3.27 \(\int \cot ^2(x) (a+b \cot ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{8 \sqrt{b}}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}+(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right ) \]

[Out]

(a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*C
ot[x])/Sqrt[a + b*Cot[x]^2]])/(8*Sqrt[b]) - ((5*a - 4*b)*Cot[x]*Sqrt[a + b*Cot[x]^2])/8 - (b*Cot[x]^3*Sqrt[a +
b*Cot[x]^2])/4

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Rubi [A]  time = 0.229799, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {3670, 477, 582, 523, 217, 206, 377, 203} \[ -\frac{\left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{8 \sqrt{b}}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}+(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2*(a + b*Cot[x]^2)^(3/2),x]

[Out]

(a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*C
ot[x])/Sqrt[a + b*Cot[x]^2]])/(8*Sqrt[b]) - ((5*a - 4*b)*Cot[x]*Sqrt[a + b*Cot[x]^2])/8 - (b*Cot[x]^3*Sqrt[a +
b*Cot[x]^2])/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 477

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*(e*x)
^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(b*e*(m + n*(p + q) + 1)), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 582

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q
+ 1) + 1)), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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])

Rule 377

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2 \left (a (4 a-3 b)+(5 a-4 b) b x^2\right )}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}+\frac{\operatorname{Subst}\left (\int \frac{a (5 a-4 b) b-b \left (3 a^2-12 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (x)\right )}{8 b}\\ &=-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}+(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (x)\right )+\frac{1}{8} \left (-3 a^2+12 a b-8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}+(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )+\frac{1}{8} \left (-3 a^2+12 a b-8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )\\ &=(a-b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )-\frac{\left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{8 \sqrt{b}}-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}\\ \end{align*}

Mathematica [A]  time = 1.14985, size = 253, normalized size = 1.99 \[ \frac{\csc (x) \sqrt{(a-b) \cos (2 x)-a-b} \left (\sqrt{a-b} \left (\sqrt{-b} \cot (x) \csc (x) \sqrt{(a-b) \cos (2 x)-a-b} \left (5 a+2 b \csc ^2(x)-6 b\right )-\sqrt{2} \left (3 a^2-12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right )\right )+8 \sqrt{2} \sqrt{-b} (a-b)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right )\right )}{8 \sqrt{2} \sqrt{-b} \sqrt{a-b} \sqrt{\csc ^2(x) (-((a-b) \cos (2 x)-a-b))}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2*(a + b*Cot[x]^2)^(3/2),x]

[Out]

(Sqrt[-a - b + (a - b)*Cos[2*x]]*Csc[x]*(8*Sqrt[2]*(a - b)^2*Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqr
t[-a - b + (a - b)*Cos[2*x]]] + Sqrt[a - b]*(-(Sqrt[2]*(3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[
x])/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]]*Cot[x]*Csc[x]*(5*a - 6*b + 2*
b*Csc[x]^2))))/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-b]*Sqrt[-((-a - b + (a - b)*Cos[2*x])*Csc[x]^2)])

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Maple [B]  time = 0.026, size = 286, normalized size = 2.3 \begin{align*} -{\frac{\cot \left ( x \right ) }{4} \left ( a+b \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a\cot \left ( x \right ) }{8}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2}}{8}\ln \left ( \cot \left ( x \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{b}}}}+{\frac{b\cot \left ( x \right ) }{2}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}+{\frac{3\,a}{2}\sqrt{b}\ln \left ( \cot \left ( x \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}} \right ) }-{b}^{{\frac{3}{2}}}\ln \left ( \cot \left ( x \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}} \right ) +{\frac{1}{a-b}\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( x \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) }-2\,{\frac{a\sqrt{{b}^{4} \left ( a-b \right ) }}{b \left ( a-b \right ) }\arctan \left ({\frac{ \left ( a-b \right ){b}^{2}\cot \left ( x \right ) }{\sqrt{{b}^{4} \left ( a-b \right ) }\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}} \right ) }+{\frac{{a}^{2}}{ \left ( a-b \right ){b}^{2}}\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( x \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2*(a+b*cot(x)^2)^(3/2),x)

[Out]

-1/4*cot(x)*(a+b*cot(x)^2)^(3/2)-3/8*a*cot(x)*(a+b*cot(x)^2)^(1/2)-3/8*a^2/b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(
x)^2)^(1/2))+1/2*b*cot(x)*(a+b*cot(x)^2)^(1/2)+3/2*b^(1/2)*a*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))-b^(3/2)*l
n(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^
2)^(1/2)*cot(x))-2*a/b*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))
+a^2*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cot \left (x\right )^{2} + a\right )}^{\frac{3}{2}} \cot \left (x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2*(a+b*cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cot(x)^2 + a)^(3/2)*cot(x)^2, x)

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Fricas [B]  time = 2.01435, size = 2689, normalized size = 21.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2*(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(8*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*co
s(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*co
s(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) -
a - 2*b)/(cos(2*x) - 1))*sin(2*x) + 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a
- b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*((3*a^2 - 12*a*b + 8*b^2 - (3*a^2 -
12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b
*cos(2*x) + b))*sin(2*x) - 4*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a +
b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b - 6*b^2)
*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/
16*(16*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos
(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)
*cos(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x
) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) - 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt((
(a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*(8*(a*b - b^2 - (a*b - b^2)*cos(2
*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x
) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan(sqrt(-b)*s
qrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b
- 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*
x))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2*(a+b*cot(x)**2)**(3/2),x)

[Out]

Integral((a + b*cot(x)**2)**(3/2)*cot(x)**2, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2*(a+b*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

Timed out