Integrand size = 19, antiderivative size = 58 \[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\csc ^2(c+d x)}{2 a d} \]
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\left (1+2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sec (c+d x)}{2 a d (1+\sec (c+d x))} \]
-1/2*((1 + 2*Cos[(c + d*x)/2]^2*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x) /2]]))*Sec[c + d*x])/(a*d*(1 + Sec[c + d*x]))
Time = 0.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {3042, 4359, 25, 25, 3042, 25, 3185, 25, 3042, 25, 3086, 15, 3091, 3042, 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.
\(\displaystyle \int \frac {\csc (c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right ) \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 4359 |
\(\displaystyle \int -\frac {\cot (c+d x)}{a (-\cos (c+d x))-a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cot (c+d x)}{\cos (c+d x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\cot (c+d x)}{a \cos (c+d x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (c+d x-\frac {\pi }{2}\right )}{a-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 c-\pi )+d x\right )}{a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
\(\Big \downarrow \) 3185 |
\(\displaystyle -\frac {\int \cot ^2(c+d x) \csc (c+d x)dx}{a}-\frac {\int -\cot (c+d x) \csc ^2(c+d x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cot (c+d x) \csc ^2(c+d x)dx}{a}-\frac {\int \cot ^2(c+d x) \csc (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \sec \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \tan \left (\frac {1}{2} (2 c-\pi )+d x\right )dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\int \csc (c+d x)d\csc (c+d x)}{a d}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\csc ^2(c+d x)}{2 a d}\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\csc ^2(c+d x)}{2 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\csc ^2(c+d x)}{2 a d}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}}{a}-\frac {\csc ^2(c+d x)}{2 a d}\) |
-1/2*Csc[c + d*x]^2/(a*d) - (ArcTanh[Cos[c + d*x]]/(2*d) - (Cot[c + d*x]*C sc[c + d*x])/(2*d))/a
3.1.62.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m _.), x_Symbol] :> Int[Cot[e + f*x]^p*(b + a*Sin[e + f*x])^m, x] /; FreeQ[{a , b, e, f, p}, x] && IntegerQ[m] && EqQ[m, p]
Time = 0.54 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}\) | \(35\) |
norman | \(-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 d a}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}\) | \(39\) |
derivativedivides | \(\frac {\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{4}-\frac {1}{2 \left (\cos \left (d x +c \right )+1\right )}-\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{4}}{d a}\) | \(43\) |
default | \(\frac {\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{4}-\frac {1}{2 \left (\cos \left (d x +c \right )+1\right )}-\frac {\ln \left (\cos \left (d x +c \right )+1\right )}{4}}{d a}\) | \(43\) |
risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )}}{a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}\) | \(72\) |
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2}{4 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
-1/4*((cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) - (cos(d*x + c) + 1)* log(-1/2*cos(d*x + c) + 1/2) + 2)/(a*d*cos(d*x + c) + a*d)
\[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\csc {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a} + \frac {2}{a \cos \left (d x + c\right ) + a}}{4 \, d} \]
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac {\cos \left (d x + c\right ) - 1}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{4 \, d} \]
1/4*(log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a + (cos(d*x + c) - 1)/(a*(cos(d*x + c) + 1)))/d
Time = 13.50 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.57 \[ \int \frac {\csc (c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {1}{2\,d\,\left (a+a\,\cos \left (c+d\,x\right )\right )}-\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{2\,a\,d} \]