3.395 \(\int \cot (x) (a+b \tan ^4(x))^{3/2} \, dx\)
Optimal. Leaf size=155 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right )+\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right ) \]
[Out]
(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/4 + ((a + b)^(3/2)*ArcTanh[(a - b*Tan[x
]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 - (a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2 + (a*Sqrt[a +
b*Tan[x]^4])/2 - ((2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/4
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Rubi [A] time = 0.266859, antiderivative size = 155, normalized size of antiderivative = 1.,
number of steps used = 13, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used =
{3670, 1252, 896, 266, 50, 63, 208, 815, 844, 217, 206, 725} \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right )+\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[Cot[x]*(a + b*Tan[x]^4)^(3/2),x]
[Out]
(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/4 + ((a + b)^(3/2)*ArcTanh[(a - b*Tan[x
]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 - (a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2 + (a*Sqrt[a +
b*Tan[x]^4])/2 - ((2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/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 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 50
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]
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 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 \tan ^4(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b x^4\right )^{3/2}}{x \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{x (1+x)} \, dx,x,\tan ^2(x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a-b x) \sqrt{a+b x^2}}{1+x} \, dx,x,\tan ^2(x)\right )\right )+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x} \, dx,x,\tan ^2(x)\right )\\ &=-\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\tan ^4(x)\right )-\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,\tan ^2(x)\right )}{4 b}\\ &=\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan ^4(x)\right )-\frac{1}{2} (a+b)^2 \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )+\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tan ^2(x)\right )\\ &=\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}+\frac{a^2 \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} (a+b)^2 \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 )+\frac{1}{4} (b (3 a+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}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )-\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right )+\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt{a+b \tan ^4(x)}\\ \end{align*}
Mathematica [A] time = 2.99893, size = 190, normalized size = 1.23 \[ \frac{1}{4} \left (-2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right )+b \tan ^2(x) \sqrt{a+b \tan ^4(x)}-2 b \sqrt{a+b \tan ^4(x)}+2 \sqrt{b} (a+b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right )+2 (a+b)^{3/2} \tanh ^{-1}\left (\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right )+\frac{\sqrt{a} \sqrt{b} \sqrt{a+b \tan ^4(x)} \sinh ^{-1}\left (\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a}}\right )}{\sqrt{\frac{b \tan ^4(x)}{a}+1}}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[x]*(a + b*Tan[x]^4)^(3/2),x]
[Out]
(2*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + 2*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)
/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - 2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]] - 2*b*Sqrt[a + b*Tan[x]
^4] + b*Tan[x]^2*Sqrt[a + b*Tan[x]^4] + (Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sqrt[a + b*Tan[x]
^4])/Sqrt[1 + (b*Tan[x]^4)/a])/4
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Maple [F] time = 0.138, size = 0, normalized size = 0. \begin{align*} \int \cot \left ( x \right ) \left ( a+b \left ( \tan \left ( x \right ) \right ) ^{4} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(x)*(a+b*tan(x)^4)^(3/2),x)
[Out]
int(cot(x)*(a+b*tan(x)^4)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \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*tan(x)^4)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*tan(x)^4 + a)^(3/2)*cot(x), x)
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Fricas [A] time = 131.972, size = 3501, normalized size = 22.59 \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)^(3/2),x, algorithm="fricas")
[Out]
[1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*(a + b)^(3/2)*l
og(((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*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4
) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), -1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b
)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*t
an(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*
tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sq
rt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan
(x)^2 + a) + 1/4*(a + b)^(3/2)*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(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b),
1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) - 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)
*sqrt(-b)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*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(b*tan(x)^4 + a)*(b*tan
(x)^2 - 2*b), 1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/8*(3*a +
2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*a^(3/2)*log((b*tan(x)^4 -
2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*(a + b)*sqr
t(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*
tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan
(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) +
1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/8*(3*a + 2*b)*sqrt(b)
*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b)
, 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a
)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2))
+ 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan ^{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*tan(x)**4)**(3/2),x)
[Out]
Integral((a + b*tan(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 \tan \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*tan(x)^4)^(3/2),x, algorithm="giac")
[Out]
integrate((b*tan(x)^4 + a)^(3/2)*cot(x), x)