Integrand size = 12, antiderivative size = 88 \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=-\frac {a \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b (a+b)^{3/2}}+\frac {x}{2 b (a+b \cos (x))^2}+\frac {\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))} \] Output:
-a*arctan((a-b)^(1/2)*tan(1/2*x)/(a+b)^(1/2))/(a-b)^(3/2)/b/(a+b)^(3/2)+1/ 2*x/b/(a+b*cos(x))^2+1/2*sin(x)/(a^2-b^2)/(a+b*cos(x))
Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=-\frac {a \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{b \left (-a^2+b^2\right )^{3/2}}+\frac {\frac {x}{b}+\frac {(a+b \cos (x)) \sin (x)}{(a-b) (a+b)}}{2 (a+b \cos (x))^2} \] Input:
Integrate[(x*Sin[x])/(a + b*Cos[x])^3,x]
Output:
-((a*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(b*(-a^2 + b^2)^(3/2))) + (x/b + ((a + b*Cos[x])*Sin[x])/((a - b)*(a + b)))/(2*(a + b*Cos[x])^2)
Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4923, 3042, 3143, 25, 27, 3042, 3138, 218}
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.
\(\displaystyle \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx\) |
\(\Big \downarrow \) 4923 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\int \frac {1}{(a+b \cos (x))^2}dx}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\int \frac {1}{\left (a+b \sin \left (x+\frac {\pi }{2}\right )\right )^2}dx}{2 b}\) |
\(\Big \downarrow \) 3143 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {-\frac {\int -\frac {a}{a+b \cos (x)}dx}{a^2-b^2}-\frac {b \sin (x)}{\left (a^2-b^2\right ) (a+b \cos (x))}}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\frac {\int \frac {a}{a+b \cos (x)}dx}{a^2-b^2}-\frac {b \sin (x)}{\left (a^2-b^2\right ) (a+b \cos (x))}}{2 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\frac {a \int \frac {1}{a+b \cos (x)}dx}{a^2-b^2}-\frac {b \sin (x)}{\left (a^2-b^2\right ) (a+b \cos (x))}}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\frac {a \int \frac {1}{a+b \sin \left (x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {b \sin (x)}{\left (a^2-b^2\right ) (a+b \cos (x))}}{2 b}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\frac {2 a \int \frac {1}{(a-b) \tan ^2\left (\frac {x}{2}\right )+a+b}d\tan \left (\frac {x}{2}\right )}{a^2-b^2}-\frac {b \sin (x)}{\left (a^2-b^2\right ) (a+b \cos (x))}}{2 b}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {x}{2 b (a+b \cos (x))^2}-\frac {\frac {2 a \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {b \sin (x)}{\left (a^2-b^2\right ) (a+b \cos (x))}}{2 b}\) |
Input:
Int[(x*Sin[x])/(a + b*Cos[x])^3,x]
Output:
x/(2*b*(a + b*Cos[x])^2) - ((2*a*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b] ])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)) - (b*Sin[x])/((a^2 - b^2)*(a + b* Cos[x])))/(2*b)
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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.) *Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*((a + b*Cos[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b*d*(n + 1))) Int[(e + f*x) ^(m - 1)*(a + b*Cos[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n }, x] && IGtQ[m, 0] && NeQ[n, -1]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.84
method | result | size |
risch | \(\frac {i \left (-2 i a^{2} x \,{\mathrm e}^{2 i x}+2 i b^{2} x \,{\mathrm e}^{2 i x}+b a \,{\mathrm e}^{3 i x}+2 \,{\mathrm e}^{2 i x} a^{2}+{\mathrm e}^{2 i x} b^{2}+3 a b \,{\mathrm e}^{i x}+b^{2}\right )}{b \left (b \,{\mathrm e}^{2 i x}+2 a \,{\mathrm e}^{i x}+b \right )^{2} \left (a^{2}-b^{2}\right )}-\frac {i a \ln \left ({\mathrm e}^{i x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) b}+\frac {i a \ln \left ({\mathrm e}^{i x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) b}\) | \(250\) |
Input:
int(x*sin(x)/(a+b*cos(x))^3,x,method=_RETURNVERBOSE)
Output:
I*(-2*I*a^2*x*exp(2*I*x)+2*I*b^2*x*exp(2*I*x)+b*a*exp(3*I*x)+2*exp(2*I*x)* a^2+exp(2*I*x)*b^2+3*a*b*exp(I*x)+b^2)/b/(b*exp(2*I*x)+2*a*exp(I*x)+b)^2/( a^2-b^2)-1/2*I/(a^2-b^2)^(1/2)*a/(a+b)/(a-b)/b*ln(exp(I*x)+(a*(a^2-b^2)^(1 /2)+a^2-b^2)/(a^2-b^2)^(1/2)/b)+1/2*I/(a^2-b^2)^(1/2)*a/(a+b)/(a-b)/b*ln(e xp(I*x)+(a*(a^2-b^2)^(1/2)-a^2+b^2)/(a^2-b^2)^(1/2)/b)
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (74) = 148\).
