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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.05, 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]

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

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 57, normalized size = 1.33 \[ -\frac {2 \, {\left (\cos \left (5 \, x\right )^{2} - 1\right )} \sin \left (5 \, x\right ) - {\left (2 \, \cos \left (5 \, x\right )^{2} - 1\right )} \sqrt {-\cos \left (5 \, x\right )^{2} + 2}}{15 \, {\left (\cos \left (5 \, x\right )^{2} - 1\right )} \sin \left (5 \, x\right )} \]

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))

________________________________________________________________________________________

giac [A]  time = 0.51, size = 48, normalized size = 1.12 \[ \frac {4 \, {\left (3 \, {\left (\sqrt {\sin \left (5 \, x\right )^{2} + 1} - \sin \left (5 \, x\right )\right )}^{2} - 1\right )}}{15 \, {\left ({\left (\sqrt {\sin \left (5 \, x\right )^{2} + 1} - \sin \left (5 \, x\right )\right )}^{2} - 1\right )}^{3}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.24, size = 38, normalized size = 0.88 \[ -\frac {\sqrt {1+\sin ^{2}\left (5 x \right )}}{15 \sin \left (5 x \right )^{3}}+\frac {2 \sqrt {1+\sin ^{2}\left (5 x \right )}}{15 \sin \left (5 x \right )} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 37, normalized size = 0.86 \[ \frac {2 \, \sqrt {\sin \left (5 \, x\right )^{2} + 1}}{15 \, \sin \left (5 \, x\right )} - \frac {\sqrt {\sin \left (5 \, x\right )^{2} + 1}}{15 \, \sin \left (5 \, x\right )^{3}} \]

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

________________________________________________________________________________________

mupad [B]  time = 3.14, size = 28, normalized size = 0.65 \[ \frac {\sqrt {{\sin \left (5\,x\right )}^2+1}\,\left (2\,{\sin \left (5\,x\right )}^2-1\right )}{15\,{\sin \left (5\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\left (5 x \right )} \csc ^{3}{\left (5 x \right )}}{\sqrt {\sin ^{2}{\left (5 x \right )} + 1}}\, dx \]

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)

________________________________________________________________________________________