3.391 \(\int \cot (x) \sqrt{a+b \tan ^4(x)} \, dx\)
Optimal. Leaf size=102 \[ \frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right ) \]
[Out]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/2 + (Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a +
b]*Sqrt[a + b*Tan[x]^4])])/2 - (Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2
________________________________________________________________________________________
Rubi [A] time = 0.170897, antiderivative size = 102, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used
= {3670, 1252, 896, 266, 63, 208, 844, 217, 206, 725} \[ \frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Cot[x]*Sqrt[a + b*Tan[x]^4],x]
[Out]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/2 + (Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a +
b]*Sqrt[a + b*Tan[x]^4])])/2 - (Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/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 1252
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
Rule 896
Int[((a_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist[(c*d^2 + a*e^2)/
(e*(e*f - d*g)), Int[(a + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)), Int[(Simp[c*d*f + a*e*g -
c*(e*f - d*g)*x, x]*(a + c*x^2)^(p - 1))/(f + g*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g,
0] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[p] && GtQ[p, 0]
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
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 \tan ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^4}}{x \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x (1+x)} \, dx,x,\tan ^2(x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{a-b x}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )\right )+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan ^4(x)\right )+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )-\frac{1}{2} (a+b) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )\\ &=-\left (\frac{1}{2} (-a-b) \operatorname{Subst}\left (\int \frac{1}{a+b-x^2} \, dx,x,\frac{a-b \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )\right )+\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan ^4(x)}\right )}{2 b}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )\\ &=\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.069431, size = 98, normalized size = 0.96 \[ \frac{1}{2} \left (\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\sqrt{a+b} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[x]*Sqrt[a + b*Tan[x]^4],x]
[Out]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*
Sqrt[a + b*Tan[x]^4])] - Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2
________________________________________________________________________________________
Maple [F] time = 0.16, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( x \right ) \sqrt{a+b \left ( \tan \left ( x \right ) \right ) ^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(x)*(a+b*tan(x)^4)^(1/2),x)
[Out]
int(cot(x)*(a+b*tan(x)^4)^(1/2),x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tan \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*tan(x)^4)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*tan(x)^4 + a)*cot(x), x)
________________________________________________________________________________________
Fricas [A] time = 3.58301, size = 2809, normalized size = 27.54 \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*tan(x)^4)^(1/2),x, algorithm="fricas")
[Out]
[1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a + b)*log(((a*b + 2*b
^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 +
2*tan(x)^2 + 1)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4), -1/2*sqrt(-
b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*ta
n(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4
*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^
4 + a)*sqrt(-a)/a) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a
+ b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^
2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) - 1/2*sqrt(-b)*arc
tan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2
- 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)), 1/2*sqrt(-
a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*ta
n(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)
^4), 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/2*sqrt(-b)*arctan(sqrt(b*
tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan
(x)^4), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sq
rt(-a - b)/(b*tan(x)^2 - a)) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a), 1/
2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)
/(b*tan(x)^2 - a)) - 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2))]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \tan ^{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*tan(x)**4)**(1/2),x)
[Out]
Integral(sqrt(a + b*tan(x)**4)*cot(x), x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tan \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*tan(x)^4)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*tan(x)^4 + a)*cot(x), x)