Integrand size = 12, antiderivative size = 100 \[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=-\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \]
2/5*sin(b*x+a)/b/c/(c*cos(b*x+a))^(5/2)+6/5*sin(b*x+a)/b/c^3/(c*cos(b*x+a) )^(1/2)-6/5*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticE(sin( 1/2*a+1/2*b*x),2^(1/2))*(c*cos(b*x+a))^(1/2)/b/c^4/cos(b*x+a)^(1/2)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=\frac {-6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+6 \sin (a+b x)+2 \sec (a+b x) \tan (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \]
(-6*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + 6*Sin[a + b*x] + 2*Sec[ a + b*x]*Tan[a + b*x])/(5*b*c^3*Sqrt[c*Cos[a + b*x]])
Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3116, 3042, 3116, 3042, 3121, 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.
\(\displaystyle \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (c \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {3 \int \frac {1}{(c \cos (a+b x))^{3/2}}dx}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \int \frac {1}{\left (c \sin \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {3 \left (\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {\int \sqrt {c \cos (a+b x)}dx}{c^2}\right )}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {\int \sqrt {c \sin \left (a+b x+\frac {\pi }{2}\right )}dx}{c^2}\right )}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {3 \left (\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {\sqrt {c \cos (a+b x)} \int \sqrt {\cos (a+b x)}dx}{c^2 \sqrt {\cos (a+b x)}}\right )}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {\sqrt {c \cos (a+b x)} \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx}{c^2 \sqrt {\cos (a+b x)}}\right )}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {3 \left (\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{b c^2 \sqrt {\cos (a+b x)}}\right )}{5 c^2}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}\) |
(2*Sin[a + b*x])/(5*b*c*(c*Cos[a + b*x])^(5/2)) + (3*((-2*Sqrt[c*Cos[a + b *x]]*EllipticE[(a + b*x)/2, 2])/(b*c^2*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b* x])/(b*c*Sqrt[c*Cos[a + b*x]])))/(5*c^2)
3.1.24.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(112)=224\).
Time = 2.34 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.66
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (24 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c}}{5 c^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {c \left (-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, b}\) | \(366\) |
-2/5*(c*(-1+2*cos(1/2*b*x+1/2*a)^2)*sin(1/2*b*x+1/2*a)^2)^(1/2)/c^4/sin(1/ 2*b*x+1/2*a)^3/(8*sin(1/2*b*x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b *x+1/2*a)^2-1)*(24*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^6-12*(2*sin(1/2*b *x+1/2*a)^2-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin(1/2*b*x+1/ 2*a)^2)^(1/2)*sin(1/2*b*x+1/2*a)^4-24*sin(1/2*b*x+1/2*a)^4*cos(1/2*b*x+1/2 *a)+12*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/ 2))*(sin(1/2*b*x+1/2*a)^2)^(1/2)*sin(1/2*b*x+1/2*a)^2+8*sin(1/2*b*x+1/2*a) ^2*cos(1/2*b*x+1/2*a)-3*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a) ^2-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2)))*(-2*sin(1/2*b*x+1/2*a)^ 4*c+sin(1/2*b*x+1/2*a)^2*c)^(1/2)/(c*(-1+2*cos(1/2*b*x+1/2*a)^2))^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=\frac {-3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, \sqrt {c \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{5 \, b c^{4} \cos \left (b x + a\right )^{3}} \]
1/5*(-3*I*sqrt(2)*sqrt(c)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 3*I*sqrt(2)*sqrt(c)*cos (b*x + a)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) + 2*sqrt(c*cos(b*x + a))*(3*cos(b*x + a)^2 + 1)*sin(b* x + a))/(b*c^4*cos(b*x + a)^3)
Timed out. \[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx=\int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]