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

Optimal. Leaf size=92 \[ \frac{(a-2 b) \tan (x) \sqrt{a+b \cot ^2(x)}}{a^2 (a-b)}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{(a-b)^{3/2}}+\frac{b \tan (x)}{a (a-b) \sqrt{a+b \cot ^2(x)}} \]

[Out]

ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) + (b*Tan[x])/(a*(a - b)*Sqrt[a + b*Cot[x]^2])
+ ((a - 2*b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(a^2*(a - b))

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Rubi [A]  time = 0.15441, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {3670, 472, 583, 12, 377, 203} \[ \frac{(a-2 b) \tan (x) \sqrt{a+b \cot ^2(x)}}{a^2 (a-b)}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{(a-b)^{3/2}}+\frac{b \tan (x)}{a (a-b) \sqrt{a+b \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) + (b*Tan[x])/(a*(a - b)*Sqrt[a + b*Cot[x]^2])
+ ((a - 2*b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(a^2*(a - b))

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 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 583

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 377

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 6.90298, size = 674, normalized size = 7.33 \[ \frac{\sin ^2(x) \tan (x) \left (\frac{8 b^2 (a-b) \cos ^2(x) \cot ^4(x) \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{7}{2}\right \},\frac{(a-b) \cos ^2(x)}{a}\right )}{15 a^3}+\frac{16 b (a-b) \cos ^2(x) \cot ^2(x) \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{7}{2}\right \},\frac{(a-b) \cos ^2(x)}{a}\right )}{15 a^2}+\frac{8 (a-b) \cos ^2(x) \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{7}{2}\right \},\frac{(a-b) \cos ^2(x)}{a}\right )}{15 a}+\frac{8 b^2 (a-b) \cos ^2(x) \cot ^4(x) \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \cos ^2(x)}{a}\right )}{5 a^3}+\frac{8 b (a-b) \cos ^2(x) \cot ^2(x) \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \cos ^2(x)}{a}\right )}{3 a^2}+\frac{16 (a-b) \cos ^2(x) \text{Hypergeometric2F1}\left (2,2,\frac{7}{2},\frac{(a-b) \cos ^2(x)}{a}\right )}{15 a}-\frac{8 b^2 \cot ^4(x) \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )}{a^2 \left (\frac{(a-b) \cos ^2(x)}{a}\right )^{3/2} \sqrt{\frac{\sin ^2(x) \left (a+b \cot ^2(x)\right )}{a}}}+\frac{8 b^2 \cot ^4(x) \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )}{a^2 \sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}}+\frac{12 b \cot ^2(x) \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )}{a \sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )}{\sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}}+\frac{8 b^2 \cot ^2(x) \csc ^2(x)}{a (a-b)}+\frac{12 b \csc ^2(x)}{a-b}+\frac{3 a \sec ^2(x)}{a-b}-\frac{12 b \cot ^2(x) \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )}{a \left (\frac{(a-b) \cos ^2(x)}{a}\right )^{3/2} \sqrt{\frac{\sin ^2(x) \left (a+b \cot ^2(x)\right )}{a}}}-\frac{3 \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )}{\left (\frac{(a-b) \cos ^2(x)}{a}\right )^{3/2} \sqrt{\frac{\sin ^2(x) \left (a+b \cot ^2(x)\right )}{a}}}\right )}{a \sqrt{a+b \cot ^2(x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sin[x]^2*((12*b*Csc[x]^2)/(a - b) + (8*b^2*Cot[x]^2*Csc[x]^2)/(a*(a - b)) + (16*(a - b)*Cos[x]^2*Hypergeometr
ic2F1[2, 2, 7/2, ((a - b)*Cos[x]^2)/a])/(15*a) + (8*(a - b)*b*Cos[x]^2*Cot[x]^2*Hypergeometric2F1[2, 2, 7/2, (
(a - b)*Cos[x]^2)/a])/(3*a^2) + (8*(a - b)*b^2*Cos[x]^2*Cot[x]^4*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[x]^
2)/a])/(5*a^3) + (8*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a) + (1
6*(a - b)*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a^2) + (8*(a -
b)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a^3) + (3*a*Sec[x]
^2)/(a - b) - (3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/((((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Cot[x]^2)*Sin[
x]^2)/a]) - (12*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*(((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Co
t[x]^2)*Sin[x]^2)/a]) - (8*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*(((a - b)*Cos[x]^2)/a)^(3/2)*
Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/Sqrt[((a - b)*Cos[x]^2*(a + b*Co
t[x]^2)*Sin[x]^2)/a^2] + (12*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*Sqrt[((a - b)*Cos[x]^2*(a + b*C
ot[x]^2)*Sin[x]^2)/a^2]) + (8*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*Sqrt[((a - b)*Cos[x]^2*(a
+ b*Cot[x]^2)*Sin[x]^2)/a^2]))*Tan[x])/(a*Sqrt[a + b*Cot[x]^2])

