Integrand size = 23, antiderivative size = 167 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b d} \]
1/3*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(2/3)/b^(1/3)/d-1/6*ln(a^(2/3)-a^(1/3 )*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(2/3)/b^(1/3)/d-1/3*ln(a+b*si n(d*x+c)^3)/b/d-1/3*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))/a^(1/3)*3^(1 /2))/a^(2/3)/b^(1/3)/d*3^(1/2)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (-a^{2/3}+(-1)^{2/3} b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )+\left (-a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-\left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} b d} \]
((-a^(2/3) + (-1)^(2/3)*b^(2/3))*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sin[c + d*x]] + (-a^(2/3) + b^(2/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] - (a^( 2/3) + (-1)^(1/3)*b^(2/3))*Log[a^(1/3) + (-1)^(2/3)*b^(1/3)*Sin[c + d*x]]) /(3*a^(2/3)*b*d)
Time = 0.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3702, 2410, 750, 16, 792, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^3}{a+b \sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1-\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2410 |
| \(\displaystyle \frac {\int \frac {1}{b \sin ^3(c+d x)+a}d\sin (c+d x)-\int \frac {\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} \sin (c+d x)+\sqrt [3]{a}}d\sin (c+d x)}{3 a^{2/3}}-\int \frac {\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}-\int \frac {\sin ^2(c+d x)}{b \sin ^3(c+d x)+a}d\sin (c+d x)+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}}{d}\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)\right )}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{b^{2/3} \sin ^2(c+d x)-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+a^{2/3}}d\sin (c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a+b \sin ^3(c+d x)\right )}{3 b}}{d}\) |
(Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*Arc Tan[(1 - (2*b^(1/3)*Sin[c + d*x])/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3 ) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(2*b^(1/3)))/(3 *a^(2/3)) - Log[a + b*Sin[c + d*x]^3]/(3*b))/d
3.4.84.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + Si mp[C Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !RationalQ[ a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x _)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.54 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(\frac {i x}{b}+\frac {2 i c}{b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 a^{2} \textit {\_Z}^{3} d^{3} b^{3}+27 a^{2} b^{2} d^{2} \textit {\_Z}^{2}+9 a^{2} d \textit {\_Z} b +a^{2}-b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (6 i a d \textit {\_R} +\frac {2 i a}{b}\right ) {\mathrm e}^{i \left (d x +c \right )}-1\right )\right )\) | \(106\) |
| derivativedivides | \(\frac {\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3 b}}{d}\) | \(133\) |
| default | \(\frac {\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3 b}}{d}\) | \(133\) |
I*x/b+2*I/b/d*c+sum(_R*ln(exp(2*I*(d*x+c))+(6*I*a*d*_R+2*I/b*a)*exp(I*(d*x +c))-1),_R=RootOf(27*_Z^3*a^2*b^3*d^3+27*_Z^2*a^2*b^2*d^2+9*_Z*a^2*b*d+a^2 -b^2))
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 1049, normalized size of antiderivative = 6.28 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
-1/12*(2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d*log(-1/2*((1/2)^(1/3)*(I*sqrt(3 ) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2 /(b*d))*a*b*d + b*sin(d*x + c) + a) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^ 3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 3*sqrt(1/3)*b*d*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2 *b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2) ^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3 *d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)) - 6)*log(1/2*((1/2)^(1/3)*(I*s qrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3 ) + 2/(b*d))*a*b*d + 3/2*sqrt(1/3)*a*b*d*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b* d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3 ) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)) + 2*b* sin(d*x + c) - a) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b* d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d - 3*sqrt(1/3)*b*d*s qrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b ^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/( b*d))*b*d + 4)/(b^2*d^2)) - 6)*log(-1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1...
Timed out. \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Timed out} \]
Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {2 \, \sqrt {3} {\left (b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, a}{b}\right )} + 2 \, a\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b} - \frac {3 \, {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{18 \, d} \]
1/18*(2*sqrt(3)*(b*(3*(a/b)^(1/3) - 2*a/b) + 2*a)*arctan(-1/3*sqrt(3)*((a/ b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/(a*b) - 3*(2*(a/b)^(2/3) + 1)*log( sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(b*(a/b)^(2/3)) - 6*((a/b)^(2/3) - 1)*log((a/b)^(1/3) + sin(d*x + c))/(b*(a/b)^(2/3)))/d
Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {2 \, \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{b} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b}}{6 \, d} \]
-1/6*(2*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/a + 2*log(abs( b*sin(d*x + c)^3 + a))/b - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((- a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a*b) - (-a*b^2)^(1/3)*log(sin( d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a*b))/d
Time = 13.99 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\sum _{k=1}^3\ln \left (\left (\mathrm {root}\left (27\,a^2\,b^3\,d^3+27\,a^2\,b^2\,d^2+9\,a^2\,b\,d-b^2+a^2,d,k\right )\,b\,3+1\right )\,\left (a+b\,\sin \left (c+d\,x\right )+\mathrm {root}\left (27\,a^2\,b^3\,d^3+27\,a^2\,b^2\,d^2+9\,a^2\,b\,d-b^2+a^2,d,k\right )\,a\,b\,3\right )\right )\,\mathrm {root}\left (27\,a^2\,b^3\,d^3+27\,a^2\,b^2\,d^2+9\,a^2\,b\,d-b^2+a^2,d,k\right )}{d} \]