Integrand size = 26, antiderivative size = 40 \[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 \sqrt [4]{-1} a \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {d} f} \] Output:
-2*(-1)^(1/4)*a*arctanh((-1)^(3/4)*(d*tan(f*x+e))^(1/2)/d^(1/2))/d^(1/2)/f
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.25 \[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 \sqrt [4]{-1} a \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (e+f x)}\right ) \sqrt {\tan (e+f x)}}{f \sqrt {d \tan (e+f x)}} \] Input:
Integrate[(a - I*a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]],x]
Output:
(-2*(-1)^(1/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Tan[e + f*x]]]*Sqrt[Tan[e + f*x]] )/(f*Sqrt[d*Tan[e + f*x]])
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {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.
\(\displaystyle \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4016 |
\(\displaystyle \frac {2 a^2 \int \frac {1}{a d+i a \tan (e+f x) d}d\sqrt {d \tan (e+f x)}}{f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \sqrt [4]{-1} a \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {d} f}\) |
Input:
Int[(a - I*a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]],x]
Output:
(-2*(-1)^(1/4)*a*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt [d]*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[((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]
Result contains complex when optimal does not.
Time = 1.58 (sec) , antiderivative size = 274, normalized size of antiderivative = 6.85
method | result | size |
derivativedivides | \(-\frac {a \left (-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}\) | \(274\) |
default | \(-\frac {a \left (-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}\) | \(274\) |
parts | \(\frac {a \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f d}-\frac {i a \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 f \left (d^{2}\right )^{\frac {1}{4}}}\) | \(276\) |
Input:
int((a-I*a*tan(f*x+e))/(d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/f*a*(-1/4/d*(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))+1/4*I/(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)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.22 \[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {1}{2} \, \sqrt {\frac {i \, a^{2}}{d f^{2}}} \log \left (\frac {d f \sqrt {\frac {i \, a^{2}}{d f^{2}}} + {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{a}\right ) - \frac {1}{2} \, \sqrt {\frac {i \, a^{2}}{d f^{2}}} \log \left (-\frac {d f \sqrt {\frac {i \, a^{2}}{d f^{2}}} - {\left (a e^{\left (2 i \, f x + 2 i \, e\right )} + a\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{a}\right ) \] Input:
integrate((a-I*a*tan(f*x+e))/(d*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
1/2*sqrt(I*a^2/(d*f^2))*log((d*f*sqrt(I*a^2/(d*f^2)) + (a*e^(2*I*f*x + 2*I *e) + a)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1))) /a) - 1/2*sqrt(I*a^2/(d*f^2))*log(-(d*f*sqrt(I*a^2/(d*f^2)) - (a*e^(2*I*f* x + 2*I*e) + a)*sqrt((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))/a)
\[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=- i a \left (\int \frac {i}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx + \int \frac {\tan {\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx\right ) \] Input:
integrate((a-I*a*tan(f*x+e))/(d*tan(f*x+e))**(1/2),x)
Output:
-I*a*(Integral(I/sqrt(d*tan(e + f*x)), x) + Integral(tan(e + f*x)/sqrt(d*t an(e + f*x)), x))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 3.85 \[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {a {\left (-\frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\left (i + 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )}}{4 \, f} \] Input:
integrate((a-I*a*tan(f*x+e))/(d*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
1/4*a*(-(2*I - 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*t an(f*x + e)))/sqrt(d))/sqrt(d) - (2*I - 2)*sqrt(2)*arctan(-1/2*sqrt(2)*(sq rt(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)/sqrt(d) - (I + 1)*sqrt(2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d))/f
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.45 \[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {2 i \, \sqrt {2} a \arctan \left (\frac {2 \, \sqrt {d \tan \left (f x + e\right )} {\left | d \right |}}{-i \, \sqrt {2} d^{\frac {3}{2}} + \sqrt {2} \sqrt {d} {\left | d \right |}}\right )}{\sqrt {d} f {\left (-\frac {i \, d}{{\left | d \right |}} + 1\right )}} \] Input:
integrate((a-I*a*tan(f*x+e))/(d*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
-2*I*sqrt(2)*a*arctan(2*sqrt(d*tan(f*x + e))*abs(d)/(-I*sqrt(2)*d^(3/2) + sqrt(2)*sqrt(d)*abs(d)))/(sqrt(d)*f*(-I*d/abs(d) + 1))
Time = 1.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=-\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,2{}\mathrm {i}}{\sqrt {d}\,f} \] Input:
int((a - a*tan(e + f*x)*1i)/(d*tan(e + f*x))^(1/2),x)
Output:
-((-1)^(1/4)*a*atan(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*2i)/(d^(1 /2)*f)
\[ \int \frac {a-i a \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx=\frac {\sqrt {d}\, a \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x -\left (\int \sqrt {\tan \left (f x +e \right )}d x \right ) i \right )}{d} \] Input:
int((a-I*a*tan(f*x+e))/(d*tan(f*x+e))^(1/2),x)
Output:
(sqrt(d)*a*(int(sqrt(tan(e + f*x))/tan(e + f*x),x) - int(sqrt(tan(e + f*x) ),x)*i))/d