Integrand size = 10, antiderivative size = 30 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \cot ^2(x)^{3/2} \tan (x)-\sqrt {\cot ^2(x)} \log (\sin (x)) \tan (x) \] Output:
-1/2*(cot(x)^2)^(3/2)*tan(x)-(cot(x)^2)^(1/2)*ln(sin(x))*tan(x)
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \sqrt {\cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \tan (x) \] Input:
Integrate[(-1 + Csc[x]^2)^(3/2),x]
Output:
-1/2*(Sqrt[Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {3042, 4609, 3042, 4141, 3042, 25, 3954, 25, 3042, 25, 3956}
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.
\(\displaystyle \int \left (\csc ^2(x)-1\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\sec \left (x+\frac {\pi }{2}\right )^2-1\right )^{3/2}dx\) |
\(\Big \downarrow \) 4609 |
\(\displaystyle \int \cot ^2(x)^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \tan (x) \sqrt {\cot ^2(x)} \int \cot ^3(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tan (x) \sqrt {\cot ^2(x)} \int -\tan \left (x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \int \tan \left (x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \left (\frac {\cot ^2(x)}{2}-\int -\cot (x)dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \left (\int \cot (x)dx+\frac {\cot ^2(x)}{2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \left (\int -\tan \left (x+\frac {\pi }{2}\right )dx+\frac {\cot ^2(x)}{2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \left (\frac {\cot ^2(x)}{2}-\int \tan \left (x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \left (\frac {\cot ^2(x)}{2}+\log (\sin (x))\right )\) |
Input:
Int[(-1 + Csc[x]^2)^(3/2),x]
Output:
-(Sqrt[Cot[x]^2]*(Cot[x]^2/2 + Log[Sin[x]])*Tan[x])
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30
method | result | size |
default | \(-\frac {\sqrt {4}\, \operatorname {csgn}\left (\cot \left (x \right )\right ) \left (4 \ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )-4 \ln \left (\frac {2}{\cos \left (x \right )+1}\right )+\cot \left (x \right )^{2}+\csc \left (x \right )^{2}\right )}{8}\) | \(39\) |
risch | \(\frac {\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (i {\mathrm e}^{4 i x} \ln \left ({\mathrm e}^{2 i x}-1\right )+{\mathrm e}^{4 i x} x -2 i {\mathrm e}^{2 i x} \ln \left ({\mathrm e}^{2 i x}-1\right )-2 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{2 i x} x +i \ln \left ({\mathrm e}^{2 i x}-1\right )+x \right )}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}\) | \(110\) |
Input:
int((csc(x)^2-1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/8*4^(1/2)*csgn(cot(x))*(4*ln(-cot(x)+csc(x))-4*ln(2/(cos(x)+1))+cot(x)^ 2+csc(x)^2)
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) - 1}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \] Input:
integrate((-1+csc(x)^2)^(3/2),x, algorithm="fricas")
Output:
1/2*(2*(cos(x)^2 - 1)*log(1/2*sin(x)) - 1)/(cos(x)^2 - 1)
\[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\int \left (\csc ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \] Input:
integrate((-1+csc(x)**2)**(3/2),x)
Output:
Integral((csc(x)**2 - 1)**(3/2), x)
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2 \, \tan \left (x\right )^{2}} + \frac {1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \log \left (\tan \left (x\right )\right ) \] Input:
integrate((-1+csc(x)^2)^(3/2),x, algorithm="maxima")
Output:
-1/2/tan(x)^2 + 1/2*log(tan(x)^2 + 1) - log(tan(x))
Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \, \tan \left (\frac {1}{2} \, x\right )^{2} - \frac {4 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{8 \, \tan \left (\frac {1}{2} \, x\right )^{2}} - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + \frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \] Input:
integrate((-1+csc(x)^2)^(3/2),x, algorithm="giac")
Output:
1/8*tan(1/2*x)^2 - 1/8*(4*tan(1/2*x)^2 - 1)/tan(1/2*x)^2 - log(tan(1/2*x)^ 2 + 1) + 1/2*log(tan(1/2*x)^2)
Timed out. \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\int {\left (\frac {1}{{\sin \left (x\right )}^2}-1\right )}^{3/2} \,d x \] Input:
int((1/sin(x)^2 - 1)^(3/2),x)
Output:
int((1/sin(x)^2 - 1)^(3/2), x)
\[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\left (\int \sqrt {\csc \left (x \right )^{2}-1}d x \right )+\int \sqrt {\csc \left (x \right )^{2}-1}\, \csc \left (x \right )^{2}d x \] Input:
int((-1+csc(x)^2)^(3/2),x)
Output:
- int(sqrt(csc(x)**2 - 1),x) + int(sqrt(csc(x)**2 - 1)*csc(x)**2,x)