Integrand size = 14, antiderivative size = 84 \[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=-\frac {2 \arctan \left (2-\sqrt {3+4 \tan (e+f x)}\right )}{5 f}+\frac {2 \arctan \left (2+\sqrt {3+4 \tan (e+f x)}\right )}{5 f}+\frac {\text {arctanh}\left (\frac {\sqrt {3+4 \tan (e+f x)}}{2+\tan (e+f x)}\right )}{5 f} \] Output:
2/5*arctan(-2+(3+4*tan(f*x+e))^(1/2))/f+2/5*arctan(2+(3+4*tan(f*x+e))^(1/2 ))/f+1/5*arctanh((3+4*tan(f*x+e))^(1/2)/(2+tan(f*x+e)))/f
Result contains complex when optimal does not.
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\frac {\left (\frac {2}{5}-\frac {i}{5}\right ) \arctan \left (\left (\frac {1}{5}+\frac {2 i}{5}\right ) \sqrt {3+4 \tan (e+f x)}\right )}{f}+\frac {\left (\frac {1}{5}-\frac {2 i}{5}\right ) \text {arctanh}\left (\left (\frac {2}{5}+\frac {i}{5}\right ) \sqrt {3+4 \tan (e+f x)}\right )}{f} \] Input:
Integrate[1/Sqrt[3 + 4*Tan[e + f*x]],x]
Output:
((2/5 - I/5)*ArcTan[(1/5 + (2*I)/5)*Sqrt[3 + 4*Tan[e + f*x]]])/f + ((1/5 - (2*I)/5)*ArcTanh[(2/5 + I/5)*Sqrt[3 + 4*Tan[e + f*x]]])/f
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 3966, 484, 1407, 1142, 27, 1083, 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.
\(\displaystyle \int \frac {1}{\sqrt {4 \tan (e+f x)+3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {4 \tan (e+f x)+3}}dx\) |
\(\Big \downarrow \) 3966 |
\(\displaystyle \frac {4 \int \frac {1}{\sqrt {4 \tan (e+f x)+3} \left (16 \tan ^2(e+f x)+16\right )}d(4 \tan (e+f x))}{f}\) |
\(\Big \downarrow \) 484 |
\(\displaystyle \frac {8 \int \frac {1}{256 \tan ^4(e+f x)-96 \tan ^2(e+f x)+25}d\sqrt {4 \tan (e+f x)+3}}{f}\) |
\(\Big \downarrow \) 1407 |
\(\displaystyle \frac {8 \left (\frac {1}{40} \int \frac {4-4 \tan (e+f x)}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}+\frac {1}{40} \int \frac {\sqrt {4 \tan (e+f x)+3}+4}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}\right )}{f}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {8 \left (\frac {1}{40} \left (2 \int \frac {1}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}-\frac {1}{2} \int -\frac {2 (2-4 \tan (e+f x))}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}\right )+\frac {1}{40} \left (2 \int \frac {1}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}+\frac {1}{2} \int \frac {2 \left (\sqrt {4 \tan (e+f x)+3}+2\right )}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}\right )\right )}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {8 \left (\frac {1}{40} \left (2 \int \frac {1}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}+\int \frac {2-4 \tan (e+f x)}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}\right )+\frac {1}{40} \left (2 \int \frac {1}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}+\int \frac {\sqrt {4 \tan (e+f x)+3}+2}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}\right )\right )}{f}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {8 \left (\frac {1}{40} \left (\int \frac {2-4 \tan (e+f x)}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}-4 \int \frac {1}{-16 \tan ^2(e+f x)-4}d(8 \tan (e+f x)-4)\right )+\frac {1}{40} \left (\int \frac {\sqrt {4 \tan (e+f x)+3}+2}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}-4 \int \frac {1}{-16 \tan ^2(e+f x)-4}d(8 \tan (e+f x)+4)\right )\right )}{f}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {8 \left (\frac {1}{40} \left (\int \frac {2-4 \tan (e+f x)}{16 \tan ^2(e+f x)-16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}+2 \arctan (2 \tan (e+f x))\right )+\frac {1}{40} \left (\int \frac {\sqrt {4 \tan (e+f x)+3}+2}{16 \tan ^2(e+f x)+16 \tan (e+f x)+5}d\sqrt {4 \tan (e+f x)+3}+2 \arctan (2 \tan (e+f x))\right )\right )}{f}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {8 \left (\frac {1}{40} \left (2 \arctan (2 \tan (e+f x))-\frac {1}{2} \log \left (16 \tan ^2(e+f x)-16 \tan (e+f x)+5\right )\right )+\frac {1}{40} \left (2 \arctan (2 \tan (e+f x))+\frac {1}{2} \log \left (16 \tan ^2(e+f x)+16 \tan (e+f x)+5\right )\right )\right )}{f}\) |
Input:
Int[1/Sqrt[3 + 4*Tan[e + f*x]],x]
Output:
(8*((2*ArcTan[2*Tan[e + f*x]] - Log[5 - 16*Tan[e + f*x] + 16*Tan[e + f*x]^ 2]/2)/40 + (2*ArcTan[2*Tan[e + f*x]] + Log[5 + 16*Tan[e + f*x] + 16*Tan[e + f*x]^2]/2)/40))/f
