Integrand size = 26, antiderivative size = 82 \[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=\frac {2 \sqrt [4]{-1} a d^{3/2} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {2 a d \sqrt {d \tan (e+f x)}}{f}-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f} \] Output:
2*(-1)^(1/4)*a*d^(3/2)*arctanh((-1)^(3/4)*(d*tan(f*x+e))^(1/2)/d^(1/2))/f+ 2*a*d*(d*tan(f*x+e))^(1/2)/f-2/3*I*a*(d*tan(f*x+e))^(3/2)/f
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=\frac {2 a \left (3 \sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (e+f x)}\right )+(3-i \tan (e+f x)) \sqrt {\tan (e+f x)}\right ) (d \tan (e+f x))^{3/2}}{3 f \tan ^{\frac {3}{2}}(e+f x)} \] Input:
Integrate[(d*Tan[e + f*x])^(3/2)*(a - I*a*Tan[e + f*x]),x]
Output:
(2*a*(3*(-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]] + (3 - I*Tan[e + f*x])*Sqrt[Tan[e + f*x]])*(d*Tan[e + f*x])^(3/2))/(3*f*Tan[e + f*x]^(3/2) )
Time = 0.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3042, 4011, 3042, 4011, 3042, 4016, 221}
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.
| \(\displaystyle \int (a-i a \tan (e+f x)) (d \tan (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a-i a \tan (e+f x)) (d \tan (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {d \tan (e+f x)} (i a d+a \tan (e+f x) d)dx-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {d \tan (e+f x)} (i a d+a \tan (e+f x) d)dx-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {i a d^2 \tan (e+f x)-a d^2}{\sqrt {d \tan (e+f x)}}dx-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i a d^2 \tan (e+f x)-a d^2}{\sqrt {d \tan (e+f x)}}dx-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f}+\frac {2 a d \sqrt {d \tan (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4016 |
| \(\displaystyle \frac {2 a^2 d^4 \int \frac {1}{-a d^3-i a \tan (e+f x) d^3}d\sqrt {d \tan (e+f x)}}{f}+\frac {2 a d \sqrt {d \tan (e+f x)}}{f}-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt [4]{-1} a d^{3/2} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {2 a d \sqrt {d \tan (e+f x)}}{f}-\frac {2 i a (d \tan (e+f x))^{3/2}}{3 f}\) |
Input:
Int[(d*Tan[e + f*x])^(3/2)*(a - I*a*Tan[e + f*x]),x]
Output:
(2*(-1)^(1/4)*a*d^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] )/f + (2*a*d*Sqrt[d*Tan[e + f*x]])/f - (((2*I)/3)*a*(d*Tan[e + f*x])^(3/2) )/f
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2*(c^2/f) Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b *Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (65 ) = 130\).
Time = 1.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.72
| method | result | size |
| parts | \(\frac {2 a d \left (\sqrt {d \tan \left (f x +e \right )}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{f}-\frac {i a \left (\frac {2 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}\) | \(305\) |
| derivativedivides | \(-\frac {a \left (\frac {2 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 d \sqrt {d \tan \left (f x +e \right )}-2 d^{2} \left (-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f}\) | \(306\) |
| default | \(-\frac {a \left (\frac {2 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 d \sqrt {d \tan \left (f x +e \right )}-2 d^{2} \left (-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f}\) | \(306\) |
Input:
int((d*tan(f*x+e))^(3/2)*(a-I*a*tan(f*x+e)),x,method=_RETURNVERBOSE)
Output:
2*a/f*d*((d*tan(f*x+e))^(1/2)-1/8*(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d ^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1 /4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4 )*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/ 2)+1)))-I*a/f*(2/3*(d*tan(f*x+e))^(3/2)-1/4*d^2/(d^2)^(1/4)*2^(1/2)*(ln((d *tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f *x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1 /2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*t an(f*x+e))^(1/2)+1)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (64) = 128\).
