Integrand size = 13, antiderivative size = 57 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=\frac {\left (a^2-b^2\right ) \log (\cos (x))}{a^3}-\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{a^3}-\frac {b \sec (x)}{a^2}+\frac {\sec ^2(x)}{2 a} \] Output:
(a^2-b^2)*ln(cos(x))/a^3-(a^2-b^2)*ln(a+b*cos(x))/a^3-b*sec(x)/a^2+1/2*sec (x)^2/a
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=\frac {2 \left (a^2-b^2\right ) (\log (\cos (x))-\log (a+b \cos (x)))-2 a b \sec (x)+a^2 \sec ^2(x)}{2 a^3} \] Input:
Integrate[Tan[x]^3/(a + b*Cos[x]),x]
Output:
(2*(a^2 - b^2)*(Log[Cos[x]] - Log[a + b*Cos[x]]) - 2*a*b*Sec[x] + a^2*Sec[ x]^2)/(2*a^3)
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 25, 3200, 522, 2009}
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.
\(\displaystyle \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\tan \left (x-\frac {\pi }{2}\right )^3 \left (a-b \sin \left (x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\left (a-b \sin \left (x-\frac {\pi }{2}\right )\right ) \tan \left (x-\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 3200 |
\(\displaystyle -\int \frac {\left (b^2-b^2 \cos ^2(x)\right ) \sec ^3(x)}{b^3 (a+b \cos (x))}d(b \cos (x))\) |
\(\Big \downarrow \) 522 |
\(\displaystyle -\int \left (\frac {\sec ^3(x)}{a b}-\frac {\sec ^2(x)}{a^2}+\frac {\left (b^2-a^2\right ) \sec (x)}{a^3 b}+\frac {a^2-b^2}{a^3 (a+b \cos (x))}\right )d(b \cos (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \sec (x)}{a^2}+\frac {\left (a^2-b^2\right ) \log (b \cos (x))}{a^3}-\frac {\left (a^2-b^2\right ) \log (a+b \cos (x))}{a^3}+\frac {\sec ^2(x)}{2 a}\) |
Input:
Int[Tan[x]^3/(a + b*Cos[x]),x]
Output:
((a^2 - b^2)*Log[b*Cos[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cos[x]])/a^3 - (b *Sec[x])/a^2 + Sec[x]^2/(2*a)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 0.65 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \cos \left (x \right )\right )}{a^{3}}-\frac {b}{a^{2} \cos \left (x \right )}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (\cos \left (x \right )\right )}{a^{3}}+\frac {1}{2 a \cos \left (x \right )^{2}}\) | \(58\) |
risch | \(-\frac {2 \left (b \,{\mathrm e}^{3 i x}-{\mathrm e}^{2 i x} a +{\mathrm e}^{i x} b \right )}{\left ({\mathrm e}^{2 i x}+1\right )^{2} a^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 a \,{\mathrm e}^{i x}}{b}+1\right ) b^{2}}{a^{3}}+\frac {\ln \left ({\mathrm e}^{2 i x}+1\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 i x}+1\right ) b^{2}}{a^{3}}\) | \(117\) |
Input:
int(tan(x)^3/(a+b*cos(x)),x,method=_RETURNVERBOSE)
Output:
-(a^2-b^2)*ln(a+b*cos(x))/a^3-b/a^2/cos(x)+(a^2-b^2)*ln(cos(x))/a^3+1/2/a/ cos(x)^2
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.16 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=-\frac {2 \, {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} \log \left (-b \cos \left (x\right ) - a\right ) - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} \log \left (-\cos \left (x\right )\right ) + 2 \, a b \cos \left (x\right ) - a^{2}}{2 \, a^{3} \cos \left (x\right )^{2}} \] Input:
integrate(tan(x)^3/(a+b*cos(x)),x, algorithm="fricas")
Output:
-1/2*(2*(a^2 - b^2)*cos(x)^2*log(-b*cos(x) - a) - 2*(a^2 - b^2)*cos(x)^2*l og(-cos(x)) + 2*a*b*cos(x) - a^2)/(a^3*cos(x)^2)
\[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=\int \frac {\tan ^{3}{\left (x \right )}}{a + b \cos {\left (x \right )}}\, dx \] Input:
integrate(tan(x)**3/(a+b*cos(x)),x)
Output:
Integral(tan(x)**3/(a + b*cos(x)), x)
