3.1.52 \(\int \frac {\tan (x)}{(a+b \cot ^2(x))^{3/2}} \, dx\) [52]

Optimal. Leaf size=84 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}} \]

[Out]

arctanh((a+b*cot(x)^2)^(1/2)/a^(1/2))/a^(3/2)-arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)+b/a/(a-b)/
(a+b*cot(x)^2)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 457, 87, 162, 65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[x]/(a + b*Cot[x]^2)^(3/2),x]

[Out]

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

Rule 65

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 87

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

Rule 162

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 214

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

Rule 457

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 3751

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]))

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.06, size = 75, normalized size = 0.89 \begin {gather*} \frac {a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \cot ^2(x)}{a-b}\right )+(-a+b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \cot ^2(x)}{a}\right )}{a (a-b) \sqrt {a+b \cot ^2(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b
*Cot[x]^2)/a])/(a*(a - b)*Sqrt[a + b*Cot[x]^2])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.66, size = 962, normalized size = 11.45

method	result	size
default	\(-\frac {\left (\left (\cos ^{2}\left (x \right )\right ) a -b \left (\cos ^{2}\left (x \right )\right )-a \right ) \left (\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticF \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, \sqrt {\frac {8 a^{\frac {3}{2}} \sqrt {a -b}-4 \sqrt {a}\, \sqrt {a -b}\, b +8 a^{2}-8 a b +b^{2}}{b^{2}}}\right ) b \sin \left (x \right )+2 \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticPi \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, \frac {b}{2 \sqrt {a}\, \sqrt {a -b}-2 a +b}, \frac {\sqrt {-\frac {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}{b}}}{\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}\right ) a \sin \left (x \right )-2 \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticPi \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, \frac {b}{2 \sqrt {a}\, \sqrt {a -b}-2 a +b}, \frac {\sqrt {-\frac {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}{b}}}{\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}\right ) b \sin \left (x \right )-2 \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticPi \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, -\frac {b}{2 \sqrt {a}\, \sqrt {a -b}-2 a +b}, \frac {\sqrt {-\frac {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}{b}}}{\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}\right ) a \sin \left (x \right )+\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}\, b \cos \left (x \right )-\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}\, b \right )}{\left (-1+\cos \left (x \right )\right ) \sin \left (x \right )^{2} \left (\frac {\left (\cos ^{2}\left (x \right )\right ) a -b \left (\cos ^{2}\left (x \right )\right )-a}{\cos ^{2}\left (x \right )-1}\right )^{\frac {3}{2}} \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}\, \left (a -b \right ) a}\)	\(962\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(x)^2*a-b*cos(x)^2-a)*(2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)-a^(1/2)*(a-b)^(1/2)+a)/(cos
(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a^(1/2)*(a-b)^(1/2)-a)/(cos(x)+1)/b)^(1/2)*E
llipticF((-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*sin(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)-a^(1/2)*(
a-b)^(1/2)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a^(1/2)*(a-b)^(1/2)-a)/(co
s(x)+1)/b)^(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*sin(x)-2*2^(1/2
)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)-a^(1/2)*(a-b)^(1/2)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2
)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a^(1/2)*(a-b)^(1/2)-a)/(cos(x)+1)/b)^(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*sin(x)-2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)-a^(
1/2)*(a-b)^(1/2)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a^(1/2)*(a-b)^(1/2)-
a)/(cos(x)+1)/b)^(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*sin(x)+(
(2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*b*cos(x)-((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*b)/(-1+cos(x))/sin(x)^
2/((cos(x)^2*a-b*cos(x)^2-a)/(cos(x)^2-1))^(3/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/(a-b)/a

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (70) = 140\).
time = 2.89, size = 863, normalized size = 10.27 \begin {gather*} \left [\frac {2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt {a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b\right ) - {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {a - b} \log \left (\frac {{\left (2 \, a - b\right )} \tan \left (x\right )^{2} + 2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2} + 1}\right )}{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 )}}, \frac {2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} - 2 \, {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a - b}\right ) + {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt {a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b\right )}{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 )}}, \frac {2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} - 2 \, {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a}\right ) - {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {a - b} \log \left (\frac {{\left (2 \, a - b\right )} \tan \left (x\right )^{2} + 2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2} + 1}\right )}{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 )}}, \frac {{\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} - {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a}\right ) - {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a - b}\right )}{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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a
*b^2)*tan(x)^2)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - (a^3*tan(
x)^2 + a^2*b)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 + 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b
)/(tan(x)^2 + 1)))/(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*(2*(a^2*b - a*b^2)*
sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - 2*(a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*
tan(x)^2 + b)/tan(x)^2)/(a - b)) + (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(a)*log(2*a*
tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b))/(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*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - 2*(a^2*b - 2*a*b^2
+ b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - (a^3*
tan(x)^2 + a^2*b)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 + 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2
+ b)/(tan(x)^2 + 1)))/(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), ((a^2*b - a*b^2)*sq
rt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(-a)*a
rctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - (a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*s
qrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)))/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan {\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (70) = 140\).
time = 0.49, size = 340, normalized size = 4.05 \begin {gather*} -\frac {{\left (2 \, \sqrt {a - b} a^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 4 \, \sqrt {a - b} a b \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + 2 \, \sqrt {a - b} b^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + \sqrt {-a^{2} + a b} \sqrt {a - b} a \log \left (b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {-a^{2} + a b} a^{3} - 2 \, \sqrt {-a^{2} + a b} a^{2} b + \sqrt {-a^{2} + a b} a b^{2}\right )}} + \frac {\frac {\sqrt {a - b} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {2 \, b \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a^{2} - a b\right )}} + \frac {2 \, \sqrt {a - b} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(a - b)*a^2*arctan(-(a - b)/sqrt(-a^2 + a*b)) - 4*sqrt(a - b)*a*b*arctan(-(a - b)/sqrt(-a^2 + a*b)
) + 2*sqrt(a - b)*b^2*arctan(-(a - b)/sqrt(-a^2 + a*b)) + sqrt(-a^2 + a*b)*sqrt(a - b)*a*log(b))*sgn(sin(x))/(
sqrt(-a^2 + a*b)*a^3 - 2*sqrt(-a^2 + a*b)*a^2*b + sqrt(-a^2 + a*b)*a*b^2) + 1/2*(sqrt(a - b)*log((sqrt(a - b)*
sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2)/(a^2 - 2*a*b + b^2) + 2*b*sin(x)/(sqrt(a*sin(x)^2 - b*sin(x)^2
+ b)*(a^2 - a*b)) + 2*sqrt(a - b)*arctan(1/2*((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - 2*a
+ b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a))/sgn(sin(x))

