3.740 \(\int \frac{\cot (5 x) \csc ^3(5 x)}{\sqrt{1+\sin ^2(5 x)}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{15} \sqrt{\sin ^2(5 x)+1} \csc (5 x)-\frac{1}{15} \sqrt{\sin ^2(5 x)+1} \csc ^3(5 x) \]

[Out]

(2*Csc[5*x]*Sqrt[1 + Sin[5*x]^2])/15 - (Csc[5*x]^3*Sqrt[1 + Sin[5*x]^2])/15

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Rubi [A]  time = 0.10563, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {271, 264} \[ \frac{2}{15} \sqrt{\sin ^2(5 x)+1} \csc (5 x)-\frac{1}{15} \sqrt{\sin ^2(5 x)+1} \csc ^3(5 x) \]

Antiderivative was successfully verified.

[In]

Int[(Cot[5*x]*Csc[5*x]^3)/Sqrt[1 + Sin[5*x]^2],x]

[Out]

(2*Csc[5*x]*Sqrt[1 + Sin[5*x]^2])/15 - (Csc[5*x]^3*Sqrt[1 + Sin[5*x]^2])/15

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\cot (5 x) \csc ^3(5 x)}{\sqrt{1+\sin ^2(5 x)}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1+x^2}} \, dx,x,\sin (5 x)\right )\\ &=-\frac{1}{15} \csc ^3(5 x) \sqrt{1+\sin ^2(5 x)}-\frac{2}{15} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x^2}} \, dx,x,\sin (5 x)\right )\\ &=\frac{2}{15} \csc (5 x) \sqrt{1+\sin ^2(5 x)}-\frac{1}{15} \csc ^3(5 x) \sqrt{1+\sin ^2(5 x)}\\ \end{align*}

Mathematica [A]  time = 0.0571662, size = 28, normalized size = 0.65 \[ -\frac{1}{15} \sqrt{\sin ^2(5 x)+1} \csc (5 x) \left (\csc ^2(5 x)-2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[5*x]*Csc[5*x]^3)/Sqrt[1 + Sin[5*x]^2],x]

[Out]

-(Csc[5*x]*(-2 + Csc[5*x]^2)*Sqrt[1 + Sin[5*x]^2])/15

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Maple [A]  time = 0.973, size = 38, normalized size = 0.9 \begin{align*} -{\frac{1}{15\, \left ( \sin \left ( 5\,x \right ) \right ) ^{3}}\sqrt{1+ \left ( \sin \left ( 5\,x \right ) \right ) ^{2}}}+{\frac{2}{15\,\sin \left ( 5\,x \right ) }\sqrt{1+ \left ( \sin \left ( 5\,x \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(5*x)*csc(5*x)^3/(1+sin(5*x)^2)^(1/2),x)

[Out]

-1/15/sin(5*x)^3*(1+sin(5*x)^2)^(1/2)+2/15/sin(5*x)*(1+sin(5*x)^2)^(1/2)

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Maxima [A]  time = 1.48258, size = 50, normalized size = 1.16 \begin{align*} \frac{2 \, \sqrt{\sin \left (5 \, x\right )^{2} + 1}}{15 \, \sin \left (5 \, x\right )} - \frac{\sqrt{\sin \left (5 \, x\right )^{2} + 1}}{15 \, \sin \left (5 \, x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(5*x)*csc(5*x)^3/(1+sin(5*x)^2)^(1/2),x, algorithm="maxima")

[Out]

2/15*sqrt(sin(5*x)^2 + 1)/sin(5*x) - 1/15*sqrt(sin(5*x)^2 + 1)/sin(5*x)^3

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Fricas [A]  time = 2.18866, size = 146, normalized size = 3.4 \begin{align*} -\frac{2 \,{\left (\cos \left (5 \, x\right )^{2} - 1\right )} \sin \left (5 \, x\right ) -{\left (2 \, \cos \left (5 \, x\right )^{2} - 1\right )} \sqrt{-\cos \left (5 \, x\right )^{2} + 2}}{15 \,{\left (\cos \left (5 \, x\right )^{2} - 1\right )} \sin \left (5 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(5*x)*csc(5*x)^3/(1+sin(5*x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/15*(2*(cos(5*x)^2 - 1)*sin(5*x) - (2*cos(5*x)^2 - 1)*sqrt(-cos(5*x)^2 + 2))/((cos(5*x)^2 - 1)*sin(5*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (5 x \right )} \csc ^{3}{\left (5 x \right )}}{\sqrt{\sin ^{2}{\left (5 x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(5*x)*csc(5*x)**3/(1+sin(5*x)**2)**(1/2),x)

[Out]

Integral(cot(5*x)*csc(5*x)**3/sqrt(sin(5*x)**2 + 1), x)

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Giac [A]  time = 1.18201, size = 65, normalized size = 1.51 \begin{align*} \frac{4 \,{\left (3 \,{\left (\sqrt{\sin \left (5 \, x\right )^{2} + 1} - \sin \left (5 \, x\right )\right )}^{2} - 1\right )}}{15 \,{\left ({\left (\sqrt{\sin \left (5 \, x\right )^{2} + 1} - \sin \left (5 \, x\right )\right )}^{2} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(5*x)*csc(5*x)^3/(1+sin(5*x)^2)^(1/2),x, algorithm="giac")

[Out]

4/15*(3*(sqrt(sin(5*x)^2 + 1) - sin(5*x))^2 - 1)/((sqrt(sin(5*x)^2 + 1) - sin(5*x))^2 - 1)^3