Integrand size = 21, antiderivative size = 51 \[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\text {arctanh}(\sin (c+d x))}{a d}+\frac {\tan (c+d x)}{a d}+\frac {\tan (c+d x)}{d (a+a \sec (c+d x))} \]
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-\text {arctanh}(\sin (c+d x))+\frac {(2+\sec (c+d x)) \tan (c+d x)}{1+\sec (c+d x)}}{a d} \]
Time = 0.45 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4277, 3042, 4276, 3042, 4257, 4281}
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.
\(\displaystyle \int \frac {\sec ^3(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3}{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 4277 |
\(\displaystyle \frac {\tan (c+d x)}{a d}-\int \frac {\sec ^2(c+d x)}{\sec (c+d x) a+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan (c+d x)}{a d}-\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\) |
\(\Big \downarrow \) 4276 |
\(\displaystyle -\frac {\int \sec (c+d x)dx}{a}+\int \frac {\sec (c+d x)}{\sec (c+d x) a+a}dx+\frac {\tan (c+d x)}{a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {\tan (c+d x)}{a d}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {\text {arctanh}(\sin (c+d x))}{a d}+\frac {\tan (c+d x)}{a d}\) |
\(\Big \downarrow \) 4281 |
\(\displaystyle -\frac {\text {arctanh}(\sin (c+d x))}{a d}+\frac {\tan (c+d x)}{a d}+\frac {\tan (c+d x)}{d (a \sec (c+d x)+a)}\) |
3.1.44.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[1/b Int[Csc[e + f*x], x], x] - Simp[a/b Int[Csc[e + f*x]/( a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[-Cot[e + f*x]/(b*f), x] - Simp[a/b Int[Csc[e + f*x]^2/(a + b *Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[-Cot[e + f*x]/(f*(b + a*Csc[e + f*x])), x] /; FreeQ[{a, b, e, f} , x] && EqQ[a^2 - b^2, 0]
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.45
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 | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (d x +c \right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \cos \left (d x +c \right )}\) | \(82\) |
risch | \(\frac {2 i \left ({\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+2\right )}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a d}\) | \(98\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(112\) |
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 = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.90 \[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right )\right )}} \]
-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))
\[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (51) = 102\).
Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.33 \[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
-(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 = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.65 \[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a}}{d} \]
-(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 = 13.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]