Integrand size = 13, antiderivative size = 84 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=-\frac {\arctan \left (\frac {1+\sqrt [3]{6} \cos (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}+\frac {\log \left (6^{2/3}-3 \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\log \left (2 \sqrt [3]{6}+6^{2/3} \cos (x)+3 \cos ^2(x)\right )}{12 \sqrt [3]{6}} \] Output:
-1/12*arctan(1/3*(1+6^(1/3)*cos(x))*3^(1/2))*2^(2/3)*3^(1/6)+1/36*ln(6^(2/ 3)-3*cos(x))*6^(2/3)-1/72*ln(2*6^(1/3)+6^(2/3)*cos(x)+3*cos(x)^2)*6^(2/3)
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=\frac {1}{72} \left (-6 2^{2/3} \sqrt [6]{3} \arctan \left (\frac {1+\sqrt [3]{6} \cos (x)}{\sqrt {3}}\right )+6^{2/3} \left (2 \log \left (2-\sqrt [3]{6} \cos (x)\right )-\log \left (4+2 \sqrt [3]{6} \cos (x)+6^{2/3} \cos ^2(x)\right )\right )\right ) \] Input:
Integrate[Sin[x]/(4 - 3*Cos[x]^3),x]
Output:
(-6*2^(2/3)*3^(1/6)*ArcTan[(1 + 6^(1/3)*Cos[x])/Sqrt[3]] + 6^(2/3)*(2*Log[ 2 - 6^(1/3)*Cos[x]] - Log[4 + 2*6^(1/3)*Cos[x] + 6^(2/3)*Cos[x]^2]))/72
Time = 0.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 25, 3702, 750, 16, 1142, 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.
| \(\displaystyle \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (x+\frac {\pi }{2}\right )}{4-3 \sin \left (x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (x+\frac {\pi }{2}\right )}{4-3 \sin \left (x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle -\int \frac {1}{4-3 \cos ^3(x)}d\cos (x)\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {\int \frac {\sqrt [3]{3} \cos (x)+2\ 2^{2/3}}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{6 \sqrt [3]{2}}-\frac {\int \frac {1}{2^{2/3}-\sqrt [3]{3} \cos (x)}d\cos (x)}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\int \frac {\sqrt [3]{3} \cos (x)+2\ 2^{2/3}}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\frac {3 \int \frac {1}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{\sqrt [3]{2}}+\frac {\int \frac {2^{2/3} \sqrt [3]{3} \left (\sqrt [3]{6} \cos (x)+1\right )}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{2 \sqrt [3]{3}}}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\frac {3 \int \frac {1}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{\sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{6} \cos (x)+1}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{\sqrt [3]{2}}}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\frac {\int \frac {\sqrt [3]{6} \cos (x)+1}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{\sqrt [3]{2}}-3^{2/3} \int \frac {1}{-\left (\sqrt [3]{6} \cos (x)+1\right )^2-3}d\left (\sqrt [3]{6} \cos (x)+1\right )}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\frac {\int \frac {\sqrt [3]{6} \cos (x)+1}{3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}}d\cos (x)}{\sqrt [3]{2}}+\sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{6} \cos (x)+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\log \left (2^{2/3}-\sqrt [3]{3} \cos (x)\right )}{6 \sqrt [3]{6}}-\frac {\sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{6} \cos (x)+1}{\sqrt {3}}\right )+\frac {\log \left (3^{2/3} \cos ^2(x)+2^{2/3} \sqrt [3]{3} \cos (x)+2 \sqrt [3]{2}\right )}{2 \sqrt [3]{3}}}{6 \sqrt [3]{2}}\) |
