Integrand size = 17, antiderivative size = 156 \[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {\sqrt [3]{a} x}{2\ 2^{2/3}}-\frac {i \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac {i \sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {3 i \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d} \]
-1/4*a^(1/3)*x*2^(1/3)+1/4*I*a^(1/3)*ln(cos(d*x+c))*2^(1/3)/d+3/4*I*a^(1/3 )*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))*2^(1/3)/d-1/2*I*a^(1/3)*arc tan(1/3*(a^(1/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/2))*3^(1/2 )*2^(1/3)/d
Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {i \sqrt [3]{a} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+\log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )\right )}{2\ 2^{2/3} d} \]
((-1/2*I)*a^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a + I*a*Tan[c + d*x])^( 1/3))/a^(1/3))/Sqrt[3]] - 2*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^( 1/3)] + Log[2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + I*a*Tan[c + d*x])^(1/3) + (a + I*a*Tan[c + d*x])^(2/3)]))/(2^(2/3)*d)
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 3962, 69, 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.
| \(\displaystyle \int \sqrt [3]{a+i a \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt [3]{a+i a \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 3962 |
| \(\displaystyle -\frac {i a \int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle -\frac {i a \left (\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2\ 2^{2/3} a^{2/3}}+\frac {3 \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {i a \left (\frac {3 \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {i a \left (-\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {i a \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{2^{2/3} a^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2\ 2^{2/3} a^{2/3}}+\frac {\log (a-i a \tan (c+d x))}{2\ 2^{2/3} a^{2/3}}\right )}{d}\) |
((-I)*a*((I*Sqrt[3]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(2/3)*a^(2/3)) - (3*Log[2^(1/3)*a^(1/3) - I*a*Tan[c + d*x]])/(2*2^(2/3)*a^(2/3)) + Log[a - I*a*Tan[c + d*x]]/(2*2^(2/3)*a^(2/3))))/d
3.3.75.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) 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[-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]
Time = 0.43 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {3 i a \left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right )}{d}\) | \(133\) |
| default | \(\frac {3 i a \left (\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {2}{3}}}-\frac {2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {2}{3}}}\right )}{d}\) | \(133\) |
3*I/d*a*(1/6*2^(1/3)/a^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))- 1/12*2^(1/3)/a^(2/3)*ln((a+I*a*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*ta n(d*x+c))^(1/3)+2^(2/3)*a^(2/3))-1/6*2^(1/3)/a^(2/3)*3^(1/2)*arctan(1/3*3^ (1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))
Time = 0.25 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21 \[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \left (-\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} - {\left (\sqrt {3} d + i \, d\right )} \left (-\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \left (-\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (\sqrt {3} d - i \, d\right )} \left (-\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) + \left (-\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + 2 i \, d \left (-\frac {i \, a}{4 \, d^{3}}\right )^{\frac {1}{3}}\right ) \]
1/2*(I*sqrt(3) - 1)*(-1/4*I*a/d^3)^(1/3)*log(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) - (sqrt(3)*d + I*d)*(-1/4*I*a/d^3)^ (1/3)) + 1/2*(-I*sqrt(3) - 1)*(-1/4*I*a/d^3)^(1/3)*log(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) + (sqrt(3)*d - I*d)*(-1/4 *I*a/d^3)^(1/3)) + (-1/4*I*a/d^3)^(1/3)*log(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*d*(-1/4*I*a/d^3)^(1/3))
\[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=\int \sqrt [3]{i a \tan {\left (c + d x \right )} + a}\, dx \]
Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87 \[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {i \, {\left (2 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) + 2^{\frac {1}{3}} a^{\frac {4}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 2 \cdot 2^{\frac {1}{3}} a^{\frac {4}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )\right )}}{4 \, a d} \]
-1/4*I*(2*sqrt(3)*2^(1/3)*a^(4/3)*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)) + 2^(1/3)*a^(4/3)*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)) - 2*2^(1/3)*a^(4/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3)))/(a*d)
\[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
Time = 4.39 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.18 \[ \int \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (18\,{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,{\left (-a\right )}^{4/3}\,d^2+a\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,9{}\mathrm {i}\right )}{d}+\frac {{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (a\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,9{}\mathrm {i}+18\,{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,{\left (-a\right )}^{4/3}\,d^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d}-\frac {{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (a\,d^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}\,9{}\mathrm {i}-18\,{\left (\frac {1}{4}{}\mathrm {i}\right )}^{1/3}\,{\left (-a\right )}^{4/3}\,d^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d} \]
((1i/4)^(1/3)*(-a)^(1/3)*log(18*(1i/4)^(1/3)*(-a)^(4/3)*d^2 + a*d^2*(a + a *tan(c + d*x)*1i)^(1/3)*9i))/d + ((1i/4)^(1/3)*(-a)^(1/3)*log(a*d^2*(a + a *tan(c + d*x)*1i)^(1/3)*9i + 18*(1i/4)^(1/3)*(-a)^(4/3)*d^2*((3^(1/2)*1i)/ 2 - 1/2))*((3^(1/2)*1i)/2 - 1/2))/d - ((1i/4)^(1/3)*(-a)^(1/3)*log(a*d^2*( a + a*tan(c + d*x)*1i)^(1/3)*9i - 18*(1i/4)^(1/3)*(-a)^(4/3)*d^2*((3^(1/2) *1i)/2 + 1/2))*((3^(1/2)*1i)/2 + 1/2))/d