3.61 \(\int \cot (x) (a+b \cot ^4(x))^{3/2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac{1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt{a+b \cot ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right ) \]

[Out]

(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]])/4 + ((a + b)^(3/2)*ArcTanh[(a - b*Cot[x
]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])])/2 - ((2*(a + b) - b*Cot[x]^2)*Sqrt[a + b*Cot[x]^4])/4 - (a + b*Cot[x
]^4)^(3/2)/6

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Rubi [A]  time = 0.20171, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {3670, 1248, 735, 815, 844, 217, 206, 725} \[ -\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac{1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt{a+b \cot ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]

[Out]

(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]])/4 + ((a + b)^(3/2)*ArcTanh[(a - b*Cot[x
]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])])/2 - ((2*(a + b) - b*Cot[x]^2)*Sqrt[a + b*Cot[x]^4])/4 - (a + b*Cot[x
]^4)^(3/2)/6

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 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

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 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]

Rubi steps

\begin{align*} \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (a+b x^4\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{1+x} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a-b x) \sqrt{a+b x^2}}{1+x} \, dx,x,\cot ^2(x)\right )\\ &=-\frac{1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt{a+b \cot ^4(x)}-\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{a b (2 a+b)-b^2 (3 a+2 b) x}{(1+x) \sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )}{4 b}\\ &=-\frac{1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt{a+b \cot ^4(x)}-\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac{1}{2} (a+b)^2 \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )+\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )\\ &=-\frac{1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt{a+b \cot ^4(x)}-\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}+\frac{1}{2} (a+b)^2 \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 )+\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )\\ &=\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )-\frac{1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt{a+b \cot ^4(x)}-\frac{1}{6} \left (a+b \cot ^4(x)\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 4.44381, size = 167, normalized size = 1.33 \[ \frac{1}{12} \left (-\sqrt{a+b \cot ^4(x)} \left (8 a+2 b \cot ^4(x)-3 b \cot ^2(x)+6 b\right )+\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \cot ^4(x)} \sinh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a}}\right )}{\sqrt{\frac{b \cot ^4(x)}{a}+1}}+6 (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )+6 \sqrt{b} (a+b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]

[Out]

(6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + 6*(a + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)
/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Cot[x]^4]*(8*a + 6*b - 3*b*Cot[x]^2 + 2*b*Cot[x]^4) + (3*Sqr
t[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Cot[x]^2)/Sqrt[a]]*Sqrt[a + b*Cot[x]^4])/Sqrt[1 + (b*Cot[x]^4)/a])/12

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Maple [B]  time = 0.033, size = 312, normalized size = 2.5 \begin{align*} -{\frac{b \left ( \cot \left ( x \right ) \right ) ^{4}}{6}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{4}}}-{\frac{2\,a}{3}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{4}}}+{\frac{b \left ( \cot \left ( x \right ) \right ) ^{2}}{4}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{4}}}+{\frac{3\,a}{4}\sqrt{b}\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}\sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{4}} \right ) }-{\frac{b}{2}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{4}}}+{\frac{1}{2}{b}^{{\frac{3}{2}}}\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}\sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{4}} \right ) }+{\frac{{a}^{2}}{2}\ln \left ({\frac{1}{1+ \left ( \cot \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{2}b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}}+{ab\ln \left ({\frac{1}{1+ \left ( \cot \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{2}b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}}+{\frac{{b}^{2}}{2}\ln \left ({\frac{1}{1+ \left ( \cot \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{2}b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)*(a+b*cot(x)^4)^(3/2),x)

[Out]

-1/6*b*cot(x)^4*(a+b*cot(x)^4)^(1/2)-2/3*a*(a+b*cot(x)^4)^(1/2)+1/4*b*cot(x)^2*(a+b*cot(x)^4)^(1/2)+3/4*a*b^(1
/2)*ln(cot(x)^2*b^(1/2)+(a+b*cot(x)^4)^(1/2))-1/2*b*(a+b*cot(x)^4)^(1/2)+1/2*b^(3/2)*ln(cot(x)^2*b^(1/2)+(a+b*
cot(x)^4)^(1/2))+1/2/(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))*a^2+1/(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))*a*b+1/2/(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))*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cot \left (x\right )^{4} + a\right )}^{\frac{3}{2}} \cot \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cot(x)^4 + a)^(3/2)*cot(x), x)

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Fricas [B]  time = 4.31602, size = 3946, normalized size = 31.32 \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)^(3/2),x, algorithm="fricas")

[Out]

[1/24*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(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)) + 3*((3*a + 2*b)*cos(2*x)^2
- 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(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)) - 2*(a - 2*b)*cos(2*x) + a + 2*b
)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*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)))/(cos(2*x)^2 - 2*cos(2*x) + 1), 1/12
*(3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-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))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + 3*((a +
b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(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)) - ((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*co
s(2*x) + 8*a + 5*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)))/(co
s(2*x)^2 - 2*cos(2*x) + 1), -1/24*(12*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(-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)))
- 3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(
2*x)^2 - 1)*sqrt(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)) - 2*
(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) + 2*((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*
x) + 8*a + 5*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)))/(cos(2*
x)^2 - 2*cos(2*x) + 1), -1/12*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(-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))) - 3*(
(3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-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))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + ((8*a + 11*b)
*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*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)))/(cos(2*x)^2 - 2*cos(2*x) + 1)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac{3}{2}} \cot{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*(a+b*cot(x)**4)**(3/2),x)

[Out]

Integral((a + b*cot(x)**4)**(3/2)*cot(x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cot \left (x\right )^{4} + a\right )}^{\frac{3}{2}} \cot \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="giac")

[Out]

integrate((b*cot(x)^4 + a)^(3/2)*cot(x), x)