Input:
Int[Sin[x]/(4 - 3*Cos[x]^3),x]
Output:
Log[2^(2/3) - 3^(1/3)*Cos[x]]/(6*6^(1/3)) - (3^(1/6)*ArcTan[(1 + 6^(1/3)*C os[x])/Sqrt[3]] + Log[2*2^(1/3) + 2^(2/3)*3^(1/3)*Cos[x] + 3^(2/3)*Cos[x]^ 2]/(2*3^(1/3)))/(6*2^(1/3))
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[((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[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 = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.42
| method | result | size |
| risch | \(-\frac {i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (162 \textit {\_Z}^{3}+i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+12 i \textit {\_R} \,{\mathrm e}^{i x}+1\right )\right )}{2}\) | \(35\) |
| derivativedivides | \(\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos \left (x \right )^{2}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cos \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}-\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cos \left (x \right )}{2}+1\right )}{3}\right )}{12}\) | \(80\) |
| default | \(\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cos \left (x \right )^{2}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cos \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}-\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cos \left (x \right )}{2}+1\right )}{3}\right )}{12}\) | \(80\) |
Input:
int(sin(x)/(4-3*cos(x)^3),x,method=_RETURNVERBOSE)
Output:
-1/2*I*sum(_R*ln(exp(2*I*x)+12*I*_R*exp(I*x)+1),_R=RootOf(162*_Z^3+I))
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=-\frac {1}{6} \cdot 6^{\frac {1}{6}} \sqrt {\frac {1}{2}} \arctan \left (\frac {1}{3} \cdot 6^{\frac {1}{6}} \sqrt {\frac {1}{2}} {\left (6^{\frac {2}{3}} \cos \left (x\right ) + 6^{\frac {1}{3}}\right )}\right ) - \frac {1}{72} \cdot 6^{\frac {2}{3}} \log \left (-3 \, \cos \left (x\right )^{2} - 6^{\frac {2}{3}} \cos \left (x\right ) - 2 \cdot 6^{\frac {1}{3}}\right ) + \frac {1}{36} \cdot 6^{\frac {2}{3}} \log \left (6^{\frac {2}{3}} - 3 \, \cos \left (x\right )\right ) \] Input:
integrate(sin(x)/(4-3*cos(x)^3),x, algorithm="fricas")
Output:
-1/6*6^(1/6)*sqrt(1/2)*arctan(1/3*6^(1/6)*sqrt(1/2)*(6^(2/3)*cos(x) + 6^(1 /3))) - 1/72*6^(2/3)*log(-3*cos(x)^2 - 6^(2/3)*cos(x) - 2*6^(1/3)) + 1/36* 6^(2/3)*log(6^(2/3) - 3*cos(x))
Time = 0.59 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=\frac {6^{\frac {2}{3}} \log {\left (\cos {\left (x \right )} - \frac {6^{\frac {2}{3}}}{3} \right )}}{36} - \frac {6^{\frac {2}{3}} \log {\left (36 \cos ^{2}{\left (x \right )} + 12 \cdot 6^{\frac {2}{3}} \cos {\left (x \right )} + 24 \cdot \sqrt [3]{6} \right )}}{72} - \frac {2^{\frac {2}{3}} \cdot \sqrt [6]{3} \operatorname {atan}{\left (\frac {\sqrt [3]{2} \cdot 3^{\frac {5}{6}} \cos {\left (x \right )}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \] Input:
integrate(sin(x)/(4-3*cos(x)**3),x)
Output:
6**(2/3)*log(cos(x) - 6**(2/3)/3)/36 - 6**(2/3)*log(36*cos(x)**2 + 12*6**( 2/3)*cos(x) + 24*6**(1/3))/72 - 2**(2/3)*3**(1/6)*atan(2**(1/3)*3**(5/6)*c os(x)/3 + sqrt(3)/3)/12
