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

Optimal. Leaf size=75 \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right )-b \sqrt{a+b \cot ^2(x)}-(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.127804, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3670, 446, 84, 156, 63, 208} \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right )-b \sqrt{a+b \cot ^2(x)}-(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[x]^2)^(3/2)*Tan[x],x]

[Out]

a^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - (a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - b*Sq
rt[a + b*Cot[x]^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 446

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

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[(f*(e + f*x)^(p -
1))/(b*d*(p - 1)), x] + Dist[1/(b*d), Int[((b*d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*(e + f*x)^(p -
2))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]

Rubi steps

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

Mathematica [A]  time = 0.0696176, size = 75, normalized size = 1. \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right )-b \sqrt{a+b \cot ^2(x)}-(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.226, size = 2628, normalized size = 35. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*sin(x)^2*((cos(x)^2*a-b*cos(x)^2-a)/(cos(x)^2-1))^(3/2)*(2*2^(1/2)*(
1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/
2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/2)*(a
-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*
a*b*sin(x)*cos(x)-2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)
)^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticF
((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*
a^2-8*a*b+b^2)/b^2)^(1/2))*b^2*sin(x)*cos(x)+2*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*
cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a
)/(cos(x)+1))^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^
(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)*co
s(x)-4*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2
/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticPi((-1+cos(x
))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+
2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a*b*sin(x)*cos(x)+2*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b
)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(
a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/
2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-
2*a+b)/b)^(1/2))*b^2*sin(x)*cos(x)-2*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*c
os(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+
1))^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b
)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)*cos(x)+2*2^(1
/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*
a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/
2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1
/2))*a*b*sin(x)-2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^
(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticF((
-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^
2-8*a*b+b^2)/b^2)^(1/2))*b^2*sin(x)+2*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*
cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)
+1))^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a
+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)-4*2^(1/2)*(
1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/
2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(
a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/
((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a*b*sin(x)+2*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(
1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*c
os(x)-a)/(cos(x)+1))^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)
*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*b^2*si
n(x)-2*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2
/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1))^(1/2)*EllipticPi((-1+cos(x
))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2
*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)+cos(x)^2*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^
(1/2)*a*b-cos(x)^2*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*b^2-((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*a*b)/(c
os(x)^2*a-b*cos(x)^2-a)^2

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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}} \tan \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 7.13679, size = 1531, normalized size = 20.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - 1/4*(a - b)^(3/2)*lo
g(-((8*a^2 - 8*a*b + b^2)*tan(x)^4 + 2*(4*a*b - 3*b^2)*tan(x)^2 + b^2 + 4*((2*a - b)*tan(x)^4 + b*tan(x)^2)*sq
rt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2), -
sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/(a*tan(x)^2 + b)) - 1/4*(a - b)^(3/2)*log(
-((8*a^2 - 8*a*b + b^2)*tan(x)^4 + 2*(4*a*b - 3*b^2)*tan(x)^2 + b^2 + 4*((2*a - b)*tan(x)^4 + b*tan(x)^2)*sqrt
(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2), 1/2
*(-a + b)^(3/2)*arctan(-2*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/((2*a - b)*tan(x)^2 + b)) + 1/
2*a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - b*sqrt((a*tan(x)^2 + b)
/tan(x)^2), -sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/(a*tan(x)^2 + b)) + 1/2*(-a +
b)^(3/2)*arctan(-2*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/((2*a - b)*tan(x)^2 + b)) - b*sqrt((
a*tan(x)^2 + b)/tan(x)^2)]

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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}} \tan{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out