\(\int \frac {\tan ^3(d+e x)}{(a+b \cot ^2(d+e x)+c \cot ^4(d+e x))^{3/2}} \, dx\) [32]

Optimal result

Integrand size = 35, antiderivative size = 478 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 a^{3/2} e}-\frac {3 b \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 a^{5/2} e}+\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}+\frac {b^2-2 a c+b c \cot ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {b^2-2 a c-b c+(b-2 c) c \cot ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\left (b^2-2 a c+b c \cot ^2(d+e x)\right ) \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right ) e} \]

[Out]

-1/2*arctanh(1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/a^(3/2)/e-3/4*b*arctanh
(1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/a^(5/2)/e+1/2*arctanh(1/2*(2*a-b+(b
-2*c)*cot(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/(a-b+c)^(3/2)/e+(b^2-2*a*c+b*c*cot(
e*x+d)^2)/a/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+(-b^2+2*a*c+b*c-(b-2*c)*c*cot(e*x+d)^2)/(a-
b+c)/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)-(b^2-2*a*c+b*c*cot(e*x+d)^2)*tan(e*x+d)^2/a/(-4*a*
c+b^2)/e/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)+1/2*(-8*a*c+3*b^2)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*ta
n(e*x+d)^2/a^2/(-4*a*c+b^2)/e

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3782, 1265, 974, 754, 820, 738, 212, 12} \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=-\frac {3 b \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 a^{5/2} e}-\frac {\text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 a^{3/2} e}+\frac {\left (3 b^2-8 a c\right ) \tan ^2(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 a^2 e \left (b^2-4 a c\right )}+\frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}}+\frac {-2 a c+b^2+b c \cot ^2(d+e x)}{a e \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {-2 a c+b^2+c (b-2 c) \cot ^2(d+e x)-b c}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}-\frac {\tan ^2(d+e x) \left (-2 a c+b^2+b c \cot ^2(d+e x)\right )}{a e \left (b^2-4 a c\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \]

[In]

Int[Tan[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]

[Out]

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

Rule 12

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

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 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 754

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

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 974

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] &&
ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

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]

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

Mathematica [A] (warning: unable to verify)

Time = 15.97 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\frac {\sqrt {2} \sqrt {(3 a+b+3 c-4 (a-c) \cos (2 (d+e x))+(a-b+c) \cos (4 (d+e x))) \csc ^4(d+e x)} \left (\frac {3 b^2 (b-c)^2-4 a^3 c+a^2 \left (b^2+8 b c-4 c^2\right )-2 a \left (b^3+5 b^2 c-10 b c^2+4 c^3\right )}{(a-b+c)^2 \left (-b^2+4 a c\right )}+\frac {8 \left (-b^5+b^4 c+2 a c^3 (a+c)-b^2 c^2 (4 a+c)+b^3 c (5 a+c)-a b c^2 (5 a+3 c)+\left (b^5-3 b^4 c+a b (5 a-9 c) c^2+b^2 (12 a-c) c^2+2 a c^3 (-3 a+c)+b^3 c (-5 a+3 c)\right ) \cos (2 (d+e x))\right )}{(a-b+c)^2 \left (-b^2+4 a c\right ) (3 a+b+3 c-4 (a-c) \cos (2 (d+e x))+(a-b+c) \cos (4 (d+e x)))}+\sec ^2(d+e x)\right )-\frac {2 \left (\frac {(2 a+3 b) (a-b+c) \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {a}}-\frac {2 a^2 \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {a-b+c}}\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{(a-b+c) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}}{8 a^2 e} \]

[In]

Integrate[Tan[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2),x]

[Out]

(Sqrt[2]*Sqrt[(3*a + b + 3*c - 4*(a - c)*Cos[2*(d + e*x)] + (a - b + c)*Cos[4*(d + e*x)])*Csc[d + e*x]^4]*((3*
b^2*(b - c)^2 - 4*a^3*c + a^2*(b^2 + 8*b*c - 4*c^2) - 2*a*(b^3 + 5*b^2*c - 10*b*c^2 + 4*c^3))/((a - b + c)^2*(
-b^2 + 4*a*c)) + (8*(-b^5 + b^4*c + 2*a*c^3*(a + c) - b^2*c^2*(4*a + c) + b^3*c*(5*a + c) - a*b*c^2*(5*a + 3*c
) + (b^5 - 3*b^4*c + a*b*(5*a - 9*c)*c^2 + b^2*(12*a - c)*c^2 + 2*a*c^3*(-3*a + c) + b^3*c*(-5*a + 3*c))*Cos[2
*(d + e*x)]))/((a - b + c)^2*(-b^2 + 4*a*c)*(3*a + b + 3*c - 4*(a - c)*Cos[2*(d + e*x)] + (a - b + c)*Cos[4*(d
+ e*x)])) + Sec[d + e*x]^2) - (2*(((2*a + 3*b)*(a - b + c)*ArcTanh[(b + 2*a*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[c
+ b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/Sqrt[a] - (2*a^2*ArcTanh[(b - 2*c + (2*a - b)*Tan[d + e*x]^2)/(2*Sq
rt[a - b + c]*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])])/Sqrt[a - b + c])*Sqrt[a + b*Cot[d + e*x]^2 + c*
Cot[d + e*x]^4]*Tan[d + e*x]^2)/((a - b + c)*Sqrt[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4]))/(8*a^2*e)

Maple [F]

\[\int \frac {\tan \left (e x +d \right )^{3}}{\left (a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}\right )^{\frac {3}{2}}}d x\]

[In]

int(tan(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x)

[Out]

int(tan(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1300 vs. \(2 (438) = 876\).

Time = 3.34 (sec) , antiderivative size = 5274, normalized size of antiderivative = 11.03 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\int \frac {\tan ^{3}{\left (d + e x \right )}}{\left (a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tan(e*x+d)**3/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(3/2),x)

[Out]

Integral(tan(d + e*x)**3/(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)**(3/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot ^2(d+e x)+c \cot ^4(d+e x)\right )^{3/2}} \, dx=\text {Hanged} \]

[In]

int(tan(d + e*x)^3/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(3/2),x)

[Out]

\text{Hanged}