Integrand size = 14, antiderivative size = 415 \[ \int (a+b \tan (c+d x))^{2/3} \, dx=-\frac {1}{4} \left (a-\sqrt {-b^2}\right )^{2/3} x-\frac {1}{4} \left (a+\sqrt {-b^2}\right )^{2/3} x-\frac {\sqrt {3} b \left (a-\sqrt {-b^2}\right )^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}+\frac {\sqrt {3} b \left (a+\sqrt {-b^2}\right )^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {b \left (a-\sqrt {-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \left (a+\sqrt {-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {3 b \left (a-\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \left (a+\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \]
-1/4*x*(a-(-b^2)^(1/2))^(2/3)-1/4*b*ln(cos(d*x+c))*(a-(-b^2)^(1/2))^(2/3)/ 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))^(2/3)/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))^(2/3)/d/(- b^2)^(1/2)-1/4*x*(a+(-b^2)^(1/2))^(2/3)+1/4*b*ln(cos(d*x+c))*(a+(-b^2)^(1/ 2))^(2/3)/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))^(2/3)/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))^ (2/3)/d/(-b^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.54 \[ \int (a+b \tan (c+d x))^{2/3} \, dx=\frac {\frac {(i a+b) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-\log (i+\tan (c+d x))+3 \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )\right )}{\sqrt [3]{a-i b}}+\frac {(-i a+b) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-\log (i-\tan (c+d x))+3 \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )\right )}{\sqrt [3]{a+i b}}}{4 d} \]
(((I*a + b)*(2*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b )^(1/3))/Sqrt[3]] - Log[I + Tan[c + d*x]] + 3*Log[(a - I*b)^(1/3) - (a + b *Tan[c + d*x])^(1/3)]))/(a - I*b)^(1/3) + (((-I)*a + b)*(2*Sqrt[3]*ArcTan[ (1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]] - Log[I - Ta n[c + d*x]] + 3*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)]))/(a + I *b)^(1/3))/(4*d)
Time = 0.58 (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.
\(\displaystyle \int (a+b \tan (c+d x))^{2/3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+b \tan (c+d x))^{2/3}dx\) |
\(\Big \downarrow \) 3966 |
\(\displaystyle \frac {b \int \frac {(a+b \tan (c+d x))^{2/3}}{\tan ^2(c+d x) b^2+b^2}d(b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 485 |
\(\displaystyle \frac {b \int \left (\frac {\sqrt {-b^2} (a+b \tan (c+d x))^{2/3}}{2 b^2 \left (\sqrt {-b^2}-b \tan (c+d x)\right )}+\frac {\sqrt {-b^2} (a+b \tan (c+d x))^{2/3}}{2 b^2 \left (b \tan (c+d x)+\sqrt {-b^2}\right )}\right )d(b \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (-\frac {\sqrt {3} \left (a-\sqrt {-b^2}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2}}+\frac {\sqrt {3} \left (a+\sqrt {-b^2}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2}}-\frac {\left (a+\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{4 \sqrt {-b^2}}+\frac {\left (a-\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{4 \sqrt {-b^2}}-\frac {3 \left (a-\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2}}+\frac {3 \left (a+\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2}}\right )}{d}\) |
(b*(-1/2*(Sqrt[3]*(a - Sqrt[-b^2])^(2/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x ])^(1/3))/(a - Sqrt[-b^2])^(1/3))/Sqrt[3]])/Sqrt[-b^2] + (Sqrt[3]*(a + Sqr t[-b^2])^(2/3)*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])^(2/3)*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(4*Sqrt[-b^2]) + ((a - Sqrt[-b^2])^(2/3)*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(4*Sqrt[-b^2]) - (3*(a - Sqrt[-b^2])^(2/3)*Log[(a - Sq rt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]) + (3*(a + Sq rt[-b^2])^(2/3)*Log[(a + Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/ (4*Sqrt[-b^2])))/d
3.7.90.3.1 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 = 0.78 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.14
method | result | size |
derivativedivides | \(\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
default | \(\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
1/2/d*b*sum(_R^4/(_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))
Leaf count of result is larger than twice the leaf count of optimal. 1011 vs. \(2 (325) = 650\).
