3.64 \(\int \frac{\cot (x)}{(a+b \cot ^4(x))^{5/2}} \, dx\)
Optimal. Leaf size=117 \[ -\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
[Out]
ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a + b*Cot[x]^2)/(6*a*(a + b)
*(a + b*Cot[x]^4)^(3/2)) - (3*a^2 + b*(5*a + 2*b)*Cot[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Cot[x]^4])
________________________________________________________________________________________
Rubi [A] time = 0.190317, antiderivative size = 117, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used =
{3670, 1248, 741, 823, 12, 725, 206} \[ -\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]
[Out]
ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a + b*Cot[x]^2)/(6*a*(a + b)
*(a + b*Cot[x]^4)^(3/2)) - (3*a^2 + b*(5*a + 2*b)*Cot[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Cot[x]^4])
Rule 3670
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rule 1248
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]
Rule 741
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 823
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 725
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1+x) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-3 a-2 b-2 b x}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot ^2(x)\right )}{6 a (a+b)}\\ &=-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}-\frac{\operatorname{Subst}\left (\int \frac{3 a^2 b}{(1+x) \sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )}{6 a^2 b (a+b)^2}\\ &=-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )}{2 (a+b)^2}\\ &=-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b-x^2} \, dx,x,\frac{a-b \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )}{2 (a+b)^2}\\ &=\frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac{a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.700434, size = 114, normalized size = 0.97 \[ \frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac{3 a^2 b \cot ^4(x)+a^2 (4 a+b)+b^2 (5 a+2 b) \cot ^6(x)+3 a b (2 a+b) \cot ^2(x)}{6 a^2 (a+b)^2 \left (a+b \cot ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]
[Out]
ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a^2*(4*a + b) + 3*a*b*(2*a +
b)*Cot[x]^2 + 3*a^2*b*Cot[x]^4 + b^2*(5*a + 2*b)*Cot[x]^6)/(6*a^2*(a + b)^2*(a + b*Cot[x]^4)^(3/2))
________________________________________________________________________________________
Maple [B] time = 0.063, size = 602, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(x)/(a+b*cot(x)^4)^(5/2),x)
[Out]
-1/24/((-a*b)^(1/2)-b)/a/(-a*b)^(1/2)/(cot(x)^2+(-a*b)^(1/2)/b)^2*((cot(x)^2+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2
)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)+1/24/((-a*b)^(1/2)-b)/a^2/(cot(x)^2+(-a*b)^(1/2)/b)*((cot(x)^2+(-a*b)^(1/2)
/b)^2*b-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)+1/2*b^2/((-a*b)^(1/2)+b)^2/((-a*b)^(1/2)-b)^2/(a+b)^(1
/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))-1/
8*(2*(-a*b)^(1/2)+b)/((-a*b)^(1/2)+b)^2/a^2/(cot(x)^2-(-a*b)^(1/2)/b)*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^
(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1/2)+1/8*(2*(-a*b)^(1/2)-b)/((-a*b)^(1/2)-b)^2/a^2/(cot(x)^2+(-a*b)^(1/2)/b)
*((cot(x)^2+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)-1/24/((-a*b)^(1/2)+b)/a/(-a*b)
^(1/2)/(cot(x)^2-(-a*b)^(1/2)/b)^2*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1
/2)-1/24/((-a*b)^(1/2)+b)/a^2/(cot(x)^2-(-a*b)^(1/2)/b)*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(cot(x)^
2-(-a*b)^(1/2)/b))^(1/2)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )}{{\left (b \cot \left (x\right )^{4} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="maxima")
[Out]
integrate(cot(x)/(b*cot(x)^4 + a)^(5/2), x)
________________________________________________________________________________________
Fricas [B] time = 4.73608, size = 3077, normalized size = 26.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2 - 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3