Time = 0.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.49 \[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=\frac {3 \, \sqrt {\frac {4 i \, a^{2} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-2 i \, a d^{2} + \sqrt {\frac {4 i \, a^{2} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{f}\right ) - 3 \, \sqrt {\frac {4 i \, a^{2} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-2 i \, a d^{2} - \sqrt {\frac {4 i \, a^{2} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{f}\right ) + 16 \, {\left (a d e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, a d\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \] Input:
integrate((d*tan(f*x+e))^(3/2)*(a-I*a*tan(f*x+e)),x, algorithm="fricas")
Output:
1/12*(3*sqrt(4*I*a^2*d^3/f^2)*(f*e^(2*I*f*x + 2*I*e) + f)*log((-2*I*a*d^2 + sqrt(4*I*a^2*d^3/f^2)*(f*e^(2*I*f*x + 2*I*e) + f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/f) - 3*sq rt(4*I*a^2*d^3/f^2)*(f*e^(2*I*f*x + 2*I*e) + f)*log((-2*I*a*d^2 - sqrt(4*I *a^2*d^3/f^2)*(f*e^(2*I*f*x + 2*I*e) + f)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/f) + 16*(a*d*e^(2*I *f*x + 2*I*e) + 2*a*d)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))/(f*e^(2*I*f*x + 2*I*e) + f)
\[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=- i a \left (\int i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\, dx\right ) \] Input:
integrate((d*tan(f*x+e))**(3/2)*(a-I*a*tan(f*x+e)),x)
Output:
-I*a*(Integral(I*(d*tan(e + f*x))**(3/2), x) + Integral((d*tan(e + f*x))** (3/2)*tan(e + f*x), x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (64) = 128\).
Time = 0.12 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.35 \[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=-\frac {3 \, a d^{3} {\left (-\frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} + 8 i \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a d - 24 \, \sqrt {d \tan \left (f x + e\right )} a d^{2}}{12 \, d f} \] Input:
integrate((d*tan(f*x+e))^(3/2)*(a-I*a*tan(f*x+e)),x, algorithm="maxima")
Output:
-1/12*(3*a*d^3*(-(2*I - 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2 *sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) - (2*I - 2)*sqrt(2)*arctan(-1/2*sq rt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) + (I + 1 )*sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/s qrt(d) - (I + 1)*sqrt(2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e)) *sqrt(d) + d)/sqrt(d)) + 8*I*(d*tan(f*x + e))^(3/2)*a*d - 24*sqrt(d*tan(f* x + e))*a*d^2)/(d*f)
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.21 \[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=-\frac {2 i \, {\left (\sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right ) - \frac {3 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{-i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{-\frac {i \, d}{{\left | d \right |}} + 1} + 3 i \, \sqrt {d \tan \left (f x + e\right )} d^{2}\right )} a}{3 \, d f} \] Input:
integrate((d*tan(f*x+e))^(3/2)*(a-I*a*tan(f*x+e)),x, algorithm="giac")
Output:
-2/3*I*(sqrt(d*tan(f*x + e))*d^2*tan(f*x + e) - 3*sqrt(2)*d^(5/2)*arctan(2 *sqrt(d*tan(f*x + e))*abs(d)/(-I*sqrt(2)*d^(3/2) + sqrt(2)*sqrt(d)*abs(d)) )/(-I*d/abs(d) + 1) + 3*I*sqrt(d*tan(f*x + e))*d^2)*a/(d*f)
Time = 1.42 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=\frac {2\,a\,d\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}-\frac {a\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,2{}\mathrm {i}}{3\,f}+\frac {{\left (-1\right )}^{1/4}\,a\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,2{}\mathrm {i}}{f} \] Input:
int((d*tan(e + f*x))^(3/2)*(a - a*tan(e + f*x)*1i),x)
Output:
(2*a*d*(d*tan(e + f*x))^(1/2))/f - (a*(d*tan(e + f*x))^(3/2)*2i)/(3*f) + ( (-1)^(1/4)*a*d^(3/2)*atan(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*2i) /f
\[ \int (d \tan (e+f x))^{3/2} (a-i a \tan (e+f x)) \, dx=\frac {\sqrt {d}\, a d \left (2 \sqrt {\tan \left (f x +e \right )}-\left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right ) f -\left (\int \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}d x \right ) f i \right )}{f} \] Input:
int((d*tan(f*x+e))^(3/2)*(a-I*a*tan(f*x+e)),x)
Output:
(sqrt(d)*a*d*(2*sqrt(tan(e + f*x)) - int(sqrt(tan(e + f*x))/tan(e + f*x),x )*f - int(sqrt(tan(e + f*x))*tan(e + f*x)**2,x)*f*i))/f