Integrand size = 26, antiderivative size = 71 \[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=\frac {4 \sqrt [4]{-1} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 i a^2}{d \sqrt {\cot (c+d x)}} \]
4*(-1)^(1/4)*a^2*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-2/3*a^2/d/cot(d*x+ c)^(3/2)+4*I*a^2/d/cot(d*x+c)^(1/2)
Time = 1.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=-\frac {2 a^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-6 (-1)^{3/4} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)} (-6 i+\tan (c+d x))\right )}{3 d} \]
(-2*a^2*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-6*(-1)^(3/4)*ArcTan[(-1)^( 3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*(-6*I + Tan[c + d*x])))/(3*d )
Time = 0.50 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4156, 3042, 4025, 27, 3042, 4012, 3042, 4016, 221}
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.
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+i a)^2}{\cot ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\int \frac {2 \left (\cot (c+d x) a^2+i a^2\right )}{\cot ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+2 \int \frac {\cot (c+d x) a^2+i a^2}{\cot ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+2 \int \frac {i a^2-a^2 \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+2 \left (\int \frac {a^2-i a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 i a^2}{d \sqrt {\cot (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+2 \left (\int \frac {i \tan \left (c+d x+\frac {\pi }{2}\right ) a^2+a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 i a^2}{d \sqrt {\cot (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+2 \left (\frac {2 a^4 \int \frac {1}{-i \cot (c+d x) a^2-a^2}d\sqrt {\cot (c+d x)}}{d}+\frac {2 i a^2}{d \sqrt {\cot (c+d x)}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+2 \left (\frac {2 \sqrt [4]{-1} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 i a^2}{d \sqrt {\cot (c+d x)}}\right )\) |
2*((2*(-1)^(1/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + ((2*I)*a^ 2)/(d*Sqrt[Cot[c + d*x]])) - (2*a^2)/(3*d*Cot[c + d*x]^(3/2))
3.8.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2*(c^2/f) Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b *Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (58 ) = 116\).
Time = 1.38 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.83
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {4 i}{\sqrt {\cot \left (d x +c \right )}}-\frac {2}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}\right )}{d}\) | \(201\) |
| default | \(\frac {a^{2} \left (\frac {4 i}{\sqrt {\cot \left (d x +c \right )}}-\frac {2}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}{1+\cot \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\cot }\left (d x +c \right )\right )\right )\right )}{2}\right )}{d}\) | \(201\) |
1/d*a^2*(4*I/cot(d*x+c)^(1/2)-2/3/cot(d*x+c)^(3/2)-1/2*2^(1/2)*(ln((1+cot( d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)))+ 2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)) )+1/2*I*2^(1/2)*(ln((1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/(1+cot(d*x+c)+ 2^(1/2)*cot(d*x+c)^(1/2)))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(- 1+2^(1/2)*cot(d*x+c)^(1/2))))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 340, normalized size of antiderivative = 4.79 \[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=-\frac {3 \, \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 3 \, \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 8 \, {\left (7 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 5 \, a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
-1/12*(3*sqrt(16*I*a^4/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I* c) + d)*log(1/2*(4*I*a^2*e^(2*I*d*x + 2*I*c) + sqrt(16*I*a^4/d^2)*(d*e^(2* I*d*x + 2*I*c) - d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/a^2) - 3*sqrt(16*I*a^4/d^2)*(d*e^(4*I*d*x + 4* I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(1/2*(4*I*a^2*e^(2*I*d*x + 2*I*c) - sqrt(16*I*a^4/d^2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt((I*e^(2*I*d*x + 2*I*c ) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/a^2) - 8*(7*a^2*e^ (4*I*d*x + 4*I*c) - 2*a^2*e^(2*I*d*x + 2*I*c) - 5*a^2)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(d*e^(4*I*d*x + 4*I*c) + 2*d*e^( 2*I*d*x + 2*I*c) + d)
\[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=- a^{2} \left (\int \frac {\tan ^{2}{\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx + \int \left (- \frac {2 i \tan {\left (c + d x \right )}}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx + \int \left (- \frac {1}{\sqrt {\cot {\left (c + d x \right )}}}\right )\, dx\right ) \]
-a**2*(Integral(tan(c + d*x)**2/sqrt(cot(c + d*x)), x) + Integral(-2*I*tan (c + d*x)/sqrt(cot(c + d*x)), x) + Integral(-1/sqrt(cot(c + d*x)), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (57) = 114\).
Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.06 \[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=-\frac {3 \, {\left (-\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} + 4 \, {\left (a^{2} - \frac {6 i \, a^{2}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{6 \, d} \]
-1/6*(3*(-(2*I - 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) - (2*I - 2)*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + (I + 1)*sqrt(2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - (I + 1)*sqrt(2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1 ))*a^2 + 4*(a^2 - 6*I*a^2/tan(d*x + c))*tan(d*x + c)^(3/2))/d
\[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{\sqrt {\cot \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^2}{\sqrt {\cot (c+d x)}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]