\(\int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [223]

Optimal result

Integrand size = 21, antiderivative size = 158 \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (a^2-6 b^2\right ) \cos (c+d x)}{a^5 d}-\frac {3 b \cos ^2(c+d x)}{2 a^4 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}-\frac {b^3 \left (a^2-b^2\right )}{2 a^6 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2-5 b^2\right )}{a^6 d (b+a \cos (c+d x))}+\frac {b \left (3 a^2-10 b^2\right ) \log (b+a \cos (c+d x))}{a^6 d} \]

[Out]

-(a^2-6*b^2)*cos(d*x+c)/a^5/d-3/2*b*cos(d*x+c)^2/a^4/d+1/3*cos(d*x+c)^3/a^3/d-1/2*b^3*(a^2-b^2)/a^6/d/(b+a*cos
(d*x+c))^2+b^2*(3*a^2-5*b^2)/a^6/d/(b+a*cos(d*x+c))+b*(3*a^2-10*b^2)*ln(b+a*cos(d*x+c))/a^6/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 908} \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {3 b \cos ^2(c+d x)}{2 a^4 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}+\frac {b^2 \left (3 a^2-5 b^2\right )}{a^6 d (a \cos (c+d x)+b)}+\frac {b \left (3 a^2-10 b^2\right ) \log (a \cos (c+d x)+b)}{a^6 d}-\frac {b^3 \left (a^2-b^2\right )}{2 a^6 d (a \cos (c+d x)+b)^2}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x)}{a^5 d} \]

[In]

Int[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]

[Out]

-(((a^2 - 6*b^2)*Cos[c + d*x])/(a^5*d)) - (3*b*Cos[c + d*x]^2)/(2*a^4*d) + Cos[c + d*x]^3/(3*a^3*d) - (b^3*(a^
2 - b^2))/(2*a^6*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 - 5*b^2))/(a^6*d*(b + a*Cos[c + d*x])) + (b*(3*a^2 -
10*b^2)*Log[b + a*Cos[c + d*x]])/(a^6*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2916

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^3(c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (a^2-x^2\right )}{a^3 (-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \frac {x^3 \left (a^2-x^2\right )}{(-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (a^2 \left (1-\frac {6 b^2}{a^2}\right )+\frac {-a^2 b^3+b^5}{(b-x)^3}+\frac {3 a^2 b^2-5 b^4}{(b-x)^2}+\frac {-3 a^2 b+10 b^3}{b-x}-3 b x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d} \\ & = -\frac {\left (a^2-6 b^2\right ) \cos (c+d x)}{a^5 d}-\frac {3 b \cos ^2(c+d x)}{2 a^4 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}-\frac {b^3 \left (a^2-b^2\right )}{2 a^6 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2-5 b^2\right )}{a^6 d (b+a \cos (c+d x))}+\frac {b \left (3 a^2-10 b^2\right ) \log (b+a \cos (c+d x))}{a^6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \left (9 a^4 (b+2 a \cos (c+d x))-(b+a \cos (c+d x))^2 \left (72 a \left (a^2-8 b^2\right ) \cos (c+d x)+\frac {-9 a^4 b+48 a^2 b^3-48 b^5}{(b+a \cos (c+d x))^2}+\frac {6 \left (3 a^4-48 a^2 b^2+80 b^4\right )}{b+a \cos (c+d x)}+72 a^2 b \cos (2 (c+d x))-8 a^3 \cos (3 (c+d x))+96 \left (-3 a^2 b+10 b^3\right ) \log (b+a \cos (c+d x))\right )\right ) \sec ^3(c+d x)}{96 a^6 d (a+b \sec (c+d x))^3} \]

[In]

Integrate[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]

[Out]

((b + a*Cos[c + d*x])*(9*a^4*(b + 2*a*Cos[c + d*x]) - (b + a*Cos[c + d*x])^2*(72*a*(a^2 - 8*b^2)*Cos[c + d*x]
+ (-9*a^4*b + 48*a^2*b^3 - 48*b^5)/(b + a*Cos[c + d*x])^2 + (6*(3*a^4 - 48*a^2*b^2 + 80*b^4))/(b + a*Cos[c + d
*x]) + 72*a^2*b*Cos[2*(c + d*x)] - 8*a^3*Cos[3*(c + d*x)] + 96*(-3*a^2*b + 10*b^3)*Log[b + a*Cos[c + d*x]]))*S
ec[c + d*x]^3)/(96*a^6*d*(a + b*Sec[c + d*x])^3)

