Integrand size = 23, antiderivative size = 58 \[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\frac {2 b \sqrt {d \sec (e+f x)}}{f}+\frac {2 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{f} \] Output:
2*b*(d*sec(f*x+e))^(1/2)/f+2*a*cos(f*x+e)^(1/2)*InverseJacobiAM(1/2*f*x+1/ 2*e,2^(1/2))*(d*sec(f*x+e))^(1/2)/f
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\frac {2 \left (b+a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )\right ) \sqrt {d \sec (e+f x)}}{f} \] Input:
Integrate[Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]),x]
Output:
(2*(b + a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[d*Sec[e + f*x ]])/f
Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3967, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))dx\) |
\(\Big \downarrow \) 3967 |
\(\displaystyle a \int \sqrt {d \sec (e+f x)}dx+\frac {2 b \sqrt {d \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \sqrt {d \csc \left (e+f x+\frac {\pi }{2}\right )}dx+\frac {2 b \sqrt {d \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle a \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx+\frac {2 b \sqrt {d \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sqrt {d \sec (e+f x)}}{f}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{f}+\frac {2 b \sqrt {d \sec (e+f x)}}{f}\) |
Input:
Int[Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]),x]
Output:
(2*b*Sqrt[d*Sec[e + f*x]])/f + (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x) /2, 2]*Sqrt[d*Sec[e + f*x]])/f
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Result contains complex when optimal does not.
Time = 9.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {\left (2 i \left (1+\cos \left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, a \operatorname {EllipticF}\left (i \left (-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right ), i\right )+2 b \right ) \sqrt {d \sec \left (f x +e \right )}}{f}\) | \(83\) |
parts | \(-\frac {2 i a \left (1+\cos \left (f x +e \right )\right ) \sqrt {d \sec \left (f x +e \right )}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \operatorname {EllipticF}\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )}{f}+\frac {2 b \sqrt {d \sec \left (f x +e \right )}}{f}\) | \(95\) |
Input:
int((d*sec(f*x+e))^(1/2)*(a+b*tan(f*x+e)),x,method=_RETURNVERBOSE)
Output:
1/f*(2*I*a*(1+cos(f*x+e))*EllipticF(I*(-csc(f*x+e)+cot(f*x+e)),I)*(1/(1+co s(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)+2*b)*(d*sec(f*x+e))^(1/ 2)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28 \[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\frac {-i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, b \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{f} \] Input:
integrate((d*sec(f*x+e))^(1/2)*(a+b*tan(f*x+e)),x, algorithm="fricas")
Output:
(-I*sqrt(2)*a*sqrt(d)*weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + I*sqrt(2)*a*sqrt(d)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*si n(f*x + e)) + 2*b*sqrt(d/cos(f*x + e)))/f
\[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\int \sqrt {d \sec {\left (e + f x \right )}} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \] Input:
integrate((d*sec(f*x+e))**(1/2)*(a+b*tan(f*x+e)),x)
Output:
Integral(sqrt(d*sec(e + f*x))*(a + b*tan(e + f*x)), x)
\[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\int { \sqrt {d \sec \left (f x + e\right )} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \] Input:
integrate((d*sec(f*x+e))^(1/2)*(a+b*tan(f*x+e)),x, algorithm="maxima")
Output:
integrate(sqrt(d*sec(f*x + e))*(b*tan(f*x + e) + a), x)
\[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\int { \sqrt {d \sec \left (f x + e\right )} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \] Input:
integrate((d*sec(f*x+e))^(1/2)*(a+b*tan(f*x+e)),x, algorithm="giac")
Output:
integrate(sqrt(d*sec(f*x + e))*(b*tan(f*x + e) + a), x)
Time = 0.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\frac {2\,\left (b+a\,\sqrt {\cos \left (e+f\,x\right )}\,\mathrm {F}\left (\frac {e}{2}+\frac {f\,x}{2}\middle |2\right )\right )\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}}{f} \] Input:
int((d/cos(e + f*x))^(1/2)*(a + b*tan(e + f*x)),x)
Output:
(2*(b + a*cos(e + f*x)^(1/2)*ellipticF(e/2 + (f*x)/2, 2))*(d/cos(e + f*x)) ^(1/2))/f
\[ \int \sqrt {d \sec (e+f x)} (a+b \tan (e+f x)) \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {\sec \left (f x +e \right )}\, b +\left (\int \sqrt {\sec \left (f x +e \right )}d x \right ) a f \right )}{f} \] Input:
int((d*sec(f*x+e))^(1/2)*(a+b*tan(f*x+e)),x)
Output:
(sqrt(d)*(2*sqrt(sec(e + f*x))*b + int(sqrt(sec(e + f*x)),x)*a*f))/f