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=-\frac {{\left (a^{2} - b^{2}\right )} \log \left (b \cos \left (x\right ) + a\right )}{a^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\cos \left (x\right )\right )}{a^{3}} - \frac {2 \, b \cos \left (x\right ) - a}{2 \, a^{2} \cos \left (x\right )^{2}} \] Input:
integrate(tan(x)^3/(a+b*cos(x)),x, algorithm="maxima")
Output:
-(a^2 - b^2)*log(b*cos(x) + a)/a^3 + (a^2 - b^2)*log(cos(x))/a^3 - 1/2*(2* b*cos(x) - a)/(a^2*cos(x)^2)
Time = 0.40 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.16 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=\frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | \cos \left (x\right ) \right |}\right )}{a^{3}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{a^{3} b} - \frac {2 \, a b \cos \left (x\right ) - a^{2}}{2 \, a^{3} \cos \left (x\right )^{2}} \] Input:
integrate(tan(x)^3/(a+b*cos(x)),x, algorithm="giac")
Output:
(a^2 - b^2)*log(abs(cos(x)))/a^3 - (a^2*b - b^3)*log(abs(b*cos(x) + a))/(a ^3*b) - 1/2*(2*a*b*cos(x) - a^2)/(a^3*cos(x)^2)
Time = 42.45 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.02 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=-\frac {2\,a^2\,\mathrm {atanh}\left (\frac {a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{-b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+a+b}\right )-2\,b^2\,\mathrm {atanh}\left (\frac {a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{-b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+a+b}\right )}{a^3}-\frac {2\,a\,b-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2+2\,b\,a\right )}{a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+a^3} \] Input:
int(tan(x)^3/(a + b*cos(x)),x)
Output:
- (2*a^2*atanh((a*tan(x/2)^2)/(a + b - b*tan(x/2)^2)) - 2*b^2*atanh((a*tan (x/2)^2)/(a + b - b*tan(x/2)^2)))/a^3 - (2*a*b - tan(x/2)^2*(2*a*b + 2*a^2 ))/(a^3 - 2*a^3*tan(x/2)^2 + a^3*tan(x/2)^4)
Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 4.65 \[ \int \frac {\tan ^3(x)}{a+b \cos (x)} \, dx=\frac {2 \cos \left (x \right ) a b +2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} a^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) \sin \left (x \right )^{2} b^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) a^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )-1\right ) b^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} a^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) \sin \left (x \right )^{2} b^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) a^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )+1\right ) b^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -\tan \left (\frac {x}{2}\right )^{2} b +a +b \right ) \sin \left (x \right )^{2} a^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -\tan \left (\frac {x}{2}\right )^{2} b +a +b \right ) \sin \left (x \right )^{2} b^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -\tan \left (\frac {x}{2}\right )^{2} b +a +b \right ) a^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -\tan \left (\frac {x}{2}\right )^{2} b +a +b \right ) b^{2}-\sin \left (x \right )^{2} a^{2}+2 \sin \left (x \right )^{2} a b -2 a b}{2 a^{3} \left (\sin \left (x \right )^{2}-1\right )} \] Input:
int(tan(x)^3/(a+b*cos(x)),x)
Output:
(2*cos(x)*a*b + 2*log(tan(x/2) - 1)*sin(x)**2*a**2 - 2*log(tan(x/2) - 1)*s in(x)**2*b**2 - 2*log(tan(x/2) - 1)*a**2 + 2*log(tan(x/2) - 1)*b**2 + 2*lo g(tan(x/2) + 1)*sin(x)**2*a**2 - 2*log(tan(x/2) + 1)*sin(x)**2*b**2 - 2*lo g(tan(x/2) + 1)*a**2 + 2*log(tan(x/2) + 1)*b**2 - 2*log(tan(x/2)**2*a - ta n(x/2)**2*b + a + b)*sin(x)**2*a**2 + 2*log(tan(x/2)**2*a - tan(x/2)**2*b + a + b)*sin(x)**2*b**2 + 2*log(tan(x/2)**2*a - tan(x/2)**2*b + a + b)*a** 2 - 2*log(tan(x/2)**2*a - tan(x/2)**2*b + a + b)*b**2 - sin(x)**2*a**2 + 2 *sin(x)**2*a*b - 2*a*b)/(2*a**3*(sin(x)**2 - 1))