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\frac {\frac {\cos \left (d x +c \right )^{3} a^{2}}{3}-\frac {3 a b \cos \left (d x +c \right )^{2}}{2}-\cos \left (d x +c \right ) a^{2}+6 \cos \left (d x +c \right ) b^{2}}{a^{5}}-\frac {b^{3} \left (a^{2}-b^{2}\right )}{2 a^{6} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (3 a^{2}-5 b^{2}\right )}{a^{6} \left (b +a \cos \left (d x +c \right )\right )}+\frac {b \left (3 a^{2}-10 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{6}}}{d}\)	\(144\)
default	\(\frac {\frac {\frac {\cos \left (d x +c \right )^{3} a^{2}}{3}-\frac {3 a b \cos \left (d x +c \right )^{2}}{2}-\cos \left (d x +c \right ) a^{2}+6 \cos \left (d x +c \right ) b^{2}}{a^{5}}-\frac {b^{3} \left (a^{2}-b^{2}\right )}{2 a^{6} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b^{2} \left (3 a^{2}-5 b^{2}\right )}{a^{6} \left (b +a \cos \left (d x +c \right )\right )}+\frac {b \left (3 a^{2}-10 b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{6}}}{d}\)	\(144\)
parallelrisch	\(\frac {72 \left (a +b \right ) \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \left (a^{2}-\frac {10 b^{2}}{3}\right ) b \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-72 \left (a +b \right ) \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \left (a^{2}-\frac {10 b^{2}}{3}\right ) b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+a \left (\left (16 a^{5}-16 a^{4} b -248 a^{3} b^{2}+120 a^{2} b^{3}+360 a \,b^{4}\right ) \cos \left (2 d x +2 c \right )-7 \left (a +b \right ) \left (a^{2}-\frac {40 b^{2}}{7}\right ) a^{2} \cos \left (3 d x +3 c \right )-5 b \,a^{3} \left (a +b \right ) \cos \left (4 d x +4 c \right )+\left (a +b \right ) a^{4} \cos \left (5 d x +5 c \right )+\left (-26 a^{5}+38 a^{4} b +424 a^{3} b^{2}-504 a^{2} b^{3}-480 a \,b^{4}+480 b^{5}\right ) \cos \left (d x +c \right )+16 a^{5}-11 a^{4} b -211 a^{3} b^{2}+344 a^{2} b^{3}+120 a \,b^{4}-480 b^{5}\right )}{24 \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) d \,a^{6} \left (a +b \right )}\)	\(368\)
risch	\(\frac {20 i b^{3} c}{a^{6} d}-\frac {3 i x b}{a^{4}}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}-\frac {3 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{3} d}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{a^{5} d}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{3} d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{a^{5} d}-\frac {3 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}-\frac {6 i b c}{a^{4} d}+\frac {10 i x \,b^{3}}{a^{6}}-\frac {2 b^{2} \left (-3 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+5 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-5 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+9 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{3} {\mathrm e}^{i \left (d x +c \right )}+5 b^{2} a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{a^{6} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{4} d}-\frac {10 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{6} d}\)	\(388\)
norman	\(\frac {\frac {\left (6 a^{4} b -12 a^{3} b^{2}-14 a^{2} b^{3}+40 a \,b^{4}-20 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a^{5} d \left (a -b \right )}+\frac {-4 a^{6}+62 a^{4} b^{2}-118 a^{2} b^{4}+60 b^{6}}{3 a^{5} d \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 \left (2 a^{6}-5 a^{5} b +18 a^{4} b^{2}-17 a^{3} b^{3}-48 a^{2} b^{4}+90 a \,b^{5}-40 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d \,a^{5} \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 \left (-10 a^{6}+25 a^{5} b +12 a^{4} b^{2}-131 a^{3} b^{3}+14 a^{2} b^{4}+270 a \,b^{5}-180 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d \,a^{5} \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (-4 a^{6}+2 a^{5} b +60 a^{4} b^{2}+154 a^{3} b^{3}-272 a^{2} b^{4}-180 a \,b^{5}+240 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d \,a^{5} \left (a^{2}-2 a b +b^{2}\right )}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}+\frac {b \left (3 a^{2}-10 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \,a^{6}}-\frac {b \left (3 a^{2}-10 b^{2}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \,a^{6}}\)	\(487\)

[In]

int(sin(d*x+c)^3/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/a^5*(1/3*cos(d*x+c)^3*a^2-3/2*a*b*cos(d*x+c)^2-cos(d*x+c)*a^2+6*cos(d*x+c)*b^2)-1/2*b^3*(a^2-b^2)/a^6/(
b+a*cos(d*x+c))^2+1/a^6*b^2*(3*a^2-5*b^2)/(b+a*cos(d*x+c))+1/a^6*b*(3*a^2-10*b^2)*ln(b+a*cos(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {4 \, a^{5} \cos \left (d x + c\right )^{5} - 10 \, a^{4} b \cos \left (d x + c\right )^{4} + 39 \, a^{2} b^{3} - 54 \, b^{5} - 4 \, {\left (3 \, a^{5} - 10 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (5 \, a^{4} b - 42 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (7 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right ) + 12 \, {\left (3 \, a^{2} b^{3} - 10 \, b^{5} + {\left (3 \, a^{4} b - 10 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} - 10 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{12 \, {\left (a^{8} d \cos \left (d x + c\right )^{2} + 2 \, a^{7} b d \cos \left (d x + c\right ) + a^{6} b^{2} d\right )}} \]

