\(\int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx\) [740]

Optimal result

Integrand size = 23, antiderivative size = 43 \[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=\frac {2}{15} \csc (5 x) \sqrt {1+\sin ^2(5 x)}-\frac {1}{15} \csc ^3(5 x) \sqrt {1+\sin ^2(5 x)} \]

[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] (verified)

Time = 0.13 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {277, 270} \[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=\frac {2}{15} \sqrt {\sin ^2(5 x)+1} \csc (5 x)-\frac {1}{15} \sqrt {\sin ^2(5 x)+1} \csc ^3(5 x) \]

[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 270

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 277

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*} \text {integral}& = \frac {1}{5} \text {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} \text {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] (verified)

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.65 \[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=-\frac {1}{15} \csc (5 x) \left (-2+\csc ^2(5 x)\right ) \sqrt {1+\sin ^2(5 x)} \]

[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])

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\sqrt {1+\sin \left (5 x \right )^{2}}}{15 \sin \left (5 x \right )^{3}}+\frac {2 \sqrt {1+\sin \left (5 x \right )^{2}}}{15 \sin \left (5 x \right )}\)	\(38\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33 \[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=-\frac {2 \, {\left (\cos \left (5 \, x\right )^{2} - 1\right )} \sin \left (5 \, x\right ) - {\left (2 \, \cos \left (5 \, x\right )^{2} - 1\right )} \sqrt {-\cos \left (5 \, x\right )^{2} + 2}}{15 \, {\left (\cos \left (5 \, x\right )^{2} - 1\right )} \sin \left (5 \, x\right )} \]

[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))

Sympy [F]

\[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=\int \frac {\cot {\left (5 x \right )} \csc ^{3}{\left (5 x \right )}}{\sqrt {\sin ^{2}{\left (5 x \right )} + 1}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=\frac {2 \, \sqrt {\sin \left (5 \, x\right )^{2} + 1}}{15 \, \sin \left (5 \, x\right )} - \frac {\sqrt {\sin \left (5 \, x\right )^{2} + 1}}{15 \, \sin \left (5 \, x\right )^{3}} \]

[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

Giac [F]

\[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=\int { \frac {\cot \left (5 \, x\right ) \csc \left (5 \, x\right )^{3}}{\sqrt {\sin \left (5 \, x\right )^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 26.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.65 \[ \int \frac {\cot (5 x) \csc ^3(5 x)}{\sqrt {1+\sin ^2(5 x)}} \, dx=\frac {\sqrt {{\sin \left (5\,x\right )}^2+1}\,\left (2\,{\sin \left (5\,x\right )}^2-1\right )}{15\,{\sin \left (5\,x\right )}^3} \]

[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)