Integrand size = 15, antiderivative size = 53 \[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}}+\frac {1}{a \sqrt {a \sec ^2(x)}} \]
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.11 \[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {-3 \arctan \left (\sqrt {-\cos ^2(x)}\right )-\sqrt {-\cos ^2(x)} \left (-4+\sin ^2(x)\right )}{3 a \sqrt {-\cos ^2(x)} \sqrt {a \sec ^2(x)}} \]
(-3*ArcTan[Sqrt[-Cos[x]^2]] - Sqrt[-Cos[x]^2]*(-4 + Sin[x]^2))/(3*a*Sqrt[- Cos[x]^2]*Sqrt[a*Sec[x]^2])
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.28, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4140, 3042, 4612, 25, 61, 61, 73, 219}
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.
\(\displaystyle \int \frac {\cot (x)}{\left (a \tan ^2(x)+a\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (x) \left (a \tan (x)^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \frac {\cot (x)}{\left (a \sec ^2(x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (x) \left (a \sec (x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4612 |
\(\displaystyle \frac {1}{2} a \int -\frac {1}{\left (a \sec ^2(x)\right )^{5/2} \left (1-\sec ^2(x)\right )}d\sec ^2(x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} a \int \frac {1}{\left (a \sec ^2(x)\right )^{5/2} \left (1-\sec ^2(x)\right )}d\sec ^2(x)\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} a \left (\frac {2}{3 a \left (a \sec ^2(x)\right )^{3/2}}-\frac {\int \frac {1}{\left (a \sec ^2(x)\right )^{3/2} \left (1-\sec ^2(x)\right )}d\sec ^2(x)}{a}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} a \left (\frac {2}{3 a \left (a \sec ^2(x)\right )^{3/2}}-\frac {\frac {\int \frac {1}{\sqrt {a \sec ^2(x)} \left (1-\sec ^2(x)\right )}d\sec ^2(x)}{a}-\frac {2}{a \sqrt {a \sec ^2(x)}}}{a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} a \left (\frac {2}{3 a \left (a \sec ^2(x)\right )^{3/2}}-\frac {\frac {2 \int \frac {1}{1-\frac {\sec ^4(x)}{a}}d\sqrt {a \sec ^2(x)}}{a^2}-\frac {2}{a \sqrt {a \sec ^2(x)}}}{a}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} a \left (\frac {2}{3 a \left (a \sec ^2(x)\right )^{3/2}}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {a \sec ^2(x)}}}{a}\right )\) |
(a*(2/(3*a*(a*Sec[x]^2)^(3/2)) - ((2*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]])/a^ (3/2) - 2/(a*Sqrt[a*Sec[x]^2]))/a))/2
3.3.81.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Simp[b/(2*f) Subst[Int[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x ], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p] && Int egerQ[(m - 1)/2]
Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\cos \left (x \right )^{2}+3+3 \ln \left (-\cot \left (x \right )+\csc \left (x \right )\right ) \sec \left (x \right )+4 \sec \left (x \right )}{3 \sqrt {a \sec \left (x \right )^{2}}\, a}\) | \(36\) |
risch | \(\frac {{\mathrm e}^{4 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{2 i x}}{8 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}+\frac {5}{8 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right ) a}+\frac {{\mathrm e}^{-2 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}-\frac {{\mathrm e}^{i x} \ln \left ({\mathrm e}^{i x}+1\right )}{a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}+\frac {{\mathrm e}^{i x} \ln \left ({\mathrm e}^{i x}-1\right )}{a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}\) | \(234\) |
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (41) = 82\).
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.77 \[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {3 \, {\left (\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1\right )} \sqrt {a} \log \left (\frac {a \tan \left (x\right )^{2} - 2 \, \sqrt {a \tan \left (x\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (x\right )^{2}}\right ) + 2 \, \sqrt {a \tan \left (x\right )^{2} + a} {\left (3 \, \tan \left (x\right )^{2} + 4\right )}}{6 \, {\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} \]
1/6*(3*(tan(x)^4 + 2*tan(x)^2 + 1)*sqrt(a)*log((a*tan(x)^2 - 2*sqrt(a*tan( x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2) + 2*sqrt(a*tan(x)^2 + a)*(3*tan(x)^2 + 4))/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)
\[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.40 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\cos \left (3 \, x\right ) + 15 \, \cos \left (x\right ) - 6 \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + 6 \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}{12 \, a^{\frac {3}{2}}} \]
1/12*(cos(3*x) + 15*cos(x) - 6*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 6 *log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1))/a^(3/2)
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a \tan \left (x\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {3 \, a \tan \left (x\right )^{2} + 4 \, a}{3 \, {\left (a \tan \left (x\right )^{2} + a\right )}^{\frac {3}{2}} a} \]
arctan(sqrt(a*tan(x)^2 + a)/sqrt(-a))/(sqrt(-a)*a) + 1/3*(3*a*tan(x)^2 + 4 *a)/((a*tan(x)^2 + a)^(3/2)*a)
Time = 11.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {\cot (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\frac {\frac {a\,{\mathrm {tan}\left (x\right )}^2+a}{a}+\frac {1}{3}}{{\left (a\,{\mathrm {tan}\left (x\right )}^2+a\right )}^{3/2}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {a\,{\mathrm {tan}\left (x\right )}^2+a}}{\sqrt {a}}\right )}{a^{3/2}} \]