Integrand size = 10, antiderivative size = 65 \[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\frac {10}{21} a \cos ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {x}{2},2\right ) \sqrt {a \sec ^3(x)}+\frac {10}{21} a \sqrt {a \sec ^3(x)} \sin (x)+\frac {2}{7} a \sec (x) \sqrt {a \sec ^3(x)} \tan (x) \]
10/21*a*cos(x)^(3/2)*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticF(sin(1/2*x), 2^(1/2))*(a*sec(x)^3)^(1/2)+10/21*a*sin(x)*(a*sec(x)^3)^(1/2)+2/7*a*sec(x) *(a*sec(x)^3)^(1/2)*tan(x)
Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\frac {2}{21} a \sec (x) \sqrt {a \sec ^3(x)} \left (5 \cos ^{\frac {5}{2}}(x) \operatorname {EllipticF}\left (\frac {x}{2},2\right )+5 \cos (x) \sin (x)+3 \tan (x)\right ) \]
(2*a*Sec[x]*Sqrt[a*Sec[x]^3]*(5*Cos[x]^(5/2)*EllipticF[x/2, 2] + 5*Cos[x]* Sin[x] + 3*Tan[x]))/21
Time = 0.41 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4611, 3042, 4255, 3042, 4255, 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.
\(\displaystyle \int \left (a \sec ^3(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec (x)^3\right )^{3/2}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \int \sec ^{\frac {9}{2}}(x)dx}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \int \csc \left (x+\frac {\pi }{2}\right )^{9/2}dx}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {5}{7} \int \sec ^{\frac {5}{2}}(x)dx+\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {5}{7} \int \csc \left (x+\frac {\pi }{2}\right )^{5/2}dx+\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\sec (x)}dx+\frac {2}{3} \sin (x) \sec ^{\frac {3}{2}}(x)\right )+\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc \left (x+\frac {\pi }{2}\right )}dx+\frac {2}{3} \sin (x) \sec ^{\frac {3}{2}}(x)\right )+\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (x)} \sqrt {\sec (x)} \int \frac {1}{\sqrt {\cos (x)}}dx+\frac {2}{3} \sin (x) \sec ^{\frac {3}{2}}(x)\right )+\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (x)} \sqrt {\sec (x)} \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )}}dx+\frac {2}{3} \sin (x) \sec ^{\frac {3}{2}}(x)\right )+\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)\right )}{\sec ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {a \sqrt {a \sec ^3(x)} \left (\frac {2}{7} \sin (x) \sec ^{\frac {7}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sin (x) \sec ^{\frac {3}{2}}(x)+\frac {2}{3} \sqrt {\cos (x)} \sqrt {\sec (x)} \operatorname {EllipticF}\left (\frac {x}{2},2\right )\right )\right )}{\sec ^{\frac {3}{2}}(x)}\) |
(a*Sqrt[a*Sec[x]^3]*((2*Sec[x]^(7/2)*Sin[x])/7 + (5*((2*Sqrt[Cos[x]]*Ellip ticF[x/2, 2]*Sqrt[Sec[x]])/3 + (2*Sec[x]^(3/2)*Sin[x])/3))/7))/Sec[x]^(3/2 )
3.1.56.3.1 Defintions of rubi rules used
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[(-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 = 4.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54
method | result | size |
default | \(-\frac {2 i \sqrt {a \sec \left (x \right )^{3}}\, a \left (5 \operatorname {EllipticF}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \cos \left (x \right )^{2}+5 \operatorname {EllipticF}\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \cos \left (x \right )+5 i \sin \left (x \right )+3 i \sec \left (x \right ) \tan \left (x \right )\right )}{21}\) | \(100\) |
-2/21*I*(a*sec(x)^3)^(1/2)*a*(5*EllipticF(I*(csc(x)-cot(x)),I)*(cos(x)/(co s(x)+1))^(1/2)*(1/(cos(x)+1))^(1/2)*cos(x)^2+5*EllipticF(I*(csc(x)-cot(x)) ,I)*(cos(x)/(cos(x)+1))^(1/2)*(1/(cos(x)+1))^(1/2)*cos(x)+5*I*sin(x)+3*I*s ec(x)*tan(x))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.14 \[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\frac {5 i \, \sqrt {2} a^{\frac {3}{2}} \cos \left (x\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 5 i \, \sqrt {2} a^{\frac {3}{2}} \cos \left (x\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{2} + 3 \, a\right )} \sqrt {\frac {a}{\cos \left (x\right )^{3}}} \sin \left (x\right )}{21 \, \cos \left (x\right )^{2}} \]
1/21*(5*I*sqrt(2)*a^(3/2)*cos(x)^2*weierstrassPInverse(-4, 0, cos(x) + I*s in(x)) - 5*I*sqrt(2)*a^(3/2)*cos(x)^2*weierstrassPInverse(-4, 0, cos(x) - I*sin(x)) + 2*(5*a*cos(x)^2 + 3*a)*sqrt(a/cos(x)^3)*sin(x))/cos(x)^2
\[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\int \left (a \sec ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\int { \left (a \sec \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
\[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\int { \left (a \sec \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a \sec ^3(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\cos \left (x\right )}^3}\right )}^{3/2} \,d x \]