Time = 0.28 (sec) , antiderivative size = 1011, normalized size of antiderivative = 2.44 \[ \int (a+b \tan (c+d x))^{2/3} \, dx=\text {Too large to display} \]
1/4*(sqrt(-3) - 1)*(-(d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6) + 2*a*b)/d^3) ^(1/3)*log(1/2*(sqrt(-3)*(a^3 - a*b^2)*d^2 + (a^3 - a*b^2)*d^2 + (sqrt(-3) *b*d^5 + b*d^5)*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6))*(-(d^3*sqrt(-(a^4 - 2* a^2*b^2 + b^4)/d^6) + 2*a*b)/d^3)^(2/3) - (a^4 - b^4)*(b*tan(d*x + c) + a) ^(1/3)) - 1/4*(sqrt(-3) + 1)*(-(d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6) + 2 *a*b)/d^3)^(1/3)*log(-1/2*(sqrt(-3)*(a^3 - a*b^2)*d^2 - (a^3 - a*b^2)*d^2 + (sqrt(-3)*b*d^5 - b*d^5)*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6))*(-(d^3*sqrt (-(a^4 - 2*a^2*b^2 + b^4)/d^6) + 2*a*b)/d^3)^(2/3) - (a^4 - b^4)*(b*tan(d* x + c) + a)^(1/3)) + 1/4*(sqrt(-3) - 1)*((d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4 )/d^6) - 2*a*b)/d^3)^(1/3)*log(1/2*(sqrt(-3)*(a^3 - a*b^2)*d^2 + (a^3 - a* b^2)*d^2 - (sqrt(-3)*b*d^5 + b*d^5)*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6))*(( d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6) - 2*a*b)/d^3)^(2/3) - (a^4 - b^4)*( b*tan(d*x + c) + a)^(1/3)) - 1/4*(sqrt(-3) + 1)*((d^3*sqrt(-(a^4 - 2*a^2*b ^2 + b^4)/d^6) - 2*a*b)/d^3)^(1/3)*log(-1/2*(sqrt(-3)*(a^3 - a*b^2)*d^2 - (a^3 - a*b^2)*d^2 - (sqrt(-3)*b*d^5 - b*d^5)*sqrt(-(a^4 - 2*a^2*b^2 + b^4) /d^6))*((d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6) - 2*a*b)/d^3)^(2/3) - (a^4 - b^4)*(b*tan(d*x + c) + a)^(1/3)) + 1/2*(-(d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6) + 2*a*b)/d^3)^(1/3)*log(-(b*d^5*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^ 6) + (a^3 - a*b^2)*d^2)*(-(d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^6) + 2*a*b) /d^3)^(2/3) - (a^4 - b^4)*(b*tan(d*x + c) + a)^(1/3)) + 1/2*((d^3*sqrt(...
\[ \int (a+b \tan (c+d x))^{2/3} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
\[ \int (a+b \tan (c+d x))^{2/3} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
\[ \int (a+b \tan (c+d x))^{2/3} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
Time = 10.11 (sec) , antiderivative size = 1229, normalized size of antiderivative = 2.96 \[ \int (a+b \tan (c+d x))^{2/3} \, dx=\text {Too large to display} \]
log((((((1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^2 + 1944*a*b ^4*(a^2 + b^2)*(-((a - b*1i)^2*1i)/d^3)^(2/3))*(-((a - b*1i)^2*1i)/d^3)^(1 /3))/2 + (972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3)*(-((a - b*1i)^2*1i)/d^3) ^(2/3))/4 + (486*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5)*(-(2 *a*b + a^2*1i - b^2*1i)/(8*d^3))^(1/3) + log(((((1944*a*b^4*(a^2 + b^2)*(- ((a*1i - b)^2*1i)/d^3)^(2/3) + (1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x) )^(1/3))/d^2)*(-((a*1i - b)^2*1i)/d^3)^(1/3))/2 + (972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3)*(-((a*1i - b)^2*1i)/d^3)^(2/3))/4 + (486*a*b^5*(a^2 + b^2 )^2*(a + b*tan(c + d*x))^(1/3))/d^5)*(-(2*a*b - a^2*1i + b^2*1i)/(8*d^3))^ (1/3) - log((((3^(1/2)*1i)/2 - 1/2)*((972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d ^3 - (((3^(1/2)*1i)/2 + 1/2)*((1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x)) ^(1/3))/d^2 + 1944*a*b^4*((3^(1/2)*1i)/2 - 1/2)*(a^2 + b^2)*(-((a - b*1i)^ 2*1i)/d^3)^(2/3))*(-((a - b*1i)^2*1i)/d^3)^(1/3))/2)*(-((a - b*1i)^2*1i)/d ^3)^(2/3))/4 + (486*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5)*( (3^(1/2)*1i)/2 + 1/2)*(-(2*a*b + a^2*1i - b^2*1i)/(8*d^3))^(1/3) + log((48 6*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5 - (((3^(1/2)*1i)/2 + 1/2)*((972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3 + (((3^(1/2)*1i)/2 - 1/2)*( (1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^2 - 1944*a*b^4*((3^( 1/2)*1i)/2 + 1/2)*(a^2 + b^2)*(-((a - b*1i)^2*1i)/d^3)^(2/3))*(-((a - b*1i )^2*1i)/d^3)^(1/3))/2)*(-((a - b*1i)^2*1i)/d^3)^(2/3))/4)*((3^(1/2)*1i)...