3.22 \(\int (-1+\csc ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=30 \[ -\frac{1}{2} \tan (x) \cot ^2(x)^{3/2}-\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x)) \]

[Out]

-((Cot[x]^2)^(3/2)*Tan[x])/2 - Sqrt[Cot[x]^2]*Log[Sin[x]]*Tan[x]

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Rubi [A]  time = 0.0218156, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4121, 3658, 3473, 3475} \[ -\frac{1}{2} \tan (x) \cot ^2(x)^{3/2}-\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Csc[x]^2)^(3/2),x]

[Out]

-((Cot[x]^2)^(3/2)*Tan[x])/2 - Sqrt[Cot[x]^2]*Log[Sin[x]]*Tan[x]

Rule 4121

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[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
]])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx &=\int \cot ^2(x)^{3/2} \, dx\\ &=\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx\\ &=-\frac{1}{2} \cot ^2(x)^{3/2} \tan (x)-\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=-\frac{1}{2} \cot ^2(x)^{3/2} \tan (x)-\sqrt{\cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0143198, size = 24, normalized size = 0.8 \[ -\frac{1}{2} \tan (x) \sqrt{\cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Csc[x]^2)^(3/2),x]

[Out]

-(Sqrt[Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])/2

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Maple [B]  time = 0.099, size = 92, normalized size = 3.1 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{8\, \left ( \cos \left ( x \right ) \right ) ^{3}} \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -4\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \right ) - \left ( \cos \left ( x \right ) \right ) ^{2}-4\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +4\,\ln \left ( 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \right ) -1 \right ) \left ( -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+csc(x)^2)^(3/2),x)

[Out]

1/8*4^(1/2)*(4*cos(x)^2*ln(-(-1+cos(x))/sin(x))-4*cos(x)^2*ln(2/(cos(x)+1))-cos(x)^2-4*ln(-(-1+cos(x))/sin(x))
+4*ln(2/(cos(x)+1))-1)*sin(x)*(-cos(x)^2/(cos(x)^2-1))^(3/2)/cos(x)^3

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Maxima [A]  time = 1.54427, size = 28, normalized size = 0.93 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+csc(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2/tan(x)^2 + 1/2*log(tan(x)^2 + 1) - log(tan(x))

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Fricas [A]  time = 0.488378, size = 80, normalized size = 2.67 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+csc(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*(2*(cos(x)^2 - 1)*log(1/2*sin(x)) - 1)/(cos(x)^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\csc ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+csc(x)**2)**(3/2),x)

[Out]

Integral((csc(x)**2 - 1)**(3/2), x)

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Giac [A]  time = 1.61731, size = 63, normalized size = 2.1 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+csc(x)^2)^(3/2),x, algorithm="giac")

[Out]

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)