Integrand size = 14, antiderivative size = 415 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=-\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \]
-1/4*x*(a-(-b^2)^(1/2))^(1/3)-1/4*b*ln(cos(d*x+c))*(a-(-b^2)^(1/2))^(1/3)/ d/(-b^2)^(1/2)-3/4*b*ln((a-(-b^2)^(1/2))^(1/3)-(a+b*tan(d*x+c))^(1/3))*(a- (-b^2)^(1/2))^(1/3)/d/(-b^2)^(1/2)+1/2*b*arctan(1/3*(1+2*(a+b*tan(d*x+c))^ (1/3)/(a-(-b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)*(a-(-b^2)^(1/2))^(1/3)/d/(- b^2)^(1/2)-1/4*x*(a+(-b^2)^(1/2))^(1/3)+1/4*b*ln(cos(d*x+c))*(a+(-b^2)^(1/ 2))^(1/3)/d/(-b^2)^(1/2)+3/4*b*ln((a+(-b^2)^(1/2))^(1/3)-(a+b*tan(d*x+c))^ (1/3))*(a+(-b^2)^(1/2))^(1/3)/d/(-b^2)^(1/2)-1/2*b*arctan(1/3*(1+2*(a+b*ta n(d*x+c))^(1/3)/(a+(-b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)*(a+(-b^2)^(1/2))^ (1/3)/d/(-b^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\frac {-i \sqrt [3]{a-i b} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a-i b)^{2/3}+\sqrt [3]{a-i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )+i \sqrt [3]{a+i b} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a+i b)^{2/3}+\sqrt [3]{a+i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )}{4 d} \]
((-I)*(a - I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3) )/(a - I*b)^(1/3))/Sqrt[3]] - 2*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x]) ^(1/3)] + Log[(a - I*b)^(2/3) + (a - I*b)^(1/3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]) + I*(a + I*b)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - 2*Log[(a + I*b )^(1/3) - (a + b*Tan[c + d*x])^(1/3)] + Log[(a + I*b)^(2/3) + (a + I*b)^(1 /3)*(a + b*Tan[c + d*x])^(1/3) + (a + b*Tan[c + d*x])^(2/3)]))/(4*d)
Time = 0.51 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3966, 485, 2009}
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.
| \(\displaystyle \int \sqrt [3]{a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt [3]{a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle \frac {b \int \frac {\sqrt [3]{a+b \tan (c+d x)}}{\tan ^2(c+d x) b^2+b^2}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 485 |
| \(\displaystyle \frac {b \int \left (\frac {\sqrt [3]{a+b \tan (c+d x)} \sqrt {-b^2}}{2 b^2 \left (\sqrt {-b^2}-b \tan (c+d x)\right )}+\frac {\sqrt [3]{a+b \tan (c+d x)} \sqrt {-b^2}}{2 b^2 \left (b \tan (c+d x)+\sqrt {-b^2}\right )}\right )d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {3} \sqrt [3]{a-\sqrt {-b^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2}}-\frac {\sqrt {3} \sqrt [3]{a+\sqrt {-b^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2}}-\frac {\sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{4 \sqrt {-b^2}}+\frac {\sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{4 \sqrt {-b^2}}-\frac {3 \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2}}+\frac {3 \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2}}\right )}{d}\) |
(b*((Sqrt[3]*(a - Sqrt[-b^2])^(1/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1 /3))/(a - Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]) - (Sqrt[3]*(a + Sqrt [-b^2])^(1/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + Sqrt[-b^2])^ (1/3))/Sqrt[3]])/(2*Sqrt[-b^2]) - ((a + Sqrt[-b^2])^(1/3)*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(4*Sqrt[-b^2]) + ((a - Sqrt[-b^2])^(1/3)*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(4*Sqrt[-b^2]) - (3*(a - Sqrt[-b^2])^(1/3)*Log[(a - Sqr t[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]) + (3*(a + Sqr t[-b^2])^(1/3)*Log[(a + Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/( 4*Sqrt[-b^2])))/d
3.7.91.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand Integrand[(c + d*x)^n, 1/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, n}, x] & & !IntegerQ[2*n]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.14
| method | result | size |