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Maple [B]  time = 0.204, size = 421, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/(-a+b)^(1/2)/((a*(a-b))^(1/2)-a+b)/a^2/((a*(a-b))^(1/2)+a-b)*(-1+cos(x))^2*(cos(x)+1)^2*(cos(x)^2*a-b*cos(x
)^2-a)*(cos(x)^2*(-(cos(x)^2*a-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(cos(x)^2*a-b*cos(
x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(cos(x)^2*a-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/
2))*a^2-cos(x)^2*(-a+b)^(1/2)*a^2+2*cos(x)^2*(-a+b)^(1/2)*a*b-2*cos(x)^2*(-a+b)^(1/2)*b^2+cos(x)*(-(cos(x)^2*a
-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(cos(x)^2*a-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*
a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(cos(x)^2*a-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^2+(-a+b)^(1/2)*a^2-(-a+b
)^(1/2)*a*b)/cos(x)/((cos(x)^2*a-b*cos(x)^2-a)/(cos(x)^2-1))^(3/2)/sin(x)^7

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

[Out]

Timed out

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Fricas [B]  time = 2.69403, size = 945, normalized size = 10.27 \begin{align*} \left [\frac{{\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt{-a + b} \log \left (-\frac{a^{2} \tan \left (x\right )^{4} - 2 \,{\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} - 4 \,{\left (a \tan \left (x\right )^{3} -{\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt{-a + b} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 4 \,{\left ({\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{3} +{\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \tan \left (x\right )\right )} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{4 \,{\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}, \frac{{\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt{a - b} \arctan \left (\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + 2 \,{\left ({\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{3} +{\left (a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right )} \tan \left (x\right )\right )} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{2 \,{\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^
2 - 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1
)) + 4*((a^3 - 2*a^2*b + a*b^2)*tan(x)^3 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/
(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), 1/2*((a^3*tan(x)^2 + a^2*b)*sqrt(a - b)*ar
ctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*((a^3 - 2*a^2*b + a*b^2)
*tan(x)^3 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^4*b - 2*a^3*b^2 + a^2*b^3 +
(a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.47444, size = 228, normalized size = 2.48 \begin{align*} \frac{{\left (a^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{a - b}}\right ) - \sqrt{a - b} a \sqrt{b} + 2 \, \sqrt{a - b} b^{\frac{3}{2}}\right )} \mathrm{sgn}\left (\tan \left (x\right )\right )}{\sqrt{a - b} a^{3} - \sqrt{a - b} a^{2} b} - \frac{b^{2}}{{\left (a^{3} \mathrm{sgn}\left (\tan \left (x\right )\right ) - a^{2} b \mathrm{sgn}\left (\tan \left (x\right )\right )\right )} \sqrt{a \tan \left (x\right )^{2} + b}} - \frac{\arctan \left (\frac{\sqrt{a \tan \left (x\right )^{2} + b}}{\sqrt{a - b}}\right )}{{\left (a \mathrm{sgn}\left (\tan \left (x\right )\right ) - b \mathrm{sgn}\left (\tan \left (x\right )\right )\right )} \sqrt{a - b}} + \frac{\sqrt{a \tan \left (x\right )^{2} + b}}{a^{2} \mathrm{sgn}\left (\tan \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*arctan(sqrt(b)/sqrt(a - b)) - sqrt(a - b)*a*sqrt(b) + 2*sqrt(a - b)*b^(3/2))*sgn(tan(x))/(sqrt(a - b)*a^3
- sqrt(a - b)*a^2*b) - b^2/((a^3*sgn(tan(x)) - a^2*b*sgn(tan(x)))*sqrt(a*tan(x)^2 + b)) - arctan(sqrt(a*tan(x
)^2 + b)/sqrt(a - b))/((a*sgn(tan(x)) - b*sgn(tan(x)))*sqrt(a - b)) + sqrt(a*tan(x)^2 + b)/(a^2*sgn(tan(x)))