Integrand size = 23, antiderivative size = 373 \[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x+\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}-\frac {9 c (c+d \tan (e+f x))^{4/3}}{28 d^2 f}+\frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f} \]
-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-1/4*(c-I*d)^(1/3)*ln(cos(f*x+ e))/f-1/4*(c+I*d)^(1/3)*ln(cos(f*x+e))/f-3/4*(c-I*d)^(1/3)*ln((c-I*d)^(1/3 )-(c+d*tan(f*x+e))^(1/3))/f-3/4*(c+I*d)^(1/3)*ln((c+I*d)^(1/3)-(c+d*tan(f* x+e))^(1/3))/f+1/2*(c-I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c -I*d)^(1/3))*3^(1/2))*3^(1/2)/f+1/2*(c+I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan (f*x+e))^(1/3)/(c+I*d)^(1/3))*3^(1/2))*3^(1/2)/f-3*(c+d*tan(f*x+e))^(1/3)/ f-9/28*c*(c+d*tan(f*x+e))^(4/3)/d^2/f+3/7*tan(f*x+e)*(c+d*tan(f*x+e))^(4/3 )/d/f
Time = 1.20 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.18 \[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {14 \sqrt {3} \sqrt [3]{c-i d} d^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )+14 \sqrt {3} \sqrt [3]{c+i d} d^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )-14 \sqrt [3]{c-i d} d^2 \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )-14 \sqrt [3]{c+i d} d^2 \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+7 \sqrt [3]{c-i d} d^2 \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+7 \sqrt [3]{c+i d} d^2 \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )-9 c^2 \sqrt [3]{c+d \tan (e+f x)}-84 d^2 \sqrt [3]{c+d \tan (e+f x)}+3 c d \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}+12 d^2 \tan ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)}}{28 d^2 f} \]
(14*Sqrt[3]*(c - I*d)^(1/3)*d^2*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3)) /(c - I*d)^(1/3))/Sqrt[3]] + 14*Sqrt[3]*(c + I*d)^(1/3)*d^2*ArcTan[(1 + (2 *(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] - 14*(c - I*d)^(1/3 )*d^2*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 14*(c + I*d)^(1/ 3)*d^2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + 7*(c - I*d)^(1/ 3)*d^2*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] + 7*(c + I*d)^(1/3)*d^2*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] - 9*c^2*(c + d*Tan[e + f*x])^(1/3) - 84*d^2*(c + d*Tan[e + f*x])^(1/3) + 3* c*d*Tan[e + f*x]*(c + d*Tan[e + f*x])^(1/3) + 12*d^2*Tan[e + f*x]^2*(c + d *Tan[e + f*x])^(1/3))/(28*d^2*f)
Time = 1.05 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.86, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 4049, 27, 3042, 4113, 27, 3042, 4011, 3042, 4022, 3042, 4020, 25, 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.
\(\displaystyle \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (e+f x)^3 \sqrt [3]{c+d \tan (e+f x)}dx\) |
\(\Big \downarrow \) 4049 |
\(\displaystyle \frac {3 \int -\frac {1}{3} \sqrt [3]{c+d \tan (e+f x)} \left (3 c \tan ^2(e+f x)+7 d \tan (e+f x)+3 c\right )dx}{7 d}+\frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\int \sqrt [3]{c+d \tan (e+f x)} \left (3 c \tan ^2(e+f x)+7 d \tan (e+f x)+3 c\right )dx}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\int \sqrt [3]{c+d \tan (e+f x)} \left (3 c \tan (e+f x)^2+7 d \tan (e+f x)+3 c\right )dx}{7 d}\) |
\(\Big \downarrow \) 4113 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\int 7 d \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}dx+\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {7 d \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}dx+\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {7 d \int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}dx+\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}}{7 d}\) |