[In]

integrate(sin(d*x+c)^3/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/12*(4*a^5*cos(d*x + c)^5 - 10*a^4*b*cos(d*x + c)^4 + 39*a^2*b^3 - 54*b^5 - 4*(3*a^5 - 10*a^3*b^2)*cos(d*x +
c)^3 - 3*(5*a^4*b - 42*a^2*b^3)*cos(d*x + c)^2 + 6*(7*a^3*b^2 + 2*a*b^4)*cos(d*x + c) + 12*(3*a^2*b^3 - 10*b^5
+ (3*a^4*b - 10*a^2*b^3)*cos(d*x + c)^2 + 2*(3*a^3*b^2 - 10*a*b^4)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^
8*d*cos(d*x + c)^2 + 2*a^7*b*d*cos(d*x + c) + a^6*b^2*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**3/(a+b*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (5 \, a^{2} b^{3} - 9 \, b^{5} + 2 \, {\left (3 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{8} \cos \left (d x + c\right )^{2} + 2 \, a^{7} b \cos \left (d x + c\right ) + a^{6} b^{2}} + \frac {2 \, a^{2} \cos \left (d x + c\right )^{3} - 9 \, a b \cos \left (d x + c\right )^{2} - 6 \, {\left (a^{2} - 6 \, b^{2}\right )} \cos \left (d x + c\right )}{a^{5}} + \frac {6 \, {\left (3 \, a^{2} b - 10 \, b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{6}}}{6 \, d} \]

[In]

integrate(sin(d*x+c)^3/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6*(3*(5*a^2*b^3 - 9*b^5 + 2*(3*a^3*b^2 - 5*a*b^4)*cos(d*x + c))/(a^8*cos(d*x + c)^2 + 2*a^7*b*cos(d*x + c) +
a^6*b^2) + (2*a^2*cos(d*x + c)^3 - 9*a*b*cos(d*x + c)^2 - 6*(a^2 - 6*b^2)*cos(d*x + c))/a^5 + 6*(3*a^2*b - 10
*b^3)*log(a*cos(d*x + c) + b)/a^6)/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.08 \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {{\left (3 \, a^{2} b - 10 \, b^{3}\right )} \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{6} d} + \frac {5 \, a^{2} b^{3} - 9 \, b^{5} + \frac {2 \, {\left (3 \, a^{3} b^{2} d - 5 \, a b^{4} d\right )} \cos \left (d x + c\right )}{d}}{2 \, {\left (a \cos \left (d x + c\right ) + b\right )}^{2} a^{6} d} + \frac {2 \, a^{6} d^{8} \cos \left (d x + c\right )^{3} - 9 \, a^{5} b d^{8} \cos \left (d x + c\right )^{2} - 6 \, a^{6} d^{8} \cos \left (d x + c\right ) + 36 \, a^{4} b^{2} d^{8} \cos \left (d x + c\right )}{6 \, a^{9} d^{9}} \]

[In]

integrate(sin(d*x+c)^3/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

(3*a^2*b - 10*b^3)*log(abs(-a*cos(d*x + c) - b))/(a^6*d) + 1/2*(5*a^2*b^3 - 9*b^5 + 2*(3*a^3*b^2*d - 5*a*b^4*d
)*cos(d*x + c)/d)/((a*cos(d*x + c) + b)^2*a^6*d) + 1/6*(2*a^6*d^8*cos(d*x + c)^3 - 9*a^5*b*d^8*cos(d*x + c)^2
- 6*a^6*d^8*cos(d*x + c) + 36*a^4*b^2*d^8*cos(d*x + c))/(a^9*d^9)

Mupad [B] (verification not implemented)

Time = 13.55 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}-\frac {\cos \left (c+d\,x\right )\,\left (5\,b^4-3\,a^2\,b^2\right )+\frac {9\,b^5-5\,a^2\,b^3}{2\,a}}{d\,\left (a^7\,{\cos \left (c+d\,x\right )}^2+2\,a^6\,b\,\cos \left (c+d\,x\right )+a^5\,b^2\right )}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^3}-\frac {6\,b^2}{a^5}\right )}{d}-\frac {3\,b\,{\cos \left (c+d\,x\right )}^2}{2\,a^4\,d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (3\,a^2\,b-10\,b^3\right )}{a^6\,d} \]

[In]

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

[Out]

cos(c + d*x)^3/(3*a^3*d) - (cos(c + d*x)*(5*b^4 - 3*a^2*b^2) + (9*b^5 - 5*a^2*b^3)/(2*a))/(d*(a^7*cos(c + d*x)
^2 + a^5*b^2 + 2*a^6*b*cos(c + d*x))) - (cos(c + d*x)*(1/a^3 - (6*b^2)/a^5))/d - (3*b*cos(c + d*x)^2)/(2*a^4*d
) + (log(b + a*cos(c + d*x))*(3*a^2*b - 10*b^3))/(a^6*d)