Integrand size = 11, antiderivative size = 31 \[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=\frac {8}{3} \csc (x) \sqrt {\sin (x) \tan (x)}-\frac {2}{3} \sin (x) \sqrt {\sin (x) \tan (x)} \] Output:
8/3*csc(x)*(sin(x)*tan(x))^(1/2)-2/3*sin(x)*(sin(x)*tan(x))^(1/2)
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=\frac {2}{3} \left (-1+4 \csc ^2(x)\right ) \sin (x) \sqrt {\sin (x) \tan (x)} \] Input:
Integrate[(-Cos[x] + Sec[x])^(3/2),x]
Output:
(2*(-1 + 4*Csc[x]^2)*Sin[x]*Sqrt[Sin[x]*Tan[x]])/3
Time = 0.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 4897, 3042, 4900, 3042, 3078, 3042, 3069}
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.
\(\displaystyle \int (\sec (x)-\cos (x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\sec (x)-\cos (x))^{3/2}dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int (\sin (x) \tan (x))^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\sin (x) \tan (x))^{3/2}dx\) |
\(\Big \downarrow \) 4900 |
\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \int \sin ^{\frac {3}{2}}(x) \tan ^{\frac {3}{2}}(x)dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \int \sin (x)^{3/2} \tan (x)^{3/2}dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\) |
\(\Big \downarrow \) 3078 |
\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {4}{3} \int \frac {\tan ^{\frac {3}{2}}(x)}{\sqrt {\sin (x)}}dx-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (x) \tan (x)} \left (\frac {4}{3} \int \frac {\tan (x)^{3/2}}{\sqrt {\sin (x)}}dx-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}\right )}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\) |
\(\Big \downarrow \) 3069 |
\(\displaystyle \frac {\left (\frac {8 \sqrt {\tan (x)}}{3 \sqrt {\sin (x)}}-\frac {2}{3} \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}\right ) \sqrt {\sin (x) \tan (x)}}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\) |
Input:
Int[(-Cos[x] + Sec[x])^(3/2),x]
Output:
(((8*Sqrt[Tan[x]])/(3*Sqrt[Sin[x]]) - (2*Sin[x]^(3/2)*Sqrt[Tan[x]])/3)*Sqr t[Sin[x]*Tan[x]])/(Sqrt[Sin[x]]*Sqrt[Tan[x]])
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f* m)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/( f*m)), x] + Simp[a^2*((m + n - 1)/m) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1 ] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])) Int[uu*vv^(m*p)*ww^(n*p), x] , x]] /; FreeQ[{m, n, p}, x] && !IntegerQ[p] && ( !InertTrigFreeQ[v] || ! InertTrigFreeQ[w])
Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(23)=46\).
Time = 1.46 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.74
method | result | size |
default | \(\frac {\csc \left (x \right ) \left (\left (-3 \cos \left (x \right )-3\right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \ln \left (\frac {4 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-2 \cos \left (x \right )+2}{\cos \left (x \right )+1}\right )+\left (3 \cos \left (x \right )+3\right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+4 \cos \left (x \right )^{2}+12\right ) \sqrt {\sin \left (x \right ) \tan \left (x \right )}}{6}\) | \(147\) |
Input:
int((-cos(x)+sec(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/6*csc(x)*((-3*cos(x)-3)*(-cos(x)/(cos(x)+1)^2)^(1/2)*ln(2*(2*cos(x)*(-co s(x)/(cos(x)+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+ 1))+(3*cos(x)+3)*(-cos(x)/(cos(x)+1)^2)^(1/2)*ln((2*cos(x)*(-cos(x)/(cos(x )+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+1))+4*cos(x )^2+12)*(sin(x)*tan(x))^(1/2)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{2} + 3\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{3 \, \sin \left (x\right )} \] Input:
integrate((-cos(x)+sec(x))^(3/2),x, algorithm="fricas")
Output:
2/3*(cos(x)^2 + 3)*sqrt(-(cos(x)^2 - 1)/cos(x))/sin(x)
\[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=\int \left (- \cos {\left (x \right )} + \sec {\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((-cos(x)+sec(x))**(3/2),x)
Output:
Integral((-cos(x) + sec(x))**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=-\frac {8 \, {\left (\frac {\sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 1\right )}}{3 \, {\left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} \] Input:
integrate((-cos(x)+sec(x))^(3/2),x, algorithm="maxima")
Output:
-8/3*(sin(x)^6/(cos(x) + 1)^6 - 1)/((sin(x)/(cos(x) + 1) + 1)^(3/2)*(-sin( x)/(cos(x) + 1) + 1)^(3/2)*(sin(x)^2/(cos(x) + 1)^2 + 1)^(3/2))
\[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=\int { {\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((-cos(x)+sec(x))^(3/2),x, algorithm="giac")
Output:
integrate((-cos(x) + sec(x))^(3/2), x)
Timed out. \[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=\int {\left (\frac {1}{\cos \left (x\right )}-\cos \left (x\right )\right )}^{3/2} \,d x \] Input:
int((1/cos(x) - cos(x))^(3/2),x)
Output:
int((1/cos(x) - cos(x))^(3/2), x)
\[ \int (-\cos (x)+\sec (x))^{3/2} \, dx=-\left (\int \sqrt {-\cos \left (x \right )+\sec \left (x \right )}\, \cos \left (x \right )d x \right )+\int \sqrt {-\cos \left (x \right )+\sec \left (x \right )}\, \sec \left (x \right )d x \] Input:
int((-cos(x)+sec(x))^(3/2),x)
Output:
- int(sqrt( - cos(x) + sec(x))*cos(x),x) + int(sqrt( - cos(x) + sec(x))*s ec(x),x)