Integrand size = 24, antiderivative size = 286 \[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=-\frac {i x}{4 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a} d}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {3}{2 d \sqrt [3]{a+i a \tan (c+d x)}} \] Output:
-1/8*I*x*2^(2/3)/a^(1/3)+3^(1/2)*arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^ (1/3))*3^(1/2)/a^(1/3))/a^(1/3)/d-1/4*3^(1/2)*arctan(1/3*(a^(1/3)+2^(2/3)* (a+I*a*tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))*2^(2/3)/a^(1/3)/d-1/8*ln(cos(d* x+c))*2^(2/3)/a^(1/3)/d-1/2*ln(tan(d*x+c))/a^(1/3)/d+3/2*ln(a^(1/3)-(a+I*a *tan(d*x+c))^(1/3))/a^(1/3)/d-3/8*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1 /3))*2^(2/3)/a^(1/3)/d+3/2/d/(a+I*a*tan(d*x+c))^(1/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.60 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.58 \[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=\frac {\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{\sqrt [3]{a}}+\frac {6}{\sqrt [3]{a+i a \tan (c+d x)}}-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},\frac {1}{2} (1+i \tan (c+d x))\right )}{\sqrt [3]{a+i a \tan (c+d x)}}}{2 d} \] Input:
Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(1/3),x]
Output:
((2*Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^( 1/3))])/a^(1/3) - Log[Tan[c + d*x]]/a^(1/3) + (3*Log[a^(1/3) - (a + I*a*Ta n[c + d*x])^(1/3)])/a^(1/3) + 6/(a + I*a*Tan[c + d*x])^(1/3) - (3*Hypergeo metric2F1[-1/3, 1, 2/3, (1 + I*Tan[c + d*x])/2])/(a + I*a*Tan[c + d*x])^(1 /3))/(2*d)
Time = 1.18 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.97, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 4045, 3042, 3960, 3042, 3962, 67, 16, 1082, 217, 4079, 27, 3042, 4082, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4045 |
| \(\displaystyle i \int \frac {1}{\sqrt [3]{i \tan (c+d x) a+a}}dx-\frac {i \int \frac {\cot (c+d x) (\tan (c+d x) a+i a)}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {1}{\sqrt [3]{i \tan (c+d x) a+a}}dx-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle i \left (\frac {\int (i \tan (c+d x) a+a)^{2/3}dx}{2 a}+\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\int (i \tan (c+d x) a+a)^{2/3}dx}{2 a}+\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 3962 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \int \frac {1}{(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{2 d}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {3 \int \frac {2}{3} \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (\tan (c+d x) a^2+i a^2\right )dx}{2 a^2}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {\int \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (\tan (c+d x) a^2+i a^2\right )dx}{a^2}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {\int \frac {(i \tan (c+d x) a+a)^{2/3} \left (\tan (c+d x) a^2+i a^2\right )}{\tan (c+d x)}dx}{a^2}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {i a \int \frac {\cot (c+d x)}{\sqrt [3]{i \tan (c+d x) a+a}}d\tan (c+d x)}{d}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {i a \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{i \tan (c+d x) a+a}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {i a \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {i a \left (-\frac {3 \int \frac {1}{-(i \tan (c+d x) a+a)^{2/3}-3}d\left (\frac {2 \sqrt [3]{i \tan (c+d x) a+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle i \left (\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}\right )-\frac {i \left (\frac {i a \left (\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a}{d \sqrt [3]{a+i a \tan (c+d x)}}\right )}{a}\) |
Input:
Int[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(1/3),x]
Output:
I*(((-1/2*I)*(((-I)*Sqrt[3]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(1/3)*a^ (1/3)) - (3*Log[2^(1/3)*a^(1/3) - I*a*Tan[c + d*x]])/(2*2^(1/3)*a^(1/3)) + Log[a - I*a*Tan[c + d*x]]/(2*2^(1/3)*a^(1/3))))/d + ((3*I)/2)/(d*(a + I*a *Tan[c + d*x])^(1/3))) - (I*((I*a*((Sqrt[3]*ArcTan[(1 + (2*(a + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[Tan[c + d*x]]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(1/3))))/d + ((3*I) *a)/(d*(a + I*a*Tan[c + d*x])^(1/3))))/a
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a) Int[(a + b*Tan[c + d*x])^ (n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d S ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b , c, d, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a/(a*c - b*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[d/(a*c - b*d) Int[(a + b*Tan[e + f*x])^m*((b + a*Tan[e + f *x])/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Ne Q[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
\[\int \frac {\cot \left (d x +c \right )}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
Input:
int(cot(d*x+c)/(a+I*a*tan(d*x+c))^(1/3),x)
Output:
int(cot(d*x+c)/(a+I*a*tan(d*x+c))^(1/3),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (206) = 412\).