\(\Big \downarrow \) 4011 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {7 d \left (\int \frac {c \tan (e+f x)-d}{(c+d \tan (e+f x))^{2/3}}dx+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )+\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {7 d \left (\int \frac {c \tan (e+f x)-d}{(c+d \tan (e+f x))^{2/3}}dx+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )+\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}}{7 d}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (\frac {1}{2} (-d+i c) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}}dx-\frac {1}{2} (d+i c) \int \frac {i \tan (e+f x)+1}{(c+d \tan (e+f x))^{2/3}}dx+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (\frac {1}{2} (-d+i c) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}}dx-\frac {1}{2} (d+i c) \int \frac {i \tan (e+f x)+1}{(c+d \tan (e+f x))^{2/3}}dx+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (-\frac {i (d+i c) \int -\frac {1}{(1-i \tan (e+f x)) (c+d \tan (e+f x))^{2/3}}d(i \tan (e+f x))}{2 f}-\frac {i (-d+i c) \int -\frac {1}{(i \tan (e+f x)+1) (c+d \tan (e+f x))^{2/3}}d(-i \tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (\frac {i (d+i c) \int \frac {1}{(1-i \tan (e+f x)) (c+d \tan (e+f x))^{2/3}}d(i \tan (e+f x))}{2 f}+\frac {i (-d+i c) \int \frac {1}{(i \tan (e+f x)+1) (c+d \tan (e+f x))^{2/3}}d(-i \tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (-\frac {i (d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c-i d}}-\frac {3 \int \frac {1}{\sqrt [3]{c-i d}-i \tan (e+f x)}d\sqrt [3]{c+d \tan (e+f x)}}{2 (c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}\right )}{2 f}-\frac {i (-d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c+i d}}-\frac {3 \int \frac {1}{i \tan (e+f x)+\sqrt [3]{c+i d}}d\sqrt [3]{c+d \tan (e+f x)}}{2 (c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}\right )}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (-\frac {i (d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c-i d}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}-\frac {i (-d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c+i d}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (-\frac {i (d+i c) \left (\frac {3 \int \frac {1}{\tan ^2(e+f x)-3}d\left (\frac {2 i \tan (e+f x)}{\sqrt [3]{c-i d}}+1\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}-\frac {i (-d+i c) \left (\frac {3 \int \frac {1}{\tan ^2(e+f x)-3}d\left (1-\frac {2 i \tan (e+f x)}{\sqrt [3]{c+i d}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {3 \tan (e+f x) (c+d \tan (e+f x))^{4/3}}{7 d f}-\frac {\frac {9 c (c+d \tan (e+f x))^{4/3}}{4 d f}+7 d \left (-\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}-\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}+\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\right )}{7 d}\) |
(3*Tan[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(7*d*f) - ((9*c*(c + d*Tan[e + f*x])^(4/3))/(4*d*f) + 7*d*(((-1/2*I)*(I*c + d)*(((-I)*Sqrt[3]*ArcTanh[Ta n[e + f*x]/Sqrt[3]])/(c - I*d)^(2/3) - Log[1 - I*Tan[e + f*x]]/(2*(c - I*d )^(2/3)) + (3*Log[(c - I*d)^(1/3) - I*Tan[e + f*x]])/(2*(c - I*d)^(2/3)))) /f - ((I/2)*(I*c - d)*((I*Sqrt[3]*ArcTanh[Tan[e + f*x]/Sqrt[3]])/(c + I*d) ^(2/3) - Log[1 + I*Tan[e + f*x]]/(2*(c + I*d)^(2/3)) + (3*Log[(c + I*d)^(1 /3) + I*Tan[e + f*x]])/(2*(c + I*d)^(2/3))))/f + (3*(c + d*Tan[e + f*x])^( 1/3))/f))/(7*d)
3.7.82.3.1 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[(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_))^(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[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.65 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.32
method | result | size |
derivativedivides | \(\frac {\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {7}{3}}}{7 d^{2}}-\frac {3 c \left (c +d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}{4 d^{2}}-3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{3} c +c^{2}+d^{2}\right ) \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2}}{f}\) | \(120\) |
default | \(\frac {\frac {3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {7}{3}}}{7 d^{2}}-\frac {3 c \left (c +d \tan \left (f x +e \right )\right )^{\frac {4}{3}}}{4 d^{2}}-3 \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 c \,\textit {\_Z}^{3}+c^{2}+d^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{3} c +c^{2}+d^{2}\right ) \ln \left (\left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} c}\right )}{2}}{f}\) | \(120\) |
1/f*(3/7/d^2*(c+d*tan(f*x+e))^(7/3)-3/4/d^2*c*(c+d*tan(f*x+e))^(4/3)-3*(c+ d*tan(f*x+e))^(1/3)+1/2*sum((-_R^3*c+c^2+d^2)/(_R^5-_R^2*c)*ln((c+d*tan(f* x+e))^(1/3)-_R),_R=RootOf(_Z^6-2*_Z^3*c+c^2+d^2)))