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[(-(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[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* d Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (8+4 \tan \left (f x +e \right )-4 \sqrt {3+4 \tan \left (f x +e \right )}\right )}{10}+\frac {2 \arctan \left (-2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{5}+\frac {\ln \left (8+4 \tan \left (f x +e \right )+4 \sqrt {3+4 \tan \left (f x +e \right )}\right )}{10}+\frac {2 \arctan \left (2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{5}}{f}\) | \(94\) |
default | \(\frac {-\frac {\ln \left (8+4 \tan \left (f x +e \right )-4 \sqrt {3+4 \tan \left (f x +e \right )}\right )}{10}+\frac {2 \arctan \left (-2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{5}+\frac {\ln \left (8+4 \tan \left (f x +e \right )+4 \sqrt {3+4 \tan \left (f x +e \right )}\right )}{10}+\frac {2 \arctan \left (2+\sqrt {3+4 \tan \left (f x +e \right )}\right )}{5}}{f}\) | \(94\) |
Input:
int(1/(3+4*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/f*(-1/10*ln(8+4*tan(f*x+e)-4*(3+4*tan(f*x+e))^(1/2))+2/5*arctan(-2+(3+4* tan(f*x+e))^(1/2))+1/10*ln(8+4*tan(f*x+e)+4*(3+4*tan(f*x+e))^(1/2))+2/5*ar ctan(2+(3+4*tan(f*x+e))^(1/2)))
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\frac {4 \, \arctan \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} + 2\right ) + 4 \, \arctan \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} - 2\right ) + \log \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} + \tan \left (f x + e\right ) + 2\right ) - \log \left (-\sqrt {4 \, \tan \left (f x + e\right ) + 3} + \tan \left (f x + e\right ) + 2\right )}{10 \, f} \] Input:
integrate(1/(3+4*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
1/10*(4*arctan(sqrt(4*tan(f*x + e) + 3) + 2) + 4*arctan(sqrt(4*tan(f*x + e ) + 3) - 2) + log(sqrt(4*tan(f*x + e) + 3) + tan(f*x + e) + 2) - log(-sqrt (4*tan(f*x + e) + 3) + tan(f*x + e) + 2))/f
\[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\int \frac {1}{\sqrt {4 \tan {\left (e + f x \right )} + 3}}\, dx \] Input:
integrate(1/(3+4*tan(f*x+e))**(1/2),x)
Output:
Integral(1/sqrt(4*tan(e + f*x) + 3), x)
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\frac {4 \, \arctan \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} + 2\right ) + 4 \, \arctan \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} - 2\right ) + \log \left (4 \, \sqrt {4 \, \tan \left (f x + e\right ) + 3} + 4 \, \tan \left (f x + e\right ) + 8\right ) - \log \left (-4 \, \sqrt {4 \, \tan \left (f x + e\right ) + 3} + 4 \, \tan \left (f x + e\right ) + 8\right )}{10 \, f} \] Input:
integrate(1/(3+4*tan(f*x+e))^(1/2),x, algorithm="maxima")
Output:
1/10*(4*arctan(sqrt(4*tan(f*x + e) + 3) + 2) + 4*arctan(sqrt(4*tan(f*x + e ) + 3) - 2) + log(4*sqrt(4*tan(f*x + e) + 3) + 4*tan(f*x + e) + 8) - log(- 4*sqrt(4*tan(f*x + e) + 3) + 4*tan(f*x + e) + 8))/f
Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\frac {2 \, \arctan \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} + 2\right )}{5 \, f} + \frac {2 \, \arctan \left (\sqrt {4 \, \tan \left (f x + e\right ) + 3} - 2\right )}{5 \, f} + \frac {\log \left (4 \, \sqrt {4 \, \tan \left (f x + e\right ) + 3} + 4 \, \tan \left (f x + e\right ) + 8\right )}{10 \, f} - \frac {\log \left (-4 \, \sqrt {4 \, \tan \left (f x + e\right ) + 3} + 4 \, \tan \left (f x + e\right ) + 8\right )}{10 \, f} \] Input:
integrate(1/(3+4*tan(f*x+e))^(1/2),x, algorithm="giac")
Output:
2/5*arctan(sqrt(4*tan(f*x + e) + 3) + 2)/f + 2/5*arctan(sqrt(4*tan(f*x + e ) + 3) - 2)/f + 1/10*log(4*sqrt(4*tan(f*x + e) + 3) + 4*tan(f*x + e) + 8)/ f - 1/10*log(-4*sqrt(4*tan(f*x + e) + 3) + 4*tan(f*x + e) + 8)/f
Time = 2.93 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\frac {\mathrm {atan}\left (\sqrt {4\,\mathrm {tan}\left (e+f\,x\right )+3}\,\left (\frac {1}{5}-\frac {2}{5}{}\mathrm {i}\right )\right )\,\left (\frac {2}{5}+\frac {1}{5}{}\mathrm {i}\right )}{f}+\frac {\mathrm {atan}\left (\sqrt {4\,\mathrm {tan}\left (e+f\,x\right )+3}\,\left (\frac {1}{5}+\frac {2}{5}{}\mathrm {i}\right )\right )\,\left (\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )}{f} \] Input:
int(1/(4*tan(e + f*x) + 3)^(1/2),x)
Output:
(atan((4*tan(e + f*x) + 3)^(1/2)*(1/5 - 2i/5))*(2/5 + 1i/5))/f + (atan((4* tan(e + f*x) + 3)^(1/2)*(1/5 + 2i/5))*(2/5 - 1i/5))/f
\[ \int \frac {1}{\sqrt {3+4 \tan (e+f x)}} \, dx=\frac {\sqrt {4 \tan \left (f x +e \right )+3}-2 \left (\int \frac {\sqrt {4 \tan \left (f x +e \right )+3}\, \tan \left (f x +e \right )^{2}}{4 \tan \left (f x +e \right )+3}d x \right ) f}{2 f} \] Input:
int(1/(3+4*tan(f*x+e))^(1/2),x)
Output:
(sqrt(4*tan(e + f*x) + 3) - 2*int((sqrt(4*tan(e + f*x) + 3)*tan(e + f*x)** 2)/(4*tan(e + f*x) + 3),x)*f)/(2*f)