Integrand size = 25, antiderivative size = 75 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {4 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \] Output:
2/3*sin(d*x+c)/d/(1-cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2)-4/3*sin(d*x+c)/d/(1 -cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \sqrt {1-\cos (c+d x)} (-1+2 \cos (c+d x)) \cot \left (\frac {1}{2} (c+d x)\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)} \] Input:
Integrate[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]
Output:
(-2*Sqrt[1 - Cos[c + d*x]]*(-1 + 2*Cos[c + d*x])*Cot[(c + d*x)/2])/(3*d*Co s[c + d*x]^(3/2))
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3251, 3042, 3250}
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.
\(\displaystyle \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {1-\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3251 |
\(\displaystyle \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int \frac {\sqrt {1-\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3250 |
\(\displaystyle \frac {2 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}-\frac {4 \sin (c+d x)}{3 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\) |
Input:
Int[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]
Output:
(2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) - (4*Sin[ c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq rt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.61 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {2 \left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3\right ) \operatorname {csgn}\left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}}{3 d \cos \left (d x +c \right )^{\frac {5}{2}}}\) | \(64\) |
Input:
int((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3/d*(8*cos(1/2*d*x+1/2*c)^4-10*cos(1/2*d*x+1/2*c)^2+3)*csgn(sin(1/2*d*x +1/2*c))*cos(1/2*d*x+1/2*c)/cos(d*x+c)^(5/2)*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1\right )} \sqrt {-\cos \left (d x + c\right ) + 1}}{3 \, d \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right )} \] Input:
integrate((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x, algorithm="fricas")
Output:
-2/3*(2*cos(d*x + c)^2 + cos(d*x + c) - 1)*sqrt(-cos(d*x + c) + 1)/(d*cos( d*x + c)^(3/2)*sin(d*x + c))
\[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {1 - \cos {\left (c + d x \right )}}}{\cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((1-cos(d*x+c))**(1/2)/cos(d*x+c)**(5/2),x)
Output:
Integral(sqrt(1 - cos(c + d*x))/cos(c + d*x)**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (63) = 126\).
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.19 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {2} - \frac {4 \, \sqrt {2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sqrt {2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} \] Input:
integrate((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x, algorithm="maxima")
Output:
-2/3*(sqrt(2) - 4*sqrt(2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sqrt(2)* sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^2/(d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/2)*(-sin(d*x + c)/(cos( d*x + c) + 1) + 1)^(5/2)*(2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1))
Time = 0.71 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 \, \sqrt {2} {\left ({\left ({\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 15\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 15\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 1\right )} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{3 \, {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1\right )}^{\frac {3}{2}} d} \] Input:
integrate((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x, algorithm="giac")
Output:
2/3*sqrt(2)*(((tan(1/4*d*x + 1/4*c)^2 - 15)*tan(1/4*d*x + 1/4*c)^2 + 15)*t an(1/4*d*x + 1/4*c)^2 - 1)*sgn(sin(1/2*d*x + 1/2*c))/((tan(1/4*d*x + 1/4*c )^4 - 6*tan(1/4*d*x + 1/4*c)^2 + 1)^(3/2)*d)
Time = 40.64 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {4\,\sqrt {1-\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )-\sin \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\right )}{3\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (3\,\cos \left (c+d\,x\right )-2\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (3\,c+3\,d\,x\right )-2\right )} \] Input:
int((1 - cos(c + d*x))^(1/2)/cos(c + d*x)^(5/2),x)
Output:
(4*(1 - cos(c + d*x))^(1/2)*(sin(c + d*x) - sin(2*c + 2*d*x) + sin(3*c + 3 *d*x)))/(3*d*cos(c + d*x)^(1/2)*(3*cos(c + d*x) - 2*cos(2*c + 2*d*x) + cos (3*c + 3*d*x) - 2))
\[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {-\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \] Input:
int((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x)
Output:
int((sqrt( - cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**3,x)