Integrand size = 12, antiderivative size = 243 \[ \int (b \tan (c+d x))^{4/3} \, dx=-\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{d}+\frac {b^{4/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}-\frac {b^{4/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 d}+\frac {\sqrt {3} b^{4/3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}-\frac {\sqrt {3} b^{4/3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 d}+\frac {3 b \sqrt [3]{b \tan (c+d x)}}{d} \]
-b^(4/3)*arctan((b*tan(d*x+c))^(1/3)/b^(1/3))/d-1/2*b^(4/3)*arctan(-3^(1/2 )+2*(b*tan(d*x+c))^(1/3)/b^(1/3))/d-1/2*b^(4/3)*arctan(3^(1/2)+2*(b*tan(d* x+c))^(1/3)/b^(1/3))/d+1/4*b^(4/3)*ln(b^(2/3)-b^(1/3)*3^(1/2)*(b*tan(d*x+c ))^(1/3)+(b*tan(d*x+c))^(2/3))*3^(1/2)/d-1/4*b^(4/3)*ln(b^(2/3)+b^(1/3)*3^ (1/2)*(b*tan(d*x+c))^(1/3)+(b*tan(d*x+c))^(2/3))*3^(1/2)/d+3*b*(b*tan(d*x+ c))^(1/3)/d
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.84 \[ \int (b \tan (c+d x))^{4/3} \, dx=\frac {b \sqrt [3]{b \tan (c+d x)} \left (-i \log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right )+i \log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right )-(-1)^{5/6} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )+(-1)^{5/6} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )-\sqrt [6]{-1} \log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )+\sqrt [6]{-1} \log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )+6 \sqrt [6]{\tan ^2(c+d x)}\right )}{2 d \sqrt [6]{\tan ^2(c+d x)}} \]
(b*(b*Tan[c + d*x])^(1/3)*((-I)*Log[1 - I*(Tan[c + d*x]^2)^(1/6)] + I*Log[ 1 + I*(Tan[c + d*x]^2)^(1/6)] - (-1)^(5/6)*Log[1 - (-1)^(1/6)*(Tan[c + d*x ]^2)^(1/6)] + (-1)^(5/6)*Log[1 + (-1)^(1/6)*(Tan[c + d*x]^2)^(1/6)] - (-1) ^(1/6)*Log[1 - (-1)^(5/6)*(Tan[c + d*x]^2)^(1/6)] + (-1)^(1/6)*Log[1 + (-1 )^(5/6)*(Tan[c + d*x]^2)^(1/6)] + 6*(Tan[c + d*x]^2)^(1/6)))/(2*d*(Tan[c + d*x]^2)^(1/6))
Time = 0.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3954, 3042, 3957, 266, 753, 27, 216, 1142, 25, 1082, 217, 1103}
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.
| \(\displaystyle \int (b \tan (c+d x))^{4/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (b \tan (c+d x))^{4/3}dx\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-b^2 \int \frac {1}{(b \tan (c+d x))^{2/3}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-b^2 \int \frac {1}{(b \tan (c+d x))^{2/3}}dx\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {b^3 \int \frac {1}{(b \tan (c+d x))^{2/3} \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \int \frac {1}{b^6 \tan ^6(c+d x)+b^2}d\sqrt [3]{b \tan (c+d x)}}{d}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\int \frac {1}{b^2 \tan ^2(c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{3 b^{4/3}}+\frac {\int \frac {2 \sqrt [3]{b}-\sqrt {3} \sqrt [3]{b \tan (c+d x)}}{2 \left (b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}\right )}d\sqrt [3]{b \tan (c+d x)}}{3 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt {3} \sqrt [3]{b \tan (c+d x)}}{2 \left (b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}\right )}d\sqrt [3]{b \tan (c+d x)}}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\int \frac {1}{b^2 \tan ^2(c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{3 b^{4/3}}+\frac {\int \frac {2 \sqrt [3]{b}-\sqrt {3} \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt {3} \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\int \frac {2 \sqrt [3]{b}-\sqrt {3} \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\int \frac {2 \sqrt [3]{b}+\sqrt {3} \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\arctan \left (b^{2/3} \tan (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\frac {1}{2} \sqrt [3]{b} \int \frac {1}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3} \sqrt [3]{b}-2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\frac {1}{2} \sqrt [3]{b} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}+2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\arctan \left (b^{2/3} \tan (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\frac {1}{2} \sqrt [3]{b} \int \frac {1}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}-2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\frac {1}{2} \sqrt [3]{b} \int \frac {1}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}+2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\arctan \left (b^{2/3} \tan (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\frac {\int \frac {1}{-b^2 \tan ^2(c+d x)-\frac {1}{3}}d\left (1-\frac {2 b^{2/3} \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}-2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}}{6 b^{5/3}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}+2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}-\frac {\int \frac {1}{-b^2 \tan ^2(c+d x)-\frac {1}{3}}d\left (\frac {2 b^{2/3} \tan (c+d x)}{\sqrt {3}}+1\right )}{\sqrt {3}}}{6 b^{5/3}}+\frac {\arctan \left (b^{2/3} \tan (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}-2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}-\arctan \left (\sqrt {3} \left (1-\frac {2 b^{2/3} \tan (c+d x)}{\sqrt {3}}\right )\right )}{6 b^{5/3}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [3]{b}+2 \sqrt [3]{b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}}d\sqrt [3]{b \tan (c+d x)}+\arctan \left (\sqrt {3} \left (\frac {2 b^{2/3} \tan (c+d x)}{\sqrt {3}}+1\right )\right )}{6 b^{5/3}}+\frac {\arctan \left (b^{2/3} \tan (c+d x)\right )}{3 b^{5/3}}\right )}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 b \sqrt [3]{b \tan (c+d x)}}{d}-\frac {3 b^3 \left (\frac {\arctan \left (b^{2/3} \tan (c+d x)\right )}{3 b^{5/3}}+\frac {-\arctan \left (\sqrt {3} \left (1-\frac {2 b^{2/3} \tan (c+d x)}{\sqrt {3}}\right )\right )-\frac {1}{2} \sqrt {3} \log \left (-\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}+b^2 \tan ^2(c+d x)\right )}{6 b^{5/3}}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 b^{2/3} \tan (c+d x)}{\sqrt {3}}+1\right )\right )+\frac {1}{2} \sqrt {3} \log \left (\sqrt {3} b^{4/3} \tan (c+d x)+b^{2/3}+b^2 \tan ^2(c+d x)\right )}{6 b^{5/3}}\right )}{d}\) |
(-3*b^3*(ArcTan[b^(2/3)*Tan[c + d*x]]/(3*b^(5/3)) + (-ArcTan[Sqrt[3]*(1 - (2*b^(2/3)*Tan[c + d*x])/Sqrt[3])] - (Sqrt[3]*Log[b^(2/3) - Sqrt[3]*b^(4/3 )*Tan[c + d*x] + b^2*Tan[c + d*x]^2])/2)/(6*b^(5/3)) + (ArcTan[Sqrt[3]*(1 + (2*b^(2/3)*Tan[c + d*x])/Sqrt[3])] + (Sqrt[3]*Log[b^(2/3) + Sqrt[3]*b^(4 /3)*Tan[c + d*x] + b^2*Tan[c + d*x]^2])/2)/(6*b^(5/3))))/d + (3*b*(b*Tan[c + d*x])^(1/3))/d
3.1.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /b]
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[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Time = 0.17 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {3 b \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d}+\frac {b \sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{4 d}-\frac {b \left (b^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 d}-\frac {b \sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{4 d}-\frac {b \left (b^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 d}-\frac {b \left (b^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{d}\) | \(215\) |