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.06 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=-\frac {1}{72} \cdot 4^{\frac {1}{3}} 3^{\frac {2}{3}} \log \left (3^{\frac {2}{3}} \cos \left (x\right )^{2} + 4^{\frac {1}{3}} 3^{\frac {1}{3}} \cos \left (x\right ) + 4^{\frac {2}{3}}\right ) + \frac {1}{36} \cdot 4^{\frac {1}{3}} 3^{\frac {2}{3}} \log \left (\frac {1}{3} \cdot 3^{\frac {2}{3}} {\left (3^{\frac {1}{3}} \cos \left (x\right ) - 4^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} 3^{\frac {1}{6}} {\left (2 \cdot 3^{\frac {2}{3}} \cos \left (x\right ) + 4^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}\right ) \] Input:
integrate(sin(x)/(4-3*cos(x)^3),x, algorithm="maxima")
Output:
-1/72*4^(1/3)*3^(2/3)*log(3^(2/3)*cos(x)^2 + 4^(1/3)*3^(1/3)*cos(x) + 4^(2 /3)) + 1/36*4^(1/3)*3^(2/3)*log(1/3*3^(2/3)*(3^(1/3)*cos(x) - 4^(1/3))) - 1/12*4^(1/3)*3^(1/6)*arctan(1/12*4^(2/3)*3^(1/6)*(2*3^(2/3)*cos(x) + 4^(1/ 3)*3^(1/3)))
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=-\frac {1}{12} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{4} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {2}{3}} {\left (\left (\frac {4}{3}\right )^{\frac {1}{3}} + 2 \, \cos \left (x\right )\right )}\right ) - \frac {1}{72} \cdot 36^{\frac {1}{3}} \log \left (\cos \left (x\right )^{2} + \left (\frac {4}{3}\right )^{\frac {1}{3}} \cos \left (x\right ) + \left (\frac {4}{3}\right )^{\frac {2}{3}}\right ) + \frac {1}{12} \, \left (\frac {4}{3}\right )^{\frac {1}{3}} \log \left (\left (\frac {4}{3}\right )^{\frac {1}{3}} - \cos \left (x\right )\right ) \] Input:
integrate(sin(x)/(4-3*cos(x)^3),x, algorithm="giac")
Output:
-1/12*sqrt(3)*(4/3)^(1/3)*arctan(1/4*sqrt(3)*(4/3)^(2/3)*((4/3)^(1/3) + 2* cos(x))) - 1/72*36^(1/3)*log(cos(x)^2 + (4/3)^(1/3)*cos(x) + (4/3)^(2/3)) + 1/12*(4/3)^(1/3)*log((4/3)^(1/3) - cos(x))
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=\frac {6^{2/3}\,\ln \left (\cos \left (x\right )-\frac {6^{2/3}}{3}\right )}{36}+\frac {6^{2/3}\,\ln \left (\cos \left (x\right )-\frac {6^{2/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{72}-\frac {6^{2/3}\,\ln \left (\cos \left (x\right )+\frac {6^{2/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{72} \] Input:
int(-sin(x)/(3*cos(x)^3 - 4),x)
Output:
(6^(2/3)*log(cos(x) - 6^(2/3)/3))/36 + (6^(2/3)*log(cos(x) - (6^(2/3)*(3^( 1/2)*1i - 1))/6)*(3^(1/2)*1i - 1))/72 - (6^(2/3)*log(cos(x) + (6^(2/3)*(3^ (1/2)*1i + 1))/6)*(3^(1/2)*1i + 1))/72
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\sin (x)}{4-3 \cos ^3(x)} \, dx=\frac {2^{\frac {2}{3}} \left (-2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2 \,3^{\frac {1}{3}} \cos \left (x \right )+2^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}\right )-\mathrm {log}\left (3^{\frac {2}{3}} \cos \left (x \right )^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \cos \left (x \right )+2 \,2^{\frac {1}{3}}\right )+2 \,\mathrm {log}\left (3^{\frac {1}{3}} \cos \left (x \right )-2^{\frac {2}{3}}\right )\right ) 3^{\frac {2}{3}}}{72} \] Input:
int(sin(x)/(4-3*cos(x)^3),x)
Output:
(2**(2/3)*( - 2*3**(1/3)*3**(1/6)*atan((2*3**(1/3)*cos(x) + 2**(2/3))/(2** (2/3)*3**(1/3)*3**(1/6))) - log(3**(2/3)*cos(x)**2 + 2**(2/3)*3**(1/3)*cos (x) + 2*2**(1/3)) + 2*log(3**(1/3)*cos(x) - 2**(2/3))))/(24*3**(1/3))