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Mupad [B]
time = 0.48, size = 1451, normalized size = 17.27 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {2\,a^2\,b^8\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}-\frac {12\,a^3\,b^7\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}+\frac {30\,a^4\,b^6\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}-\frac {38\,a^5\,b^5\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}+\frac {24\,a^6\,b^4\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}-\frac {6\,a^7\,b^3\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}\right )}{\sqrt {a^3}}-\frac {b}{\left (a\,b-a^2\right )\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}-\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7-2\,a^4\,b^8-28\,a^6\,b^6+32\,a^7\,b^5-18\,a^8\,b^4+4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )\,1{}\mathrm {i}}{{\left (a-b\right )}^3}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8-12\,a^5\,b^7+28\,a^6\,b^6-32\,a^7\,b^5+18\,a^8\,b^4-4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )\,1{}\mathrm {i}}{{\left (a-b\right )}^3}}{2\,a^3\,b^6-6\,a^4\,b^5+6\,a^5\,b^4-2\,a^6\,b^3-\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7-2\,a^4\,b^8-28\,a^6\,b^6+32\,a^7\,b^5-18\,a^8\,b^4+4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )}{{\left (a-b\right )}^3}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8-12\,a^5\,b^7+28\,a^6\,b^6-32\,a^7\,b^5+18\,a^8\,b^4-4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )}{{\left (a-b\right )}^3}}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{{\left (a-b\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(a + b*cot(x)^2)^(3/2),x)

[Out]

atanh((2*a^2*b^8*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5
*b^4 - 6*a^6*b^3)) - (12*a^3*b^7*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*
a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) + (30*a^4*b^6*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 +
30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) - (38*a^5*b^5*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8
- 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) + (24*a^6*b^4*(a + b/tan(x)^2)^(1/2))/((a^3
)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) - (6*a^7*b^3*(a + b/tan(x)^
2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)))/(a^3)^(1/2)
- (atan(((((a - b)^3)^(1/2)*(((a + b/tan(x)^2)^(1/2)*(2*a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a
^7*b^3 - 4*a^8*b^2))/2 + (((a - b)^3)^(1/2)*(12*a^5*b^7 - 2*a^4*b^8 - 28*a^6*b^6 + 32*a^7*b^5 - 18*a^8*b^4 + 4
*a^9*b^3 + ((a + b/tan(x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200
*a^9*b^4 - 88*a^10*b^3 + 16*a^11*b^2))/(4*(a - b)^3)))/(2*(a - b)^3))*1i)/(a - b)^3 + (((a - b)^3)^(1/2)*(((a
+ b/tan(x)^2)^(1/2)*(2*a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a^7*b^3 - 4*a^8*b^2))/2 + (((a - b)
^3)^(1/2)*(2*a^4*b^8 - 12*a^5*b^7 + 28*a^6*b^6 - 32*a^7*b^5 + 18*a^8*b^4 - 4*a^9*b^3 + ((a + b/tan(x)^2)^(1/2)
*((a - b)^3)^(1/2)*(8*a^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200*a^9*b^4 - 88*a^10*b^3 + 16*a^11*b
^2))/(4*(a - b)^3)))/(2*(a - b)^3))*1i)/(a - b)^3)/(2*a^3*b^6 - 6*a^4*b^5 + 6*a^5*b^4 - 2*a^6*b^3 - (((a - b)^
3)^(1/2)*(((a + b/tan(x)^2)^(1/2)*(2*a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a^7*b^3 - 4*a^8*b^2))
/2 + (((a - b)^3)^(1/2)*(12*a^5*b^7 - 2*a^4*b^8 - 28*a^6*b^6 + 32*a^7*b^5 - 18*a^8*b^4 + 4*a^9*b^3 + ((a + b/t
an(x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200*a^9*b^4 - 88*a^10*b
^3 + 16*a^11*b^2))/(4*(a - b)^3)))/(2*(a - b)^3)))/(a - b)^3 + (((a - b)^3)^(1/2)*(((a + b/tan(x)^2)^(1/2)*(2*
a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a^7*b^3 - 4*a^8*b^2))/2 + (((a - b)^3)^(1/2)*(2*a^4*b^8 -
12*a^5*b^7 + 28*a^6*b^6 - 32*a^7*b^5 + 18*a^8*b^4 - 4*a^9*b^3 + ((a + b/tan(x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a
^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200*a^9*b^4 - 88*a^10*b^3 + 16*a^11*b^2))/(4*(a - b)^3)))/(2
*(a - b)^3)))/(a - b)^3))*((a - b)^3)^(1/2)*1i)/(a - b)^3 - b/((a*b - a^2)*(a + b/tan(x)^2)^(1/2))

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