3.60 \(\int \cot (x) \sqrt{a+b \cot ^4(x)} \, dx\)
Optimal. Leaf size=90 \[ -\frac{1}{2} \sqrt{a+b \cot ^4(x)}+\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right ) \]
[Out]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]])/2 + (Sqrt[a + b]*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a +
b]*Sqrt[a + b*Cot[x]^4])])/2 - Sqrt[a + b*Cot[x]^4]/2
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Rubi [A] time = 0.13532, antiderivative size = 90, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used =
{3670, 1248, 735, 844, 217, 206, 725} \[ -\frac{1}{2} \sqrt{a+b \cot ^4(x)}+\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Cot[x]*Sqrt[a + b*Cot[x]^4],x]
[Out]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]])/2 + (Sqrt[a + b]*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a +
b]*Sqrt[a + b*Cot[x]^4])])/2 - Sqrt[a + b*Cot[x]^4]/2
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 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) \sqrt{a+b \cot ^4(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x \sqrt{a+b x^4}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{1+x} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac{1}{2} \sqrt{a+b \cot ^4(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{a-b x}{(1+x) \sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )\\ &=-\frac{1}{2} \sqrt{a+b \cot ^4(x)}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )-\frac{1}{2} (a+b) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\cot ^2(x)\right )\\ &=-\frac{1}{2} \sqrt{a+b \cot ^4(x)}-\frac{1}{2} (-a-b) \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}{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}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )-\frac{1}{2} \sqrt{a+b \cot ^4(x)}\\ \end{align*}
Mathematica [A] time = 0.150861, size = 86, normalized size = 0.96 \[ \frac{1}{2} \left (-\sqrt{a+b \cot ^4(x)}+\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right )+\sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[x]*Sqrt[a + b*Cot[x]^4],x]
[Out]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + Sqrt[a + b]*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*
Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Cot[x]^4])/2
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Maple [A] time = 0.057, size = 139, normalized size = 1.5 \begin{align*} -{\frac{1}{2}\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}}+{\frac{1}{2}\sqrt{b}\ln \left ({( \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b-b){\frac{1}{\sqrt{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 ) }+{\frac{1}{2}\sqrt{a+b}\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(x)*(a+b*cot(x)^4)^(1/2),x)
[Out]
-1/2*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2)+1/2*b^(1/2)*ln(((1+cot(x)^2)*b-b)/b^(1/2)+((1+cot(x)^2)^2*b
-2*(1+cot(x)^2)*b+a+b)^(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))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cot \left (x\right )^{4} + a} \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)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*cot(x)^4 + a)*cot(x), x)
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Fricas [B] time = 3.62138, size = 2849, normalized size = 31.66 \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)^(1/2),x, algorithm="fricas")
[Out]
[1/4*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*co
s(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)) + 1/4*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)/(c
os(2*x)^2 - 2*cos(2*x) + 1)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(
2*x) + 1)), 1/2*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)) + 1/4*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)) - 1/2*sqrt(((a + b)*cos(2*x)^2
- 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)), -1/2*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))) + 1/4*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)) - 1/2*sqrt(((a + b)*
cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)), -1/2*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))) + 1/2*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))*(co
s(2*x) - 1)/(b*cos(2*x) + b)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(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 \sqrt{a + b \cot ^{4}{\left (x \right )}} \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)**(1/2),x)
[Out]
Integral(sqrt(a + b*cot(x)**4)*cot(x), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cot \left (x\right )^{4} + a} \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)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*cot(x)^4 + a)*cot(x), x)