\(\int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx\) [582]

Optimal result

Integrand size = 23, antiderivative size = 58 \[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=-\frac {2 b}{f \sqrt {d \sec (e+f x)}}+\frac {2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}} \]

[Out]

-2*b/f/(d*sec(f*x+e))^(1/2)+2*a*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2
^(1/2))/f/cos(f*x+e)^(1/2)/(d*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3567, 3856, 2719} \[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\frac {2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}-\frac {2 b}{f \sqrt {d \sec (e+f x)}} \]

[In]

Int[(a + b*Tan[e + f*x])/Sqrt[d*Sec[e + f*x]],x]

[Out]

(-2*b)/(f*Sqrt[d*Sec[e + f*x]]) + (2*a*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3567

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] + Dist[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])

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b}{f \sqrt {d \sec (e+f x)}}+a \int \frac {1}{\sqrt {d \sec (e+f x)}} \, dx \\ & = -\frac {2 b}{f \sqrt {d \sec (e+f x)}}+\frac {a \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}} \\ & = -\frac {2 b}{f \sqrt {d \sec (e+f x)}}+\frac {2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\frac {-2 b \sqrt {\cos (e+f x)}+2 a E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}} \]

[In]

Integrate[(a + b*Tan[e + f*x])/Sqrt[d*Sec[e + f*x]],x]

[Out]

(-2*b*Sqrt[Cos[e + f*x]] + 2*a*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 10.44 (sec) , antiderivative size = 306, normalized size of antiderivative = 5.28

method	result	size
risch	\(-\frac {i \left (-i b +a \right ) \sqrt {2}}{f \sqrt {\frac {d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {i a \left (-\frac {2 \left ({\mathrm e}^{2 i \left (f x +e \right )} d +d \right )}{d \sqrt {{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )} d +d \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (f x +e \right )}}\, \left (-2 i E\left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i F\left (\sqrt {-i \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {d \,{\mathrm e}^{3 i \left (f x +e \right )}+d \,{\mathrm e}^{i \left (f x +e \right )}}}\right ) \sqrt {2}\, \sqrt {d \,{\mathrm e}^{i \left (f x +e \right )} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}}{f \sqrt {\frac {d \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\)	\(306\)
parts	\(\frac {2 a \left (i \cos \left (f x +e \right ) E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-i \cos \left (f x +e \right ) F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-2 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+i \sec \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+\sin \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {d \sec \left (f x +e \right )}}-\frac {2 b}{f \sqrt {d \sec \left (f x +e \right )}}\)	\(405\)
default	\(\text {Expression too large to display}\)	\(779\)

[In]

int((a+b*tan(f*x+e))/(d*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-I*(a-I*b)/f*2^(1/2)/(d*exp(I*(f*x+e))/(exp(I*(f*x+e))^2+1))^(1/2)-I*a/f*(-2*(exp(I*(f*x+e))^2*d+d)/d/(exp(I*(
f*x+e))*(exp(I*(f*x+e))^2*d+d))^(1/2)+I*(-I*(exp(I*(f*x+e))+I))^(1/2)*2^(1/2)*(I*(exp(I*(f*x+e))-I))^(1/2)*(I*
exp(I*(f*x+e)))^(1/2)/(d*exp(I*(f*x+e))^3+d*exp(I*(f*x+e)))^(1/2)*(-2*I*EllipticE((-I*(exp(I*(f*x+e))+I))^(1/2
),1/2*2^(1/2))+I*EllipticF((-I*(exp(I*(f*x+e))+I))^(1/2),1/2*2^(1/2))))*2^(1/2)/(d*exp(I*(f*x+e))/(exp(I*(f*x+
e))^2+1))^(1/2)*(d*exp(I*(f*x+e))*(exp(I*(f*x+e))^2+1))^(1/2)/(exp(I*(f*x+e))^2+1)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.53 \[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\frac {i \, \sqrt {2} a \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - i \, \sqrt {2} a \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, b \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{d f} \]

[In]

integrate((a+b*tan(f*x+e))/(d*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

(I*sqrt(2)*a*sqrt(d)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - I*sqr
t(2)*a*sqrt(d)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - 2*b*sqrt(d/
cos(f*x + e))*cos(f*x + e))/(d*f)

Sympy [F]

\[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\int \frac {a + b \tan {\left (e + f x \right )}}{\sqrt {d \sec {\left (e + f x \right )}}}\, dx \]

[In]

integrate((a+b*tan(f*x+e))/(d*sec(f*x+e))**(1/2),x)

[Out]

Integral((a + b*tan(e + f*x))/sqrt(d*sec(e + f*x)), x)

Maxima [F]

\[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\sqrt {d \sec \left (f x + e\right )}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))/(d*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)/sqrt(d*sec(f*x + e)), x)

Giac [F]

\[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\sqrt {d \sec \left (f x + e\right )}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))/(d*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e) + a)/sqrt(d*sec(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \tan (e+f x)}{\sqrt {d \sec (e+f x)}} \, dx=\int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int((a + b*tan(e + f*x))/(d/cos(e + f*x))^(1/2),x)

[Out]

int((a + b*tan(e + f*x))/(d/cos(e + f*x))^(1/2), x)