Integrand size = 21, antiderivative size = 53 \[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\text {arctanh}(\sin (c+d x))}{a d}+\frac {2 \tan (c+d x)}{a d}-\frac {\tan (c+d x)}{d (a+a \cos (c+d x))} \]
Leaf count is larger than twice the leaf count of optimal. \(188\) vs. \(2(53)=106\).
Time = 0.78 (sec) , antiderivative size = 188, normalized size of antiderivative = 3.55 \[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (\sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )\right )}{a d (1+\cos (c+d x))} \]
(2*Cos[(c + d*x)/2]*(Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(1 + Cos [c + d*x]))
Time = 0.45 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3247, 25, 3042, 3227, 3042, 4254, 24, 4257}
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.
\(\displaystyle \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )}dx\) |
\(\Big \downarrow \) 3247 |
\(\displaystyle -\frac {\int -\left ((2 a-a \cos (c+d x)) \sec ^2(c+d x)\right )dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int (2 a-a \cos (c+d x)) \sec ^2(c+d x)dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {2 a-a \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {2 a \int \sec ^2(c+d x)dx-a \int \sec (c+d x)dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx-a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {-\frac {2 a \int 1d(-\tan (c+d x))}{d}-a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {2 a \tan (c+d x)}{d}-a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\frac {2 a \tan (c+d x)}{d}-\frac {a \text {arctanh}(\sin (c+d x))}{d}}{a^2}-\frac {\tan (c+d x)}{d (a \cos (c+d x)+a)}\) |
-(Tan[c + d*x]/(d*(a + a*Cos[c + d*x]))) + (-((a*ArcTanh[Sin[c + d*x]])/d) + (2*a*Tan[c + d*x])/d)/a^2
3.1.50.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Simp[d/(a*(b*c - a*d)) Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[ c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.93 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(74\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(74\) |
parallelrisch | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+2 \cos \left (d x +c \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \cos \left (d x +c \right )}\) | \(82\) |
norman | \(\frac {\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(93\) |
risch | \(\frac {2 i \left ({\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+2\right )}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}\) | \(98\) |
1/d/a*(tan(1/2*d*x+1/2*c)-1/(tan(1/2*d*x+1/2*c)-1)+ln(tan(1/2*d*x+1/2*c)-1 )-1/(tan(1/2*d*x+1/2*c)+1)-ln(tan(1/2*d*x+1/2*c)+1))
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.83 \[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right )\right )}} \]
-1/2*((cos(d*x + c)^2 + cos(d*x + c))*log(sin(d*x + c) + 1) - (cos(d*x + c )^2 + cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(2*cos(d*x + c) + 1)*sin(d* x + c))/(a*d*cos(d*x + c)^2 + a*d*cos(d*x + c))
\[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\int \frac {\sec ^{2}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (53) = 106\).
Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.25 \[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
-(log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a - log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a - 2*sin(d*x + c)/((a - a*sin(d*x + c)^2/(cos(d*x + c) + 1) ^2)*(cos(d*x + c) + 1)) - sin(d*x + c)/(a*(cos(d*x + c) + 1)))/d
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.58 \[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a}}{d} \]
-(log(abs(tan(1/2*d*x + 1/2*c) + 1))/a - log(abs(tan(1/2*d*x + 1/2*c) - 1) )/a - tan(1/2*d*x + 1/2*c)/a + 2*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2* c)^2 - 1)*a))/d
Time = 14.80 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \frac {\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]