3.361 \(\int \cot ^4(x) \csc ^3(x) \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot ^3(x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{16} \cot (x) \csc (x) \]

[Out]

-ArcTanh[Cos[x]]/16 - (Cot[x]*Csc[x])/16 + (Cot[x]*Csc[x]^3)/8 - (Cot[x]^3*Csc[x]^3)/6

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Rubi [A]  time = 0.0530602, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ -\frac{1}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot ^3(x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{16} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^4*Csc[x]^3,x]

[Out]

-ArcTanh[Cos[x]]/16 - (Cot[x]*Csc[x])/16 + (Cot[x]*Csc[x]^3)/8 - (Cot[x]^3*Csc[x]^3)/6

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rubi steps

\begin{align*} \int \cot ^4(x) \csc ^3(x) \, dx &=-\frac{1}{6} \cot ^3(x) \csc ^3(x)-\frac{1}{2} \int \cot ^2(x) \csc ^3(x) \, dx\\ &=\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{6} \cot ^3(x) \csc ^3(x)+\frac{1}{8} \int \csc ^3(x) \, dx\\ &=-\frac{1}{16} \cot (x) \csc (x)+\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{6} \cot ^3(x) \csc ^3(x)+\frac{1}{16} \int \csc (x) \, dx\\ &=-\frac{1}{16} \tanh ^{-1}(\cos (x))-\frac{1}{16} \cot (x) \csc (x)+\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{6} \cot ^3(x) \csc ^3(x)\\ \end{align*}

Mathematica [B]  time = 0.0192547, size = 95, normalized size = 2.5 \[ -\frac{1}{384} \csc ^6\left (\frac{x}{2}\right )+\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )-\frac{1}{64} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{384} \sec ^6\left (\frac{x}{2}\right )-\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^2\left (\frac{x}{2}\right )+\frac{1}{16} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{16} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^4*Csc[x]^3,x]

[Out]

-Csc[x/2]^2/64 + Csc[x/2]^4/64 - Csc[x/2]^6/384 - Log[Cos[x/2]]/16 + Log[Sin[x/2]]/16 + Sec[x/2]^2/64 - Sec[x/
2]^4/64 + Sec[x/2]^6/384

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Maple [A]  time = 0.016, size = 52, normalized size = 1.4 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{6\, \left ( \sin \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{24\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{48\, \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{48}}+{\frac{\cos \left ( x \right ) }{16}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^4*csc(x)^3,x)

[Out]

-1/6/sin(x)^6*cos(x)^5-1/24/sin(x)^4*cos(x)^5+1/48/sin(x)^2*cos(x)^5+1/48*cos(x)^3+1/16*cos(x)+1/16*ln(csc(x)-
cot(x))

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Maxima [A]  time = 0.932084, size = 73, normalized size = 1.92 \begin{align*} \frac{3 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{48 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac{1}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^4*csc(x)^3,x, algorithm="maxima")

[Out]

1/48*(3*cos(x)^5 + 8*cos(x)^3 - 3*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1) - 1/32*log(cos(x) + 1) + 1/
32*log(cos(x) - 1)

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Fricas [B]  time = 2.55986, size = 297, normalized size = 7.82 \begin{align*} \frac{6 \, \cos \left (x\right )^{5} + 16 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 6 \, \cos \left (x\right )}{96 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^4*csc(x)^3,x, algorithm="fricas")

[Out]

1/96*(6*cos(x)^5 + 16*cos(x)^3 - 3*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^
6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(-1/2*cos(x) + 1/2) - 6*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)

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Sympy [A]  time = 0.159786, size = 56, normalized size = 1.47 \begin{align*} \frac{3 \cos ^{5}{\left (x \right )} + 8 \cos ^{3}{\left (x \right )} - 3 \cos{\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{32} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**4*csc(x)**3,x)

[Out]

(3*cos(x)**5 + 8*cos(x)**3 - 3*cos(x))/(48*cos(x)**6 - 144*cos(x)**4 + 144*cos(x)**2 - 48) + log(cos(x) - 1)/3
2 - log(cos(x) + 1)/32

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Giac [A]  time = 1.0869, size = 59, normalized size = 1.55 \begin{align*} \frac{3 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{48 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{3}} - \frac{1}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{32} \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^4*csc(x)^3,x, algorithm="giac")

[Out]

1/48*(3*cos(x)^5 + 8*cos(x)^3 - 3*cos(x))/(cos(x)^2 - 1)^3 - 1/32*log(cos(x) + 1) + 1/32*log(-cos(x) + 1)