Integrand size = 24, antiderivative size = 65 \[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 (-1)^{3/4} a \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a}{d \sqrt {\cot (c+d x)}} \] Output:
-2*(-1)^(3/4)*a*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d+2/3*I*a/d/cot(d*x+c )^(3/2)+2*a/d/cot(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58 \[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 i a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-i \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \] Input:
Integrate[(a + I*a*Tan[c + d*x])/Cot[c + d*x]^(3/2),x]
Output:
(((2*I)/3)*a*Hypergeometric2F1[-3/2, 1, -1/2, (-I)*Cot[c + d*x]])/(d*Cot[c + d*x]^(3/2))
Time = 0.51 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 4156, 3042, 4012, 3042, 4012, 25, 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.
\(\displaystyle \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i a \tan (c+d x)}{\cot (c+d x)^{3/2}}dx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle \int \frac {a \cot (c+d x)+i a}{\cot ^{\frac {5}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int \frac {a-i a \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \tan \left (c+d x+\frac {\pi }{2}\right ) a+a}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int -\frac {\cot (c+d x) a+i a}{\sqrt {\cot (c+d x)}}dx+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a}{d \sqrt {\cot (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cot (c+d x) a+i a}{\sqrt {\cot (c+d x)}}dx+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a}{d \sqrt {\cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {i a-a \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a}{d \sqrt {\cot (c+d x)}}\) |
\(\Big \downarrow \) 4016 |
\(\displaystyle \frac {2 a^2 \int \frac {1}{a \cot (c+d x)-i a}d\sqrt {\cot (c+d x)}}{d}+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a}{d \sqrt {\cot (c+d x)}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 (-1)^{3/4} a \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {2 i a}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a}{d \sqrt {\cot (c+d x)}}\) |
Input:
Int[(a + I*a*Tan[c + d*x])/Cot[c + d*x]^(3/2),x]
Output:
(-2*(-1)^(3/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (((2*I)/3)*a) /(d*Cot[c + d*x]^(3/2)) + (2*a)/(d*Sqrt[Cot[c + d*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[(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. 199 vs. \(2 (52 ) = 104\).
Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 3.08
method | result | size |
derivativedivides | \(-\frac {a \left (-\frac {2}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 i}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(200\) |
default | \(-\frac {a \left (-\frac {2}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 i}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}\right )}{d}\) | \(200\) |
Input:
int((a+I*a*tan(d*x+c))/cot(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/d*a*(-2/cot(d*x+c)^(1/2)-2/3*I/cot(d*x+c)^(3/2)-1/4*I*2^(1/2)*(ln((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)+1) )+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2 )))-1/4*2^(1/2)*(ln((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)+1))+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. 329 vs. \(2 (51) = 102\).
Time = 0.09 (sec) , antiderivative size = 329, normalized size of antiderivative = 5.06 \[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {3 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \log \left (\frac {{\left ({\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} + 2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a}\right ) - 3 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \log \left (\frac {{\left ({\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} + 2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a}\right ) + 16 \, {\left (2 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate((a+I*a*tan(d*x+c))/cot(d*x+c)^(3/2),x, algorithm="fricas")
Output:
-1/12*(3*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-4*I*a ^2/d^2)*log(((I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(-4*I*a^2/d^2)*sqrt((I*e^ (2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)) + 2*I*a*e^(2*I*d*x + 2*I *c))*e^(-2*I*d*x - 2*I*c)/a) - 3*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(-4*I*a^2/d^2)*log(((-I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt (-4*I*a^2/d^2)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)) + 2*I*a*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/a) + 16*(2*I*a*e^(4*I*d *x + 4*I*c) - I*a*e^(2*I*d*x + 2*I*c) - I*a)*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)
\[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=i a \left (\int \left (- \frac {i}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right ) \] Input:
integrate((a+I*a*tan(d*x+c))/cot(d*x+c)**(3/2),x)
Output:
I*a*(Integral(-I/cot(c + d*x)**(3/2), x) + Integral(tan(c + d*x)/cot(c + d *x)**(3/2), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (51) = 102\).
Time = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.18 \[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {8 \, {\left (i \, a + \frac {3 \, a}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} - 3 \, {\left (-\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a}{12 \, d} \] Input:
integrate((a+I*a*tan(d*x+c))/cot(d*x+c)^(3/2),x, algorithm="maxima")
Output:
1/12*(8*(I*a + 3*a/tan(d*x + c))*tan(d*x + c)^(3/2) - 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)*l og(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)/d
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {i \, {\left (-\left (3 i - 3\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) - 2 \, \tan \left (d x + c\right )^{\frac {3}{2}} + 6 i \, \sqrt {\tan \left (d x + c\right )}\right )} a}{3 \, d} \] Input:
integrate((a+I*a*tan(d*x+c))/cot(d*x+c)^(3/2),x, algorithm="giac")
Output:
-1/3*I*(-(3*I - 3)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c) )) - 2*tan(d*x + c)^(3/2) + 6*I*sqrt(tan(d*x + c)))*a/d
Timed out. \[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((a + a*tan(c + d*x)*1i)/cot(c + d*x)^(3/2),x)
Output:
int((a + a*tan(c + d*x)*1i)/cot(c + d*x)^(3/2), x)
\[ \int \frac {a+i a \tan (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=a \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{2}}d x +\left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )}{\cot \left (d x +c \right )^{2}}d x \right ) i \right ) \] Input:
int((a+I*a*tan(d*x+c))/cot(d*x+c)^(3/2),x)
Output:
a*(int(sqrt(cot(c + d*x))/cot(c + d*x)**2,x) + int((sqrt(cot(c + d*x))*tan (c + d*x))/cot(c + d*x)**2,x)*i)