^2 + 2*a*d^4*tan(1/2*c)^2 + a*d^4)

Mupad [B] (verification not implemented)

Time = 3.22 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {3\,f^3\,\sin \left (2\,c+2\,d\,x\right )}{2}-48\,f^3\,\cos \left (c+d\,x\right )+8\,d^3\,e^3\,\sin \left (c+d\,x\right )+2\,d^3\,e^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,e^2\,f\,\sin \left (2\,c+2\,d\,x\right )+24\,d^2\,f^3\,x^2\,\cos \left (c+d\,x\right )+8\,d^3\,f^3\,x^3\,\sin \left (c+d\,x\right )-48\,d\,e\,f^2\,\sin \left (c+d\,x\right )-48\,d\,f^3\,x\,\sin \left (c+d\,x\right )+2\,d^3\,f^3\,x^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,f^3\,x^2\,\sin \left (2\,c+2\,d\,x\right )-3\,d\,e\,f^2\,\cos \left (2\,c+2\,d\,x\right )+24\,d^2\,e^2\,f\,\cos \left (c+d\,x\right )-3\,d\,f^3\,x\,\cos \left (2\,c+2\,d\,x\right )+48\,d^2\,e\,f^2\,x\,\cos \left (c+d\,x\right )+24\,d^3\,e^2\,f\,x\,\sin \left (c+d\,x\right )+6\,d^3\,e^2\,f\,x\,\cos \left (2\,c+2\,d\,x\right )-6\,d^2\,e\,f^2\,x\,\sin \left (2\,c+2\,d\,x\right )+24\,d^3\,e\,f^2\,x^2\,\sin \left (c+d\,x\right )+6\,d^3\,e\,f^2\,x^2\,\cos \left (2\,c+2\,d\,x\right )}{8\,a\,d^4} \]

[In]

int((cos(c + d*x)^3*(e + f*x)^3)/(a + a*sin(c + d*x)),x)

[Out]

((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*sin(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)