\(\int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) [5]

Optimal result

Integrand size = 33, antiderivative size = 501 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e} \]

[Out]

1/8*(-4*a*c+3*b^2)*arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/a^(5/2)/e-arc
tanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/e/a^(1/2)-1/2*arctanh(1/2*(a-c+b*co
t(e*x+d)-(a^2-2*a*c+b^2+c^2)^(1/2))*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(a-c-(a^2-2*a*c+b^2+c^2)^(1/
2))^(1/2))*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/2)+1/2*arctanh(1/2*(a-c+b*co
t(e*x+d)+(a^2-2*a*c+b^2+c^2)^(1/2))*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(a-c+(a^2-2*a*c+b^2+c^2)^(1/
2))^(1/2))*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/e*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/2)-3/4*b*(a+b*cot(e*x+d)+c*c
ot(e*x+d)^2)^(1/2)*tan(e*x+d)/a^2/e+1/2*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)^2/a/e

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3782, 6857, 758, 820, 738, 212, 1050, 1044, 214} \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {3 b \tan (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a^2 e}-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\tan ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{2 a e} \]

[In]

Int[Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]

[Out]

-(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(Sqrt[a]*e)) + ((3*b
^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(5/2
)*e) - (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d +
e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sq
rt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^
2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d
+ e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (3*b*Sqrt[a + b*Cot[d + e*x] + c*Co
t[d + e*x]^2]*Tan[d + e*x])/(4*a^2*e) + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(2*a*e)

Rule 212

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

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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 1044

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1050

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]

Rule 3782

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Dist[-f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2
)), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{x^3 \sqrt {a+b x+c x^2}}-\frac {1}{x \sqrt {a+b x+c x^2}}+\frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}}\right ) \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e} \\ & = \frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}-\frac {2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}+\frac {\text {Subst}\left (\int \frac {\frac {3 b}{2}+c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 a e}+\frac {\text {Subst}\left (\int \frac {-b+\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\text {Subst}\left (\int \frac {-b+\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e} \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}-\frac {\left (3 b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{8 a^2 e}+\frac {\left (b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\left (b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e} \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}+\frac {\left (3 b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{4 a^2 e} \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.50 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.84 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\sqrt {a-i b-c} \sqrt {a+i b-c} \left (8 a^2-3 b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 \sqrt {a} \left (2 i a^2 \sqrt {a-i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a+i b-c} \left (2 i a^2 \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a-i b-c} (3 b-2 a \tan (d+e x)) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )\right )}{8 a^{5/2} \sqrt {a-i b-c} \sqrt {a+i b-c} e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \]

[In]

Integrate[Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2],x]

[Out]

-1/8*(Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*(Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*(8*a^2 - 3
*b^2 + 4*a*c)*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + 2*Sqrt
[a]*((2*I)*a^2*Sqrt[a - I*b - c]*ArcTan[(I*b - 2*c + ((2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c +
b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + Sqrt[a + I*b - c]*((2*I)*a^2*ArcTan[(I*b + 2*c + ((2*I)*a + b)*Tan[d +
e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + Sqrt[a - I*b - c]*(3*b - 2*a*Tan[d
+ e*x])*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]))))/(a^(5/2)*Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*e*Sqrt[a
+ b*Cot[d + e*x] + c*Cot[d + e*x]^2])

Maple [F(-1)]

Timed out.

hanged

[In]

int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)

[Out]

int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5247 vs. \(2 (442) = 884\).

Time = 1.63 (sec) , antiderivative size = 10495, normalized size of antiderivative = 20.95 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \]

[In]

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {\tan ^{3}{\left (d + e x \right )}}{\sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \]

[In]

integrate(tan(e*x+d)**3/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(1/2),x)

[Out]

Integral(tan(d + e*x)**3/sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2), x)

Maxima [F]

\[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )^{3}}{\sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}} \,d x } \]

[In]

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(tan(e*x + d)^3/sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^3}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a}} \,d x \]

[In]

int(tan(d + e*x)^3/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2),x)

[Out]

int(tan(d + e*x)^3/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(1/2), x)