Integrand size = 21, antiderivative size = 221 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {9 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{4 f}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f} \]
9/4*arctanh((1+tan(f*x+e))^(1/2))/f-1/2*arctan(1/2*(4-3*2^(1/2)+(2-2^(1/2) )*tan(f*x+e))/(-7+5*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))*(-2+2*2^(1/2))^(1 /2)/f-1/2*arctanh(1/2*(4+3*2^(1/2)+(2+2^(1/2))*tan(f*x+e))/(7+5*2^(1/2))^( 1/2)/(1+tan(f*x+e))^(1/2))*(2+2*2^(1/2))^(1/2)/f-1/4*cot(f*x+e)*(1+tan(f*x +e))^(1/2)/f-1/2*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2)/f
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.56 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {-9 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+4 \sqrt {1-i} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+4 \sqrt {1+i} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+\cot (e+f x) \sqrt {1+\tan (e+f x)}+2 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \]
-1/4*(-9*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 4*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 4*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/ Sqrt[1 + I]] + Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 2*Cot[e + f*x]^2*Sqrt [1 + Tan[e + f*x]])/f
Time = 1.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 4051, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4019, 25, 3042, 4018, 216, 220, 4117, 73, 220}
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.
\(\displaystyle \int \sqrt {\tan (e+f x)+1} \cot ^3(e+f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\tan (e+f x)+1}}{\tan (e+f x)^3}dx\) |
\(\Big \downarrow \) 4051 |
\(\displaystyle -\frac {1}{2} \int -\frac {\cot ^2(e+f x) \left (-3 \tan ^2(e+f x)-4 \tan (e+f x)+1\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {\cot ^2(e+f x) \left (-3 \tan ^2(e+f x)-4 \tan (e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \int \frac {-3 \tan (e+f x)^2-4 \tan (e+f x)+1}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {1}{4} \left (-\int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+8 \tan (e+f x)+9\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+8 \tan (e+f x)+9\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {\tan (e+f x)^2+8 \tan (e+f x)+9}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\int \frac {8 (1-\tan (e+f x))}{\sqrt {\tan (e+f x)+1}}dx-9 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \int \frac {1-\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-9 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \int \frac {1-\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 4019 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (\frac {\int \frac {\left (2-\sqrt {2}\right ) \tan (e+f x)+\sqrt {2}}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (\frac {\int \frac {\left (2-\sqrt {2}\right ) \tan (e+f x)+\sqrt {2}}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \left (\frac {\int \frac {\left (2-\sqrt {2}\right ) \tan (e+f x)+\sqrt {2}}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 4018 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (-\frac {\sqrt {2} \left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4\right )^2}{\tan (e+f x)+1}-4 \left (7-5 \sqrt {2}\right )}d\left (-\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{\sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4\right )^2}{\tan (e+f x)+1}-4 \left (7+5 \sqrt {2}\right )}d\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{\sqrt {\tan (e+f x)+1}}}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}-\frac {\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4\right )^2}{\tan (e+f x)+1}-4 \left (7+5 \sqrt {2}\right )}d\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{\sqrt {\tan (e+f x)+1}}}{f}\right )-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {9 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {18 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {18 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\) |
-1/2*(Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/f + (((18*ArcTanh[Sqrt[1 + Ta n[e + f*x]]])/f - 8*(((3 - 2*Sqrt[2])*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2] )*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/(Sqrt[2* (-7 + 5*Sqrt[2])]*f) + ((3 + 2*Sqrt[2])*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt [2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/(Sqrt[ 2*(7 + 5*Sqrt[2])]*f)))/2 - (Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f)/4
3.4.82.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[-2*(d^2/f) Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0 ] && NeQ[c^2 + d^2, 0] && EqQ[2*a*c*d - b*(c^2 - d^2), 0]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( f_.)*(x_)]], x_Symbol] :> With[{q = Rt[a^2 + b^2, 2]}, Simp[1/(2*q) Int[( a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x], x] - Simp[1/(2*q) Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f *x])/Sqrt[a + b*Tan[e + f*x]], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && N eQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2*a*c*d - b*(c^2 - d^2), 0] && NiceSqrtQ[a^2 + b^2]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2 )) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c *(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && Int egerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(3818\) vs. \(2(173)=346\).
Time = 163.28 (sec) , antiderivative size = 3819, normalized size of antiderivative = 17.28
-1/8/f*(20*2^(1/2)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(f*x+e)^2*((c os(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f* x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4 *((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*s in(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1) *(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f* x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))+8*2^(1/2)*(-2+2*2^ (1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+ e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*s in(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(c os(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f* x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1 /2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2 ))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))-18*2^(1/2)*(1+2^(1/2))^(1/2)*ln(2*co t(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot (f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^ 2)^(1/2))*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*cos(f*x+e)^2+8*2^(1/2)*(1+2^(1/2 ))^(1/2)*cos(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2 )^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+4*2^(1/2)*(1+2^(1/2))^(1/2)*cos(f* x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(...
Time = 0.26 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.53 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {4 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} + 4 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{2} + 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{2} + 2 \, {\left (\tan \left (f x + e\right ) + 2\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{8 \, f \tan \left (f x + e\right )^{2}} \]
-1/8*(4*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 4*f*sqrt((f^2*sqrt(-1/f ^4) + 1)/f^2)*log(-f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 + 4*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(f*sqrt(-( f^2*sqrt(-1/f^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 4*f* sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(-f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 9*log(sqrt(tan(f*x + e) + 1) + 1)*tan(f*x + e)^2 + 9*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^2 + 2* (tan(f*x + e) + 2)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^2)
\[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\int \sqrt {\tan {\left (e + f x \right )} + 1} \cot ^{3}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\int { \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )^{3} \,d x } \]
\[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\int { \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )^{3} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.67 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{4\,f}-\frac {\frac {\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{4}+\frac {{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{4}}{f-2\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]
atan(f*((1/4 + 1i/4)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*(1 + 1i))*((1/4 + 1i/4)/f^2)^(1/2)*2i - ((tan(e + f*x) + 1)^(1/2)/4 + (tan(e + f*x) + 1)^(3 /2)/4)/(f - 2*f*(tan(e + f*x) + 1) + f*(tan(e + f*x) + 1)^2) - atan(f*((1/ 4 - 1i/4)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*(1 - 1i))*((1/4 - 1i/4)/f^2) ^(1/2)*2i - (atan((tan(e + f*x) + 1)^(1/2)*1i)*9i)/(4*f)