Time = 0.10 (sec) , antiderivative size = 417, normalized size of antiderivative = 4.74 \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=\left [\frac {{\left (a b^{2} \cos \left (x\right )^{2} + 2 \, a^{2} b \cos \left (x\right ) + a^{3}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (x\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right ) + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x + 2 \, {\left (a^{3} b - a b^{3} + {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{4 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )\right )}}, -\frac {{\left (a b^{2} \cos \left (x\right )^{2} + 2 \, a^{2} b \cos \left (x\right ) + a^{3}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x\right )}\right ) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x - {\left (a^{3} b - a b^{3} + {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} + {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )\right )}}\right ] \] Input:
integrate(x*sin(x)/(a+b*cos(x))^3,x, algorithm="fricas")
Output:
[1/4*((a*b^2*cos(x)^2 + 2*a^2*b*cos(x) + a^3)*sqrt(-a^2 + b^2)*log((2*a*b* cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) + 2*(a^4 - 2*a^2*b^2 + b^4)*x + 2*(a^3*b - a*b^3 + (a^2*b^2 - b^4)*cos(x))*sin(x))/(a^6*b - 2*a ^4*b^3 + a^2*b^5 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(x)^2 + 2*(a^5*b^2 - 2*a ^3*b^4 + a*b^6)*cos(x)), -1/2*((a*b^2*cos(x)^2 + 2*a^2*b*cos(x) + a^3)*sqr t(a^2 - b^2)*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) - (a^4 - 2*a ^2*b^2 + b^4)*x - (a^3*b - a*b^3 + (a^2*b^2 - b^4)*cos(x))*sin(x))/(a^6*b - 2*a^4*b^3 + a^2*b^5 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(x)^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(x))]
Timed out. \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=\text {Timed out} \] Input:
integrate(x*sin(x)/(a+b*cos(x))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*sin(x)/(a+b*cos(x))^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
\[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=\int { \frac {x \sin \left (x\right )}{{\left (b \cos \left (x\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(x*sin(x)/(a+b*cos(x))^3,x, algorithm="giac")
Output:
integrate(x*sin(x)/(b*cos(x) + a)^3, x)
Timed out. \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=\int \frac {x\,\sin \left (x\right )}{{\left (a+b\,\cos \left (x\right )\right )}^3} \,d x \] Input:
int((x*sin(x))/(a + b*cos(x))^3,x)
Output:
int((x*sin(x))/(a + b*cos(x))^3, x)
Time = 0.16 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.12 \[ \int \frac {x \sin (x)}{(a+b \cos (x))^3} \, dx=\frac {-2 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a -\tan \left (\frac {x}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (x \right )^{2} a \,b^{2}-4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a -\tan \left (\frac {x}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (x \right ) a^{2} b -2 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) a -\tan \left (\frac {x}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3}+\cos \left (x \right ) \sin \left (x \right ) a^{2} b^{2}-\cos \left (x \right ) \sin \left (x \right ) b^{4}+\sin \left (x \right ) a^{3} b -\sin \left (x \right ) a \,b^{3}+a^{4} x -2 a^{2} b^{2} x +b^{4} x}{2 b \left (\cos \left (x \right )^{2} a^{4} b^{2}-2 \cos \left (x \right )^{2} a^{2} b^{4}+\cos \left (x \right )^{2} b^{6}+2 \cos \left (x \right ) a^{5} b -4 \cos \left (x \right ) a^{3} b^{3}+2 \cos \left (x \right ) a \,b^{5}+a^{6}-2 a^{4} b^{2}+a^{2} b^{4}\right )} \] Input:
int(x*sin(x)/(a+b*cos(x))^3,x)
Output:
( - 2*sqrt(a**2 - b**2)*atan((tan(x/2)*a - tan(x/2)*b)/sqrt(a**2 - b**2))* cos(x)**2*a*b**2 - 4*sqrt(a**2 - b**2)*atan((tan(x/2)*a - tan(x/2)*b)/sqrt (a**2 - b**2))*cos(x)*a**2*b - 2*sqrt(a**2 - b**2)*atan((tan(x/2)*a - tan( x/2)*b)/sqrt(a**2 - b**2))*a**3 + cos(x)*sin(x)*a**2*b**2 - cos(x)*sin(x)* b**4 + sin(x)*a**3*b - sin(x)*a*b**3 + a**4*x - 2*a**2*b**2*x + b**4*x)/(2 *b*(cos(x)**2*a**4*b**2 - 2*cos(x)**2*a**2*b**4 + cos(x)**2*b**6 + 2*cos(x )*a**5*b - 4*cos(x)*a**3*b**3 + 2*cos(x)*a*b**5 + a**6 - 2*a**4*b**2 + a** 2*b**4))