Time = 0.29 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.46 \[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {14 \, d^{2} f \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (f \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + 14 \, d^{2} f \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (f \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 7 \, {\left (\sqrt {-3} d^{2} f + d^{2} f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} f + f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + 7 \, {\left (\sqrt {-3} d^{2} f - d^{2} f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} f - f\right )} \left (-\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} + c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) - 7 \, {\left (\sqrt {-3} d^{2} f + d^{2} f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} f + f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + 7 \, {\left (\sqrt {-3} d^{2} f - d^{2} f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} f - f\right )} \left (\frac {f^{3} \sqrt {-\frac {d^{2}}{f^{6}}} - c}{f^{3}}\right )^{\frac {1}{3}} + {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, d^{2} \tan \left (f x + e\right )^{2} + c d \tan \left (f x + e\right ) - 3 \, c^{2} - 28 \, d^{2}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}}}{28 \, d^{2} f} \]
1/28*(14*d^2*f*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(f*(-(f^3*sqrt(-d^ 2/f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + 14*d^2*f*((f^3*sqrt (-d^2/f^6) - c)/f^3)^(1/3)*log(f*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3) + (d *tan(f*x + e) + c)^(1/3)) - 7*(sqrt(-3)*d^2*f + d^2*f)*(-(f^3*sqrt(-d^2/f^ 6) + c)/f^3)^(1/3)*log(-1/2*(sqrt(-3)*f + f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^ 3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + 7*(sqrt(-3)*d^2*f - d^2*f)*(-(f^3 *sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(1/2*(sqrt(-3)*f - f)*(-(f^3*sqrt(-d^2/ f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) - 7*(sqrt(-3)*d^2*f + d ^2*f)*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(-1/2*(sqrt(-3)*f + f)*((f^3 *sqrt(-d^2/f^6) - c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + 7*(sqrt(-3 )*d^2*f - d^2*f)*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(1/2*(sqrt(-3)*f - f)*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + 3*(4*d^2*tan(f*x + e)^2 + c*d*tan(f*x + e) - 3*c^2 - 28*d^2)*(d*tan(f*x + e) + c)^(1/3))/(d^2*f)
\[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\, dx \]
\[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \tan \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Timed out} \]
Time = 13.23 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.39 \[ \int \tan ^3(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \]
log((c + d*tan(e + f*x))^(1/3) + f*(-(c - d*1i)/f^3)^(1/3))*(-(c - d*1i)/( 8*f^3))^(1/3) + log((c + d*tan(e + f*x))^(1/3) + f*(-(c + d*1i)/f^3)^(1/3) )*(-(c + d*1i)/(8*f^3))^(1/3) + ((3*c^2)/(d^2*f) - (3*(c^2 + d^2))/(d^2*f) )*(c + d*tan(e + f*x))^(1/3) + (3*(c + d*tan(e + f*x))^(7/3))/(7*d^2*f) + log((((3^(1/2)*1i)/2 - 1/2)*((972*(d^8 - c^4*d^4))/f^3 + (((3^(1/2)*1i)/2 + 1/2)*((3888*c*d^4*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f + 3888*c*d^4 *((3^(1/2)*1i)/2 - 1/2)*(-(c - d*1i)/f^3)^(1/3)*(c^2 + d^2))*(-(c - d*1i)/ f^3)^(2/3))/4)*(-(c - d*1i)/f^3)^(1/3))/2 + (486*(d^8 - c^4*d^4)*(c + d*ta n(e + f*x))^(1/3))/f^4)*((3^(1/2)*1i)/2 - 1/2)*(-(c - d*1i)/(8*f^3))^(1/3) + log((((3^(1/2)*1i)/2 - 1/2)*((972*(d^8 - c^4*d^4))/f^3 + (((3^(1/2)*1i) /2 + 1/2)*((3888*c*d^4*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f + 3888*c* d^4*((3^(1/2)*1i)/2 - 1/2)*(-(c + d*1i)/f^3)^(1/3)*(c^2 + d^2))*(-(c + d*1 i)/f^3)^(2/3))/4)*(-(c + d*1i)/f^3)^(1/3))/2 + (486*(d^8 - c^4*d^4)*(c + d *tan(e + f*x))^(1/3))/f^4)*((3^(1/2)*1i)/2 - 1/2)*(-(c + d*1i)/(8*f^3))^(1 /3) - log((486*(d^8 - c^4*d^4)*(c + d*tan(e + f*x))^(1/3))/f^4 - (((3^(1/2 )*1i)/2 + 1/2)*((972*(d^8 - c^4*d^4))/f^3 - (((3^(1/2)*1i)/2 - 1/2)*((3888 *c*d^4*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/3))/f - 3888*c*d^4*((3^(1/2)*1i )/2 + 1/2)*(-(c - d*1i)/f^3)^(1/3)*(c^2 + d^2))*(-(c - d*1i)/f^3)^(2/3))/4 )*(-(c - d*1i)/f^3)^(1/3))/2)*((3^(1/2)*1i)/2 + 1/2)*(-(c - d*1i)/(8*f^3)) ^(1/3) - log((486*(d^8 - c^4*d^4)*(c + d*tan(e + f*x))^(1/3))/f^4 - (((...