| derivativedivides | \(\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
| default | \(\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
1/2/d*b*sum(_R^3/(_R^5-_R^2*a)*ln((a+b*tan(d*x+c))^(1/3)-_R),_R=RootOf(_Z^ 6-2*_Z^3*a+a^2+b^2))
Time = 0.25 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.29 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} d^{4} + d^{4}\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} d^{4} - d^{4}\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} d^{4} + d^{4}\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} d^{4} - d^{4}\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{2} \, \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \log \left (d^{4} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{2} \, \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \log \left (-d^{4} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) \]
-1/4*(sqrt(-3) + 1)*(-(d^3*sqrt(-a^2/d^6) + b)/d^3)^(1/3)*log(-1/2*(sqrt(- 3)*d^4 + d^4)*(-(d^3*sqrt(-a^2/d^6) + b)/d^3)^(1/3)*sqrt(-a^2/d^6) + (b*ta n(d*x + c) + a)^(1/3)*a) + 1/4*(sqrt(-3) - 1)*(-(d^3*sqrt(-a^2/d^6) + b)/d ^3)^(1/3)*log(1/2*(sqrt(-3)*d^4 - d^4)*(-(d^3*sqrt(-a^2/d^6) + b)/d^3)^(1/ 3)*sqrt(-a^2/d^6) + (b*tan(d*x + c) + a)^(1/3)*a) - 1/4*(sqrt(-3) + 1)*((d ^3*sqrt(-a^2/d^6) - b)/d^3)^(1/3)*log(1/2*(sqrt(-3)*d^4 + d^4)*((d^3*sqrt( -a^2/d^6) - b)/d^3)^(1/3)*sqrt(-a^2/d^6) + (b*tan(d*x + c) + a)^(1/3)*a) + 1/4*(sqrt(-3) - 1)*((d^3*sqrt(-a^2/d^6) - b)/d^3)^(1/3)*log(-1/2*(sqrt(-3 )*d^4 - d^4)*((d^3*sqrt(-a^2/d^6) - b)/d^3)^(1/3)*sqrt(-a^2/d^6) + (b*tan( d*x + c) + a)^(1/3)*a) + 1/2*(-(d^3*sqrt(-a^2/d^6) + b)/d^3)^(1/3)*log(d^4 *(-(d^3*sqrt(-a^2/d^6) + b)/d^3)^(1/3)*sqrt(-a^2/d^6) + (b*tan(d*x + c) + a)^(1/3)*a) + 1/2*((d^3*sqrt(-a^2/d^6) - b)/d^3)^(1/3)*log(-d^4*((d^3*sqrt (-a^2/d^6) - b)/d^3)^(1/3)*sqrt(-a^2/d^6) + (b*tan(d*x + c) + a)^(1/3)*a)
\[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\int \sqrt [3]{a + b \tan {\left (c + d x \right )}}\, dx \]
\[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
\[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
Time = 7.96 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.08 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
log((a + b*tan(c + d*x))^(1/3) + d*(-(a*1i + b)/d^3)^(1/3)*1i)*(-(a*1i + b )/(8*d^3))^(1/3) + log(b*(a + b*tan(c + d*x))^(1/3)*1i - a*(a + b*tan(c + d*x))^(1/3) + d^4*((a*1i - b)/d^3)^(4/3) + 2*b*d*((a*1i - b)/d^3)^(1/3))*( (a*1i - b)/(8*d^3))^(1/3) - log((486*(b^8 - a^4*b^4)*(a + b*tan(c + d*x))^ (1/3))/d^4 - (((((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/d^3)^(2/3)*((3888*b^5* (a^2 + b^2)*(a + b*tan(c + d*x))^(1/3))/d - 3888*a*b^4*((3^(1/2)*1i)/2 + 1 /2)*(-(a*1i + b)/d^3)^(1/3)*(a^2 + b^2)))/4 + (1944*a*b^5*(a^2 + b^2))/d^3 )*((3^(1/2)*1i)/2 + 1/2)*(-(a*1i + b)/d^3)^(1/3))/2)*((3^(1/2)*1i)/2 + 1/2 )*(-(a*1i + b)/(8*d^3))^(1/3) + log((486*(b^8 - a^4*b^4)*(a + b*tan(c + d* x))^(1/3))/d^4 - (((((3^(1/2)*1i)/2 + 1/2)*(-(a*1i + b)/d^3)^(2/3)*((3888* b^5*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/3))/d + 3888*a*b^4*((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/d^3)^(1/3)*(a^2 + b^2)))/4 - (1944*a*b^5*(a^2 + b^2)) /d^3)*((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/d^3)^(1/3))/2)*((3^(1/2)*1i)/2 - 1/2)*(-(a*1i + b)/(8*d^3))^(1/3) - log((((a*1i - b)/d^3)^(1/3)*((3^(1/2)* 1i)/2 + 1/2)*((((a*1i - b)/d^3)^(2/3)*((3^(1/2)*1i)/2 - 1/2)*((3888*b^5*(a ^2 + b^2)*(a + b*tan(c + d*x))^(1/3))/d - 3888*a*b^4*((a*1i - b)/d^3)^(1/3 )*((3^(1/2)*1i)/2 + 1/2)*(a^2 + b^2)))/4 + (1944*a*b^5*(a^2 + b^2))/d^3))/ 2 - (486*(b^8 - a^4*b^4)*(a + b*tan(c + d*x))^(1/3))/d^4)*((3^(1/2)*1i)/2 + 1/2)*((a*1i - b)/(8*d^3))^(1/3) + log((486*(b^8 - a^4*b^4)*(a + b*tan(c + d*x))^(1/3))/d^4 - (((a*1i - b)/d^3)^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(((...