| default | \(\frac {3 b \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d}+\frac {b \sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{4 d}-\frac {b \left (b^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 d}-\frac {b \sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{4 d}-\frac {b \left (b^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 d}-\frac {b \left (b^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{d}\) | \(215\) |
3*b*(b*tan(d*x+c))^(1/3)/d+1/4/d*b*3^(1/2)*(b^2)^(1/6)*ln((b*tan(d*x+c))^( 2/3)-3^(1/2)*(b^2)^(1/6)*(b*tan(d*x+c))^(1/3)+(b^2)^(1/3))-1/2/d*b*(b^2)^( 1/6)*arctan(2*(b*tan(d*x+c))^(1/3)/(b^2)^(1/6)-3^(1/2))-1/4/d*b*3^(1/2)*(b ^2)^(1/6)*ln((b*tan(d*x+c))^(2/3)+3^(1/2)*(b^2)^(1/6)*(b*tan(d*x+c))^(1/3) +(b^2)^(1/3))-1/2/d*b*(b^2)^(1/6)*arctan(2*(b*tan(d*x+c))^(1/3)/(b^2)^(1/6 )+3^(1/2))-1/d*b*(b^2)^(1/6)*arctan((b*tan(d*x+c))^(1/3)/(b^2)^(1/6))
Time = 0.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.28 \[ \int (b \tan (c+d x))^{4/3} \, dx=-\frac {\left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d + d\right )} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b + \frac {1}{2} \, \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d + d\right )}\right ) - \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d + d\right )} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b - \frac {1}{2} \, \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d + d\right )}\right ) + \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d - d\right )} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b + \frac {1}{2} \, \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d - d\right )}\right ) - \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d - d\right )} \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b - \frac {1}{2} \, \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} d - d\right )}\right ) + 2 \, \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} d \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b + \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} d\right ) - 2 \, \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} d \log \left (\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b - \left (-\frac {b^{8}}{d^{6}}\right )^{\frac {1}{6}} d\right ) - 12 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b}{4 \, d} \]
-1/4*((-b^8/d^6)^(1/6)*(sqrt(-3)*d + d)*log((b*tan(d*x + c))^(1/3)*b + 1/2 *(-b^8/d^6)^(1/6)*(sqrt(-3)*d + d)) - (-b^8/d^6)^(1/6)*(sqrt(-3)*d + d)*lo g((b*tan(d*x + c))^(1/3)*b - 1/2*(-b^8/d^6)^(1/6)*(sqrt(-3)*d + d)) + (-b^ 8/d^6)^(1/6)*(sqrt(-3)*d - d)*log((b*tan(d*x + c))^(1/3)*b + 1/2*(-b^8/d^6 )^(1/6)*(sqrt(-3)*d - d)) - (-b^8/d^6)^(1/6)*(sqrt(-3)*d - d)*log((b*tan(d *x + c))^(1/3)*b - 1/2*(-b^8/d^6)^(1/6)*(sqrt(-3)*d - d)) + 2*(-b^8/d^6)^( 1/6)*d*log((b*tan(d*x + c))^(1/3)*b + (-b^8/d^6)^(1/6)*d) - 2*(-b^8/d^6)^( 1/6)*d*log((b*tan(d*x + c))^(1/3)*b - (-b^8/d^6)^(1/6)*d) - 12*(b*tan(d*x + c))^(1/3)*b)/d
\[ \int (b \tan (c+d x))^{4/3} \, dx=\int \left (b \tan {\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.76 \[ \int (b \tan (c+d x))^{4/3} \, dx=-\frac {\sqrt {3} b^{\frac {7}{3}} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right ) - \sqrt {3} b^{\frac {7}{3}} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right ) + 2 \, b^{\frac {7}{3}} \arctan \left (\frac {\sqrt {3} b^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) + 2 \, b^{\frac {7}{3}} \arctan \left (-\frac {\sqrt {3} b^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) + 4 \, b^{\frac {7}{3}} \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) - 12 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{2}}{4 \, b d} \]