Time = 0.09 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.03 \[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))^(1/3),x, algorithm="fricas")
Output:
1/4*(2*(1/2)^(1/3)*a*d*(-1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c)*log(-2*(1/2) ^(2/3)*a*d^2*(-1/(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1 /3)*e^(2/3*I*d*x + 2/3*I*c)) + 4*a*d*(1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c) *log(-a*d^2*(1/(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3 )*e^(2/3*I*d*x + 2/3*I*c)) - (1/2)^(1/3)*(I*sqrt(3)*a*d + a*d)*(-1/(a*d^3) )^(1/3)*e^(2*I*d*x + 2*I*c)*log(-(1/2)^(2/3)*(I*sqrt(3)*a*d^2 - a*d^2)*(-1 /(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - (1/2)^(1/3)*(-I*sqrt(3)*a*d + a*d)*(-1/(a*d^3))^(1/3)*e^(2* I*d*x + 2*I*c)*log(-(1/2)^(2/3)*(-I*sqrt(3)*a*d^2 - a*d^2)*(-1/(a*d^3))^(2 /3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 3*2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(e^(2*I*d*x + 2*I*c) + 1) *e^(4/3*I*d*x + 4/3*I*c) - 2*(-I*sqrt(3)*a*d + a*d)*(1/(a*d^3))^(1/3)*e^(2 *I*d*x + 2*I*c)*log(1/2*(I*sqrt(3)*a*d^2 + a*d^2)*(1/(a*d^3))^(2/3) + 2^(1 /3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 2*(I*sq rt(3)*a*d + a*d)*(1/(a*d^3))^(1/3)*e^(2*I*d*x + 2*I*c)*log(1/2*(-I*sqrt(3) *a*d^2 + a*d^2)*(1/(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^ (1/3)*e^(2/3*I*d*x + 2/3*I*c)))*e^(-2*I*d*x - 2*I*c)/(a*d)
\[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))**(1/3),x)
Output:
Integral(cot(c + d*x)/(I*a*(tan(c + d*x) - I))**(1/3), x)
Time = 0.12 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.87 \[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=-\frac {\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {8 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} + \frac {4 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} - \frac {8 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {12}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}}{8 \, d} \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))^(1/3),x, algorithm="maxima")
Output:
-1/8*(2*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I *a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(1/3) - 2^(2/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^( 2/3))/a^(1/3) + 2*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1 /3))/a^(1/3) - 8*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3 ) + a^(1/3))/a^(1/3))/a^(1/3) + 4*log((I*a*tan(d*x + c) + a)^(2/3) + (I*a* tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(1/3) - 8*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(1/3) - 12/(I*a*tan(d*x + c) + a)^(1/3))/d
Time = 0.33 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.87 \[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=-\frac {\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {8 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} + \frac {4 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} - \frac {8 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {12}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}}{8 \, d} \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))^(1/3),x, algorithm="giac")
Output:
-1/8*(2*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I *a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(1/3) - 2^(2/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^( 2/3))/a^(1/3) + 2*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1 /3))/a^(1/3) - 8*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3 ) + a^(1/3))/a^(1/3))/a^(1/3) + 4*log((I*a*tan(d*x + c) + a)^(2/3) + (I*a* tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(1/3) - 8*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(1/3) - 12/(I*a*tan(d*x + c) + a)^(1/3))/d
Time = 0.28 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.95 \[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:
int(cot(c + d*x)/(a + a*tan(c + d*x)*1i)^(1/3),x)
Output:
3/(2*d*(a + a*tan(c + d*x)*1i)^(1/3)) + log((746496*a^7*d^9*(1/(a*d^3))^(2 /3) - 528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3))*(1/(a*d^3))^(1/3) - 21 7728*a^6*d^6)*(1/(a*d^3))^(1/3) + log((746496*a^7*d^9*(-1/(16*a*d^3))^(2/3 ) - 528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3))*(-1/(16*a*d^3))^(1/3) - 217728*a^6*d^6)*(-1/(16*a*d^3))^(1/3) + (log(217728*a^6*d^6 + ((3^(1/2)*1i - 1)*(528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 186624*a^7*d^9*(3^(1 /2)*1i - 1)^2*(1/(a*d^3))^(2/3))*(1/(a*d^3))^(1/3))/2)*(3^(1/2)*1i - 1)*(1 /(a*d^3))^(1/3))/2 - (log(217728*a^6*d^6 - ((3^(1/2)*1i + 1)*(528768*a^6*d ^7*(a + a*tan(c + d*x)*1i)^(1/3) - 186624*a^7*d^9*(3^(1/2)*1i + 1)^2*(1/(a *d^3))^(2/3))*(1/(a*d^3))^(1/3))/2)*(3^(1/2)*1i + 1)*(1/(a*d^3))^(1/3))/2 + log(217728*a^6*d^6 + ((3^(1/2)*1i)/2 - 1/2)*(528768*a^6*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 746496*a^7*d^9*((3^(1/2)*1i)/2 - 1/2)^2*(-1/(16*a*d^3) )^(2/3))*(-1/(16*a*d^3))^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(-1/(16*a*d^3))^(1/ 3) - log(217728*a^6*d^6 - ((3^(1/2)*1i)/2 + 1/2)*(528768*a^6*d^7*(a + a*ta n(c + d*x)*1i)^(1/3) - 746496*a^7*d^9*((3^(1/2)*1i)/2 + 1/2)^2*(-1/(16*a*d ^3))^(2/3))*(-1/(16*a*d^3))^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(-1/(16*a*d^3))^ (1/3)
\[ \int \frac {\cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=\frac {\int \frac {\cot \left (d x +c \right )}{\left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}}}d x}{a^{\frac {1}{3}}} \] Input:
int(cot(d*x+c)/(a+I*a*tan(d*x+c))^(1/3),x)
Output:
int(cot(c + d*x)/(tan(c + d*x)*i + 1)**(1/3),x)/a**(1/3)