Integrand size = 10, antiderivative size = 42 \[ \int \sqrt {a \sec ^3(x)} \, dx=-2 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right ) \sqrt {a \sec ^3(x)}+2 \cos (x) \sqrt {a \sec ^3(x)} \sin (x) \]
-2*cos(x)^(3/2)*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/ 2))*(a*sec(x)^3)^(1/2)+2*cos(x)*sin(x)*(a*sec(x)^3)^(1/2)
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int \sqrt {a \sec ^3(x)} \, dx=2 \cos (x) \sqrt {a \sec ^3(x)} \left (-\sqrt {\cos (x)} E\left (\left .\frac {x}{2}\right |2\right )+\sin (x)\right ) \]
Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17, 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, 4255, 3042, 4258, 3042, 3119}
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.
\(\displaystyle \int \sqrt {a \sec ^3(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a \sec (x)^3}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \int \sec ^{\frac {3}{2}}(x)dx}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \int \csc \left (x+\frac {\pi }{2}\right )^{3/2}dx}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \left (2 \sin (x) \sqrt {\sec (x)}-\int \frac {1}{\sqrt {\sec (x)}}dx\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \left (2 \sin (x) \sqrt {\sec (x)}-\int \frac {1}{\sqrt {\csc \left (x+\frac {\pi }{2}\right )}}dx\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \left (2 \sin (x) \sqrt {\sec (x)}-\sqrt {\cos (x)} \sqrt {\sec (x)} \int \sqrt {\cos (x)}dx\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \left (2 \sin (x) \sqrt {\sec (x)}-\sqrt {\cos (x)} \sqrt {\sec (x)} \int \sqrt {\sin \left (x+\frac {\pi }{2}\right )}dx\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sqrt {a \sec ^3(x)} \left (2 \sin (x) \sqrt {\sec (x)}-2 \sqrt {\cos (x)} \sqrt {\sec (x)} E\left (\left .\frac {x}{2}\right |2\right )\right )}{\sec ^{\frac {3}{2}}(x)}\) |
(Sqrt[a*Sec[x]^3]*(-2*Sqrt[Cos[x]]*EllipticE[x/2, 2]*Sqrt[Sec[x]] + 2*Sqrt [Sec[x]]*Sin[x]))/Sec[x]^(3/2)
3.1.57.3.1 Defintions of rubi rules used
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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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 = 1.76 (sec) , antiderivative size = 244, normalized size of antiderivative = 5.81
method | result | size |
default | \(\frac {2 \left (i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \cos \left (x \right )^{2}-i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \cos \left (x \right )^{2}+2 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \cos \left (x \right )-2 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \cos \left (x \right )+i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )-i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+\sin \left (x \right )\right ) \sqrt {a \sec \left (x \right )^{3}}\, \cos \left (x \right )}{\cos \left (x \right )+1}\) | \(244\) |
2*(I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(csc(x)-co t(x)),I)*cos(x)^2-I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*Ellipti cE(I*(csc(x)-cot(x)),I)*cos(x)^2+2*I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+ 1))^(1/2)*EllipticF(I*(csc(x)-cot(x)),I)*cos(x)-2*I*(1/(cos(x)+1))^(1/2)*( cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(csc(x)-cot(x)),I)*cos(x)+I*(1/(cos(x )+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(csc(x)-cot(x)),I)-I*(1/ (cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(csc(x)-cot(x)),I) +sin(x))*(a*sec(x)^3)^(1/2)*cos(x)/(cos(x)+1)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.36 \[ \int \sqrt {a \sec ^3(x)} \, dx=2 \, \sqrt {\frac {a}{\cos \left (x\right )^{3}}} \cos \left (x\right ) \sin \left (x\right ) + i \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - i \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) \]
2*sqrt(a/cos(x)^3)*cos(x)*sin(x) + I*sqrt(2)*sqrt(a)*weierstrassZeta(-4, 0 , weierstrassPInverse(-4, 0, cos(x) + I*sin(x))) - I*sqrt(2)*sqrt(a)*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(x) - I*sin(x)))
\[ \int \sqrt {a \sec ^3(x)} \, dx=\int \sqrt {a \sec ^{3}{\left (x \right )}}\, dx \]
\[ \int \sqrt {a \sec ^3(x)} \, dx=\int { \sqrt {a \sec \left (x\right )^{3}} \,d x } \]
\[ \int \sqrt {a \sec ^3(x)} \, dx=\int { \sqrt {a \sec \left (x\right )^{3}} \,d x } \]
Timed out. \[ \int \sqrt {a \sec ^3(x)} \, dx=\int \sqrt {\frac {a}{{\cos \left (x\right )}^3}} \,d x \]