\(\int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx\) [238]

Optimal result

Integrand size = 33, antiderivative size = 61 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {d} \sqrt {c+d} f} \]

[Out]

2*arctan(a^(1/2)*d^(1/2)*tan(f*x+e)/(c+d)^(1/2)/(a+a*sec(f*x+e))^(1/2))*a^(1/2)/f/d^(1/2)/(c+d)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4052, 211} \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {d} f \sqrt {c+d}} \]

[In]

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

[Out]

(2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[d]*Sqrt[c + d]
*f)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 4052

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
), x_Symbol] :> Dist[-2*(b/f), Subst[Int[1/(b*c + a*d + d*x^2), x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]
])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(2 a) \text {Subst}\left (\int \frac {1}{a c+a d+d x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{f} \\ & = \frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {d} \sqrt {c+d} f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d} \sqrt {\cos (e+f x)}}\right ) \sqrt {\cos (e+f x)} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{\sqrt {d} \sqrt {c+d} f} \]

[In]

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

[Out]

(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]])]*Sqrt[Cos[e + f*x]]*Sec[(e
+ f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(Sqrt[d]*Sqrt[c + d]*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(423\) vs. \(2(47)=94\).

Time = 19.38 (sec) , antiderivative size = 424, normalized size of antiderivative = 6.95

method	result	size
default	\(\frac {\sqrt {2}\, \left (\ln \left (-\frac {2 \left (\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, c -\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-c +d \right )}{-c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )-\ln \left (\frac {2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, c -2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d -2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-2 c +2 d}{c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}}{2 f \sqrt {\frac {d}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}}\)	\(424\)

[In]

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

[Out]

1/2/f*2^(1/2)/(d/(c-d))^(1/2)/((c+d)*(c-d))^(1/2)*(ln(-2*(((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*2^(1/2)*(d/(
c-d))^(1/2)*c-2^(1/2)*(d/(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*d+((c+d)*(c-d))^(1/2)*(-cot(f*x+
e)+csc(f*x+e))-c+d)/(-c*(-cot(f*x+e)+csc(f*x+e))+(-cot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d))^(1/2)))-ln(2*(((1-co
s(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*2^(1/2)*(d/(c-d))^(1/2)*c-2^(1/2)*(d/(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+
e)^2-1)^(1/2)*d-((c+d)*(c-d))^(1/2)*(-cot(f*x+e)+csc(f*x+e))-c+d)/(c*(-cot(f*x+e)+csc(f*x+e))-(-cot(f*x+e)+csc
(f*x+e))*d+((c+d)*(c-d))^(1/2))))*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2
-1))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (47) = 94\).

Time = 0.44 (sec) , antiderivative size = 343, normalized size of antiderivative = 5.62 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\left [\frac {\sqrt {-\frac {a}{c d + d^{2}}} \log \left (-\frac {{\left (a c^{2} + 8 \, a c d + 8 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} + a d^{2} + {\left (a c^{2} + 2 \, a c d\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - {\left (6 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )}{c^{2} \cos \left (f x + e\right )^{3} + {\left (c^{2} + 2 \, c d\right )} \cos \left (f x + e\right )^{2} + d^{2} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )}\right )}{2 \, f}, \frac {\sqrt {\frac {a}{c d + d^{2}}} \arctan \left (\frac {2 \, {\left (c d + d^{2}\right )} \sqrt {\frac {a}{c d + d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{{\left (a c + 2 \, a d\right )} \cos \left (f x + e\right )^{2} - a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}\right )}{f}\right ] \]

[In]

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

[Out]

[1/2*sqrt(-a/(c*d + d^2))*log(-((a*c^2 + 8*a*c*d + 8*a*d^2)*cos(f*x + e)^3 + a*d^2 + (a*c^2 + 2*a*c*d)*cos(f*x
+ e)^2 - 4*((c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e)^2 - (c*d^2 + d^3)*cos(f*x + e))*sqrt(-a/(c*d + d^2))*sqrt(
(a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) - (6*a*c*d + 7*a*d^2)*cos(f*x + e))/(c^2*cos(f*x + e)^3 + (c^2
+ 2*c*d)*cos(f*x + e)^2 + d^2 + (2*c*d + d^2)*cos(f*x + e)))/f, sqrt(a/(c*d + d^2))*arctan(2*(c*d + d^2)*sqrt
(a/(c*d + d^2))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e)/((a*c + 2*a*d)*cos(f*x + e)^
2 - a*d + (a*c + a*d)*cos(f*x + e)))/f]

Sympy [F]

\[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sec {\left (e + f x \right )}}{c + d \sec {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a*(sec(e + f*x) + 1))*sec(e + f*x)/(c + d*sec(e + f*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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