Integrand size = 28, antiderivative size = 102 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {(1+i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d} \]
(1+I)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2))*a^( 1/2)*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d-2*cot(d*x+c)^(1/2)*(a+I*a*tan(d*x +c))^(1/2)/d
Time = 0.37 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}-2 \sqrt {a+i a \tan (c+d x)}\right )}{d} \]
(Sqrt[Cot[c + d*x]]*(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt [a + I*a*Tan[c + d*x]]]*Sqrt[I*a*Tan[c + d*x]] - 2*Sqrt[a + I*a*Tan[c + d* x]]))/d
Time = 0.50 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4729, 3042, 4031, 3042, 4027, 218}
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.
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{3/2} \sqrt {a+i a \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4031 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (i \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (i \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 a^2 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {(1+i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*S qrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*Sqrt[a + I* a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))
3.8.53.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-d)*(a + b*Tan[e + f*x])^m*((c + d*Ta n[e + f*x])^(n + 1)/(f*m*(c^2 + d^2))), x] + Simp[a/(a*c - b*d) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d , e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^ 2, 0] && EqQ[m + n + 1, 0] && !LtQ[m, -1]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (84 ) = 168\).
Time = 39.45 (sec) , antiderivative size = 493, normalized size of antiderivative = 4.83
| method | result | size |
| default | \(\frac {i \sec \left (d x +c \right ) \left (i \cos \left (d x +c \right )+i+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (2 i \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sin \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )+2 i \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sin \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )+i \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sin \left (d x +c \right ) \ln \left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1}{-\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-\csc \left (d x +c \right )+\cot \left (d x +c \right )+1}\right )+\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sin \left (d x +c \right ) \ln \left (\frac {-\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-\csc \left (d x +c \right )+\cot \left (d x +c \right )+1}{\cot \left (d x +c \right )-\csc \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1}\right )+2 \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sin \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )+2 \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sin \left (d x +c \right ) \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )-2 i \sqrt {2}\, \cos \left (d x +c \right )+2 i \sqrt {2}+2 \sin \left (d x +c \right ) \sqrt {2}\right ) \sqrt {2}}{4 d}\) | \(493\) |
1/4*I/d*sec(d*x+c)*(I*cos(d*x+c)+I+sin(d*x+c))*(a*(1+I*tan(d*x+c)))^(1/2)* cot(d*x+c)^(3/2)*(2*I*(cot(d*x+c)-csc(d*x+c))^(1/2)*sin(d*x+c)*arctan((cot (d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-1)+2*I*(cot(d*x+c)-csc(d*x+c))^(1/2)*sin (d*x+c)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1)+I*(cot(d*x+c)-csc( d*x+c))^(1/2)*sin(d*x+c)*ln((cot(d*x+c)-csc(d*x+c)+(cot(d*x+c)-csc(d*x+c)) ^(1/2)*2^(1/2)+1)/(-(cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-csc(d*x+c)+cot(d *x+c)+1))+(cot(d*x+c)-csc(d*x+c))^(1/2)*sin(d*x+c)*ln((-(cot(d*x+c)-csc(d* x+c))^(1/2)*2^(1/2)-csc(d*x+c)+cot(d*x+c)+1)/(cot(d*x+c)-csc(d*x+c)+(cot(d *x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1))+2*(cot(d*x+c)-csc(d*x+c))^(1/2)*sin(d* x+c)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-1)+2*(cot(d*x+c)-csc(d*x +c))^(1/2)*sin(d*x+c)*arctan((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)+1)-2*I* 2^(1/2)*cos(d*x+c)+2*I*2^(1/2)+2*sin(d*x+c)*2^(1/2))*2^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (78) = 156\).
Time = 0.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.76 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {8 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (i \, d x + i \, c\right )} - d \sqrt {\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + d \sqrt {\frac {8 i \, a}{d^{2}}} \log \left (-{\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} - 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{4 \, d} \]
-1/4*(8*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*e^(2*I*d*x + 2*I *c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*e^(I*d*x + I*c) - d*sqrt(8*I*a/d^2)*lo g((sqrt(2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*s qrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(8*I*a/d^2) + 4*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + d*sqrt(8*I*a/d^2)*log(-(sqrt (2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I* e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(8*I*a/d^2) - 4*I* a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)))/d
\[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (78) = 156\).
Time = 0.41 (sec) , antiderivative size = 540, normalized size of antiderivative = 5.29 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \sqrt {a} {\left (\left (2 i - 2\right ) \, \arctan \left (2 \, {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right ) + 2 \, \sin \left (d x + c\right ), 2 \, {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right ) + 2 \, \cos \left (d x + c\right )\right ) + \left (i + 1\right ) \, \log \left (4 \, \cos \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right )^{2} + 4 \, \sqrt {\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right )^{2}\right )} + 8 \, {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (d x + c\right ) \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right ) + \sin \left (d x + c\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right )\right )}\right )\right )} - 4 \, {\left ({\left (\left (i + 1\right ) \, \cos \left (d x + c\right ) + \left (i - 1\right ) \, \sin \left (d x + c\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right ) + {\left (-\left (i - 1\right ) \, \cos \left (d x + c\right ) + \left (i + 1\right ) \, \sin \left (d x + c\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) - 1\right )\right )\right )} \sqrt {a}}{2 \, {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} - 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} d} \]
1/2*((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1 /4)*sqrt(a)*((2*I - 2)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d* x + 2*c) - 1)) + 2*sin(d*x + c), 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^ 2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2* d*x + 2*c) - 1)) + 2*cos(d*x + c)) + (I + 1)*log(4*cos(d*x + c)^2 + 4*sin( d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(d*x + c)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c) *sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))))) - 4*(((I + 1) *cos(d*x + c) + (I - 1)*sin(d*x + c))*cos(1/2*arctan2(sin(2*d*x + 2*c), co s(2*d*x + 2*c) - 1)) + (-(I - 1)*cos(d*x + c) + (I + 1)*sin(d*x + c))*sin( 1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a))/((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*d)
\[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]