Integrand size = 10, antiderivative size = 44 \[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {x}{2},2\right )}{3 \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}+\frac {2 \tan (x)}{3 \sqrt {a \sec ^3(x)}} \] Output:
2/3*InverseJacobiAM(1/2*x,2^(1/2))/cos(x)^(3/2)/(a*sec(x)^3)^(1/2)+2/3*tan (x)/(a*sec(x)^3)^(1/2)
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\frac {2 \left (\frac {\operatorname {EllipticF}\left (\frac {x}{2},2\right )}{\cos ^{\frac {3}{2}}(x)}+\tan (x)\right )}{3 \sqrt {a \sec ^3(x)}} \] Input:
Integrate[1/Sqrt[a*Sec[x]^3],x]
Output:
(2*(EllipticF[x/2, 2]/Cos[x]^(3/2) + Tan[x]))/(3*Sqrt[a*Sec[x]^3])
Time = 0.35 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4611, 3042, 4256, 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.
\(\displaystyle \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a \sec (x)^3}}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \int \frac {1}{\sec ^{\frac {3}{2}}(x)}dx}{\sqrt {a \sec ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \int \frac {1}{\csc \left (x+\frac {\pi }{2}\right )^{3/2}}dx}{\sqrt {a \sec ^3(x)}}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \left (\frac {1}{3} \int \sqrt {\sec (x)}dx+\frac {2 \sin (x)}{3 \sqrt {\sec (x)}}\right )}{\sqrt {a \sec ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \left (\frac {1}{3} \int \sqrt {\csc \left (x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (x)}{3 \sqrt {\sec (x)}}\right )}{\sqrt {a \sec ^3(x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \left (\frac {1}{3} \sqrt {\cos (x)} \sqrt {\sec (x)} \int \frac {1}{\sqrt {\cos (x)}}dx+\frac {2 \sin (x)}{3 \sqrt {\sec (x)}}\right )}{\sqrt {a \sec ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \left (\frac {1}{3} \sqrt {\cos (x)} \sqrt {\sec (x)} \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (x)}{3 \sqrt {\sec (x)}}\right )}{\sqrt {a \sec ^3(x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\sec ^{\frac {3}{2}}(x) \left (\frac {2 \sin (x)}{3 \sqrt {\sec (x)}}+\frac {2}{3} \sqrt {\cos (x)} \sqrt {\sec (x)} \operatorname {EllipticF}\left (\frac {x}{2},2\right )\right )}{\sqrt {a \sec ^3(x)}}\) |
Input:
Int[1/Sqrt[a*Sec[x]^3],x]
Output:
(Sec[x]^(3/2)*((2*Sqrt[Cos[x]]*EllipticF[x/2, 2]*Sqrt[Sec[x]])/3 + (2*Sin[ x])/(3*Sqrt[Sec[x]])))/Sqrt[a*Sec[x]^3]
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
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]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {-\frac {2 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \left (\sec \left (x \right )+\sec \left (x \right )^{2}\right )}{3}+\frac {2 \tan \left (x \right )}{3}}{\sqrt {a \sec \left (x \right )^{3}}}\) | \(57\) |
Input:
int(1/(a*sec(x)^3)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-2/3*I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(csc(x) -cot(x)),I)*(sec(x)+sec(x)^2)+2/3*tan(x))/(a*sec(x)^3)^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\frac {2 \, \sqrt {\frac {a}{\cos \left (x\right )^{3}}} \cos \left (x\right )^{2} \sin \left (x\right ) + i \, \sqrt {2} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - i \, \sqrt {2} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{3 \, a} \] Input:
integrate(1/(a*sec(x)^3)^(1/2),x, algorithm="fricas")
Output:
1/3*(2*sqrt(a/cos(x)^3)*cos(x)^2*sin(x) + I*sqrt(2)*sqrt(a)*weierstrassPIn verse(-4, 0, cos(x) + I*sin(x)) - I*sqrt(2)*sqrt(a)*weierstrassPInverse(-4 , 0, cos(x) - I*sin(x)))/a
\[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \sec ^{3}{\left (x \right )}}}\, dx \] Input:
integrate(1/(a*sec(x)**3)**(1/2),x)
Output:
Integral(1/sqrt(a*sec(x)**3), x)
\[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \sec \left (x\right )^{3}}} \,d x } \] Input:
integrate(1/(a*sec(x)^3)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(a*sec(x)^3), x)
\[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \sec \left (x\right )^{3}}} \,d x } \] Input:
integrate(1/(a*sec(x)^3)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(a*sec(x)^3), x)
Timed out. \[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\cos \left (x\right )}^3}}} \,d x \] Input:
int(1/(a/cos(x)^3)^(1/2),x)
Output:
int(1/(a/cos(x)^3)^(1/2), x)
\[ \int \frac {1}{\sqrt {a \sec ^3(x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (x \right )}}{\sec \left (x \right )^{2}}d x \right )}{a} \] Input:
int(1/(a*sec(x)^3)^(1/2),x)
Output:
(sqrt(a)*int(sqrt(sec(x))/sec(x)**2,x))/a