\(\int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c-d \sec (e+f x)} \, dx\) [184]

Optimal result

Integrand size = 34, antiderivative size = 65 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c-d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c-d} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {c-d} \sqrt {d} f} \]

[Out]

2*arctanh(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/(c-d)^(1/2)/d^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {4052, 214} \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c-d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c-d} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {d} f \sqrt {c-d}} \]

[In]

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

[Out]

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

Rule 214

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]

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} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c-d} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {c-d} \sqrt {d} f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.51 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c-d \sec (e+f x)} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c-d} \sqrt {\cos (e+f x)}}\right ) \sqrt {\cos (e+f x)} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{\sqrt {c-d} \sqrt {d} f} \]

[In]

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

[Out]

(Sqrt[2]*ArcTanh[(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[c - d]*Sqrt[d]*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(401\) vs. \(2(51)=102\).

Time = 18.88 (sec) , antiderivative size = 402, normalized size of antiderivative = 6.18

method	result	size
default	\(\frac {\left (\ln \left (-\frac {2 \left (\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {-\frac {2 d}{c +d}}\, c +\sqrt {-\frac {2 d}{c +d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-c -d \right )}{-c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )-\ln \left (-\frac {2 \left (-\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {-\frac {2 d}{c +d}}\, c -\sqrt {-\frac {2 d}{c +d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+c +d \right )}{c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}}{f \sqrt {-\frac {2 d}{c +d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}}\)	\(402\)

[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/f/(-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*d/(c+d))^(1/2)*
c+(-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-cos(f*x+e))^2*csc
(f*x+e)^2-1)^(1/2)*(-2*d/(c+d))^(1/2)*c-(-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. 120 vs. \(2 (51) = 102\).

Time = 0.42 (sec) , antiderivative size = 357, normalized size of antiderivative = 5.49 \[ \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{c-d \sec (e+f x)} \, dx=\left [\frac {\sqrt {\frac {a}{c d - d^{2}}} \log \left (-\frac {{\left (a c^{2} - 8 \, a c d + 8 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} + a d^{2} + {\left (a c^{2} - 2 \, a c d\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (c^{2} d - 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c d^{2} - d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a}{c d - d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + {\left (6 \, a c d - 7 \, a d^{2}\right )} \cos \left (f x + e\right )}{c^{2} \cos \left (f x + e\right )^{3} + {\left (c^{2} - 2 \, c d\right )} \cos \left (f x + e\right )^{2} + d^{2} - {\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )}\right )}{2 \, f}, -\frac {\sqrt {-\frac {a}{c d - d^{2}}} \arctan \left (\frac {2 \, {\left (c d - d^{2}\right )} \sqrt {-\frac {a}{c d - d^{2}}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{{\left (a c - 2 \, a d\right )} \cos \left (f x + e\right )^{2} + a d + {\left (a c - a d\right )} \cos \left (f x + e\right )}\right )}{f}\right ] \]

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

[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)/(d - c*cos(e + f*x)), x)