Integrand size = 35, antiderivative size = 65 \[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=-\frac {3 a^2 \text {arctanh}(\sin (e+f x))}{c f}+\frac {4 a^2 \sin (e+f x)}{c f (1-\cos (e+f x))}-\frac {a^2 \tan (e+f x)}{c f} \]
Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(65)=130\).
Time = 1.12 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.98 \[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=\frac {2 a^2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (4 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+\sin \left (\frac {1}{2} (e+f x)\right ) \left (-3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {\sin (f x)}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )}{c f (-1+\cos (e+f x))} \]
(2*a^2*Sin[(e + f*x)/2]*(4*Csc[e/2]*Sin[(f*x)/2] + Sin[(e + f*x)/2]*(-3*Lo g[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 3*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Sin[f*x]/((Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))))/(c *f*(-1 + Cos[e + f*x]))
Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3042, 3431, 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.
| \(\displaystyle \int \frac {\sec ^2(e+f x) (a \cos (e+f x)+a)^2}{c \cos (e+f x)-c} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (e+f x+\frac {\pi }{2}\right )+a\right )^2}{\sin \left (e+f x+\frac {\pi }{2}\right )^2 \left (c \sin \left (e+f x+\frac {\pi }{2}\right )-c\right )}dx\) |
| \(\Big \downarrow \) 3431 |
| \(\displaystyle \int \left (\frac {4 a^2}{c (\cos (e+f x)-1)}-\frac {a^2 \sec ^2(e+f x)}{c}-\frac {3 a^2 \sec (e+f x)}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a^2 \text {arctanh}(\sin (e+f x))}{c f}-\frac {a^2 \tan (e+f x)}{c f}+\frac {4 a^2 \sin (e+f x)}{c f (1-\cos (e+f x))}\) |
(-3*a^2*ArcTanh[Sin[e + f*x]])/(c*f) + (4*a^2*Sin[e + f*x])/(c*f*(1 - Cos[ e + f*x])) - (a^2*Tan[e + f*x])/(c*f)
3.1.1.3.1 Defintions of rubi rules used
Int[((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[Exp andTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])^n, x ], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (Int egersQ[m, n] || IntegersQ[m, p] || IntegersQ[n, p]) && NeQ[p, 2]
Time = 1.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26
| method | result | size |
| derivativedivides | \(\frac {4 a^{2} \left (\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {1}{4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4}+\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}+\frac {1}{4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+4}-\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}\right )}{f c}\) | \(82\) |
| default | \(\frac {4 a^{2} \left (\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {1}{4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-4}+\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}+\frac {1}{4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+4}-\frac {3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}\right )}{f c}\) | \(82\) |
| parallelrisch | \(-\frac {a^{2} \left (-5 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (f x +e \right )-3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \cos \left (f x +e \right )+3 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \cos \left (f x +e \right )+\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f \cos \left (f x +e \right )}\) | \(87\) |
| risch | \(\frac {2 i a^{2} \left (4 \,{\mathrm e}^{2 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}+5\right )}{f c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{c f}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{c f}\) | \(112\) |
| norman | \(\frac {-\frac {4 a^{2}}{c f}-\frac {2 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {8 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {6 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+\frac {3 a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{c f}-\frac {3 a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{c f}\) | \(168\) |
4/f*a^2/c*(1/tan(1/2*f*x+1/2*e)+1/4/(tan(1/2*f*x+1/2*e)-1)+3/4*ln(tan(1/2* f*x+1/2*e)-1)+1/4/(tan(1/2*f*x+1/2*e)+1)-3/4*ln(tan(1/2*f*x+1/2*e)+1))
Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.66 \[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=-\frac {3 \, a^{2} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 3 \, a^{2} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 10 \, a^{2} \cos \left (f x + e\right )^{2} - 8 \, a^{2} \cos \left (f x + e\right ) + 2 \, a^{2}}{2 \, c f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \]
-1/2*(3*a^2*cos(f*x + e)*log(sin(f*x + e) + 1)*sin(f*x + e) - 3*a^2*cos(f* x + e)*log(-sin(f*x + e) + 1)*sin(f*x + e) - 10*a^2*cos(f*x + e)^2 - 8*a^2 *cos(f*x + e) + 2*a^2)/(c*f*cos(f*x + e)*sin(f*x + e))
\[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=\frac {a^{2} \left (\int \frac {\sec ^{2}{\left (e + f x \right )}}{\cos {\left (e + f x \right )} - 1}\, dx + \int \frac {2 \cos {\left (e + f x \right )} \sec ^{2}{\left (e + f x \right )}}{\cos {\left (e + f x \right )} - 1}\, dx + \int \frac {\cos ^{2}{\left (e + f x \right )} \sec ^{2}{\left (e + f x \right )}}{\cos {\left (e + f x \right )} - 1}\, dx\right )}{c} \]
a**2*(Integral(sec(e + f*x)**2/(cos(e + f*x) - 1), x) + Integral(2*cos(e + f*x)*sec(e + f*x)**2/(cos(e + f*x) - 1), x) + Integral(cos(e + f*x)**2*se c(e + f*x)**2/(cos(e + f*x) - 1), x))/c
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (63) = 126\).
Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.46 \[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=-\frac {a^{2} {\left (\frac {\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1}{\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c}\right )} + 2 \, a^{2} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c} - \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} - \frac {a^{2} {\left (\cos \left (f x + e\right ) + 1\right )}}{c \sin \left (f x + e\right )}}{f} \]
-(a^2*((3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 1)/(c*sin(f*x + e)/(cos(f* x + e) + 1) - c*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + log(sin(f*x + e)/(c os(f*x + e) + 1) + 1)/c - log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c) + 2* a^2*(log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/c - log(sin(f*x + e)/(cos(f* x + e) + 1) - 1)/c - (cos(f*x + e) + 1)/(c*sin(f*x + e))) - a^2*(cos(f*x + e) + 1)/(c*sin(f*x + e)))/f
Time = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=-\frac {\frac {3 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{c} - \frac {3 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{c} - \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} c}}{f} \]
-(3*a^2*log(abs(tan(1/2*f*x + 1/2*e) + 1))/c - 3*a^2*log(abs(tan(1/2*f*x + 1/2*e) - 1))/c - 2*(3*a^2*tan(1/2*f*x + 1/2*e)^2 - 2*a^2)/((tan(1/2*f*x + 1/2*e)^3 - tan(1/2*f*x + 1/2*e))*c))/f
Time = 0.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.18 \[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx=\frac {6\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-4\,a^2}{c\,f\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}-\frac {6\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{c\,f} \]