Integrand size = 23, antiderivative size = 49 \[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{(a+b) d} \] Output:
2*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*sec(d* x+c)^(1/2)/(a+b)/d
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\frac {2 \cot (c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) \sqrt {-\tan ^2(c+d x)}}{a d} \] Input:
Integrate[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x]),x]
Output:
(2*Cot[c + d*x]*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-( a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(a*d)
Time = 0.39 (sec) , antiderivative size = 49, 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, 3717, 3042, 4336, 3042, 3284}
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.
\(\displaystyle \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3717 |
\(\displaystyle \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a \sec (c+d x)+b}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a \csc \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
\(\Big \downarrow \) 4336 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}\) |
Input:
Int[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x]),x]
Output:
(2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d)
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(149\) vs. \(2(47)=94\).
Time = 2.58 (sec) , antiderivative size = 150, normalized size of antiderivative = 3.06
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), -\frac {2 b}{a -b}, \sqrt {2}\right )}{\left (a -b \right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(150\) |
Input:
int(sec(d*x+c)^(1/2)/(a+cos(d*x+c)*b),x,method=_RETURNVERBOSE)
Output:
-2*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticPi(cos(1/2*d*x+1/2 *c),-2*b/(a-b),2^(1/2))/(a-b)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^ 2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Timed out. \[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^(1/2)/(a+b*cos(d*x+c)),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\int \frac {\sqrt {\sec {\left (c + d x \right )}}}{a + b \cos {\left (c + d x \right )}}\, dx \] Input:
integrate(sec(d*x+c)**(1/2)/(a+b*cos(d*x+c)),x)
Output:
Integral(sqrt(sec(c + d*x))/(a + b*cos(c + d*x)), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{b \cos \left (d x + c\right ) + a} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)/(a+b*cos(d*x+c)),x, algorithm="maxima")
Output:
integrate(sqrt(sec(d*x + c))/(b*cos(d*x + c) + a), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\int { \frac {\sqrt {\sec \left (d x + c\right )}}{b \cos \left (d x + c\right ) + a} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)/(a+b*cos(d*x+c)),x, algorithm="giac")
Output:
integrate(sqrt(sec(d*x + c))/(b*cos(d*x + c) + a), x)
Timed out. \[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:
int((1/cos(c + d*x))^(1/2)/(a + b*cos(c + d*x)),x)
Output:
int((1/cos(c + d*x))^(1/2)/(a + b*cos(c + d*x)), x)
\[ \int \frac {\sqrt {\sec (c+d x)}}{a+b \cos (c+d x)} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}}{\cos \left (d x +c \right ) b +a}d x \] Input:
int(sec(d*x+c)^(1/2)/(a+b*cos(d*x+c)),x)
Output:
int(sqrt(sec(c + d*x))/(cos(c + d*x)*b + a),x)