*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*c
os(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(
2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 4*((2*a^4 + a^3*
b - 5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 + 5*a*b^3 + b^4 - 2*(4*a^4 + 2*a^3*b -
a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12*(a^4 + a^3*b)*cos(2*x)^2 - 2*(4*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3
- b^4)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^7 +
5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^
4 + a^2*b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)^3 + 2*(3*a^
7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*cos(2*x)^2 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4
*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)), -1/6*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2
- 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqr
t(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)
*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2
- b^2)*cos(2*x))) + 2*((2*a^4 + a^3*b - 5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 +
5*a*b^3 + b^4 - 2*(4*a^4 + 2*a^3*b - a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12*(a^4 + a^3*b)*cos(2*x)^2 - 2*(4
*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3 - b^4)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(
cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b
+ 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^
3*b^4 - a^2*b^5)*cos(2*x)^3 + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*cos(2*x)^2 -
4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x))]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(x)/(a+b*cot(x)**4)**(5/2),x)
[Out]
Integral(cot(x)/(a + b*cot(x)**4)**(5/2), x)
________________________________________________________________________________________
Giac [B] time = 14.2831, size = 564, normalized size = 4.82 \begin{align*} \frac{{\left (a^{5} b^{4} + a^{4} b^{5}\right )} \log \left ({\left | -{\left (\sqrt{a + b} \cos \left (x\right )^{2} - \sqrt{a \cos \left (x\right )^{4} + b \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}\right )}{\left (a + b\right )} + \sqrt{a + b} a \right |}\right )}{2 \,{\left (a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}\right )} \sqrt{a + b}} + \frac{{\left (2 \,{\left (\frac{{\left (2 \, a^{6} b^{4} + a^{5} b^{5} - 5 \, a^{4} b^{6} - 5 \, a^{3} b^{7} - a^{2} b^{8}\right )} \cos \left (x\right )^{2}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}} - \frac{3 \,{\left (2 \, a^{6} b^{4} + a^{5} b^{5} - 2 \, a^{4} b^{6} - a^{3} b^{7}\right )}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}\right )} \cos \left (x\right )^{2} + \frac{3 \,{\left (4 \, a^{6} b^{4} + 3 \, a^{5} b^{5} - 2 \, a^{4} b^{6} - a^{3} b^{7}\right )}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}\right )} \cos \left (x\right )^{2} - \frac{4 \, a^{6} b^{4} + 5 \, a^{5} b^{5} + a^{4} b^{6}}{a^{7} b^{4} + 3 \, a^{6} b^{5} + 3 \, a^{5} b^{6} + a^{4} b^{7}}}{6 \,{\left (a \cos \left (x\right )^{4} + b \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="giac")
[Out]
1/2*(a^5*b^4 + a^4*b^5)*log(abs(-(sqrt(a + b)*cos(x)^2 - sqrt(a*cos(x)^4 + b*cos(x)^4 - 2*a*cos(x)^2 + a))*(a
+ b) + sqrt(a + b)*a))/((a^7*b^4 + 3*a^6*b^5 + 3*a^5*b^6 + a^4*b^7)*sqrt(a + b)) + 1/6*((2*((2*a^6*b^4 + a^5*b
^5 - 5*a^4*b^6 - 5*a^3*b^7 - a^2*b^8)*cos(x)^2/(a^7*b^4 + 3*a^6*b^5 + 3*a^5*b^6 + a^4*b^7) - 3*(2*a^6*b^4 + a^
5*b^5 - 2*a^4*b^6 - a^3*b^7)/(a^7*b^4 + 3*a^6*b^5 + 3*a^5*b^6 + a^4*b^7))*cos(x)^2 + 3*(4*a^6*b^4 + 3*a^5*b^5
- 2*a^4*b^6 - a^3*b^7)/(a^7*b^4 + 3*a^6*b^5 + 3*a^5*b^6 + a^4*b^7))*cos(x)^2 - (4*a^6*b^4 + 5*a^5*b^5 + a^4*b^
6)/(a^7*b^4 + 3*a^6*b^5 + 3*a^5*b^6 + a^4*b^7))/(a*cos(x)^4 + b*cos(x)^4 - 2*a*cos(x)^2 + a)^(3/2)