-1/4*(sqrt(3)*b^(7/3)*log(sqrt(3)*(b*tan(d*x + c))^(1/3)*b^(1/3) + (b*tan( d*x + c))^(2/3) + b^(2/3)) - sqrt(3)*b^(7/3)*log(-sqrt(3)*(b*tan(d*x + c)) ^(1/3)*b^(1/3) + (b*tan(d*x + c))^(2/3) + b^(2/3)) + 2*b^(7/3)*arctan((sqr t(3)*b^(1/3) + 2*(b*tan(d*x + c))^(1/3))/b^(1/3)) + 2*b^(7/3)*arctan(-(sqr t(3)*b^(1/3) - 2*(b*tan(d*x + c))^(1/3))/b^(1/3)) + 4*b^(7/3)*arctan((b*ta n(d*x + c))^(1/3)/b^(1/3)) - 12*(b*tan(d*x + c))^(1/3)*b^2)/(b*d)
Time = 0.44 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.86 \[ \int (b \tan (c+d x))^{4/3} \, dx=-\frac {1}{4} \, b {\left (\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{d} - \frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{d} + \frac {2 \, {\left | b \right |}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{d} + \frac {2 \, {\left | b \right |}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{d} + \frac {4 \, {\left | b \right |}^{\frac {1}{3}} \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{d} - \frac {12 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{d}\right )} \]
-1/4*b*(sqrt(3)*abs(b)^(1/3)*log(sqrt(3)*(b*tan(d*x + c))^(1/3)*abs(b)^(1/ 3) + (b*tan(d*x + c))^(2/3) + abs(b)^(2/3))/d - sqrt(3)*abs(b)^(1/3)*log(- sqrt(3)*(b*tan(d*x + c))^(1/3)*abs(b)^(1/3) + (b*tan(d*x + c))^(2/3) + abs (b)^(2/3))/d + 2*abs(b)^(1/3)*arctan((sqrt(3)*abs(b)^(1/3) + 2*(b*tan(d*x + c))^(1/3))/abs(b)^(1/3))/d + 2*abs(b)^(1/3)*arctan(-(sqrt(3)*abs(b)^(1/3 ) - 2*(b*tan(d*x + c))^(1/3))/abs(b)^(1/3))/d + 4*abs(b)^(1/3)*arctan((b*t an(d*x + c))^(1/3)/abs(b)^(1/3))/d - 12*(b*tan(d*x + c))^(1/3)/d)
Time = 3.34 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.02 \[ \int (b \tan (c+d x))^{4/3} \, dx=\frac {3\,b\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{d}-\frac {{\left (-1\right )}^{1/6}\,b^{4/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{5/6}\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{b^{1/3}}\right )\,1{}\mathrm {i}}{d}-\frac {{\left (-1\right )}^{1/6}\,b^{4/3}\,\ln \left ({\left (-1\right )}^{1/6}\,b^{1/3}+2\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,b^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}-\frac {{\left (-1\right )}^{1/6}\,b^{4/3}\,\ln \left (2\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}-{\left (-1\right )}^{1/6}\,b^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,b^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,d}+\frac {{\left (-1\right )}^{1/6}\,b^{4/3}\,\ln \left ({\left (-1\right )}^{1/6}\,b^{1/3}-2\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,b^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d}+\frac {{\left (-1\right )}^{1/6}\,b^{4/3}\,\ln \left ({\left (-1\right )}^{1/6}\,b^{1/3}+2\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}-{\left (-1\right )}^{2/3}\,\sqrt {3}\,b^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{d} \]
(3*b*(b*tan(c + d*x))^(1/3))/d - ((-1)^(1/6)*b^(4/3)*atan(((-1)^(5/6)*(b*t an(c + d*x))^(1/3)*1i)/b^(1/3))*1i)/d - ((-1)^(1/6)*b^(4/3)*log((-1)^(1/6) *b^(1/3) + 2*(b*tan(c + d*x))^(1/3) + (-1)^(2/3)*3^(1/2)*b^(1/3))*((3^(1/2 )*1i)/2 + 1/2))/(2*d) - ((-1)^(1/6)*b^(4/3)*log(2*(b*tan(c + d*x))^(1/3) - (-1)^(1/6)*b^(1/3) + (-1)^(2/3)*3^(1/2)*b^(1/3))*((3^(1/2)*1i)/2 - 1/2))/ (2*d) + ((-1)^(1/6)*b^(4/3)*log((-1)^(1/6)*b^(1/3) - 2*(b*tan(c + d*x))^(1 /3) + (-1)^(2/3)*3^(1/2)*b^(1/3))*((3^(1/2)*1i)/4 + 1/4))/d + ((-1)^(1/6)* b^(4/3)*log((-1)^(1/6)*b^(1/3) + 2*(b*tan(c + d*x))^(1/3) - (-1)^(2/3)*3^( 1/2)*b^(1/3))*((3^(1/2)*1i)/4 - 1/4))/d