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

Optimal. Leaf size=33 \[ \frac{1}{2} \sqrt{-\cot ^2(x)} \cot (x)+\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0272457, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4121, 3658, 3473, 3475} \[ \frac{1}{2} \sqrt{-\cot ^2(x)} \cot (x)+\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Csc[x]^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0168073, size = 26, normalized size = 0.79 \[ \frac{1}{2} \tan (x) \sqrt{-\cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \]

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

________________________________________________________________________________________

Maple [B]  time = 0.125, size = 91, normalized size = 2.8 \begin{align*}{\frac{\sin \left ( x \right ) \sqrt{4}}{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*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)*4^(1/2)*(cos(x)^2/(cos(x)^2-1))^(3/2)/cos(x)^3

________________________________________________________________________________________

Maxima [C]  time = 1.51738, size = 28, normalized size = 0.85 \begin{align*} \frac{i}{2 \, \tan \left (x\right )^{2}} - \frac{1}{2} i \, \log \left (\tan \left (x\right )^{2} + 1\right ) + i \, \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*I/tan(x)^2 - 1/2*I*log(tan(x)^2 + 1) + I*log(tan(x))

________________________________________________________________________________________

Fricas [A]  time = 0.487482, size = 36, normalized size = 1.09 \begin{align*} x + \arctan \left (\frac{\cos \left (x\right )}{\sin \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="fricas")

[Out]

x + arctan(cos(x)/sin(x))

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [C]  time = 2.11824, size = 63, normalized size = 1.91 \begin{align*} \frac{1}{8} i \, \tan \left (\frac{1}{2} \, x\right )^{2} - \frac{4 i \, \tan \left (\frac{1}{2} \, x\right )^{2} - i}{8 \, \tan \left (\frac{1}{2} \, x\right )^{2}} - i \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + \frac{1}{2} i \, \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*I*tan(1/2*x)^2 - 1/8*(4*I*tan(1/2*x)^2 - I)/tan(1/2*x)^2 - I*log(tan(1/2*x)^2 + 1) + 1/2*I*log(tan(1/2*x)^
2)