3.1.50 \(\int \frac {\cot ^3(d+e x)}{(a+b \tan ^2(d+e x)+c \tan ^4(d+e x))^{3/2}} \, dx\) [50]
Optimal. Leaf size=477 \[ \frac {\tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 a^{3/2} e}+\frac {3 b \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 a^{5/2} e}-\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\cot ^2(d+e x) \left (b^2-2 a c+b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) e} \]
[Out]
1/2*arctanh(1/2*(2*a+b*tan(e*x+d)^2)/a^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))/a^(3/2)/e+3/4*b*arctanh(
1/2*(2*a+b*tan(e*x+d)^2)/a^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))/a^(5/2)/e-1/2*arctanh(1/2*(2*a-b+(b-
2*c)*tan(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2))/(a-b+c)^(3/2)/e-1/2*(-8*a*c+3*b^2)*c
ot(e*x+d)^2*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)/a^2/(-4*a*c+b^2)/e+(-b^2+2*a*c-b*c*tan(e*x+d)^2)/a/(-4*a*c
+b^2)/e/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)+cot(e*x+d)^2*(b^2-2*a*c+b*c*tan(e*x+d)^2)/a/(-4*a*c+b^2)/e/(a+
b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)+(b^2-2*a*c-b*c+(b-2*c)*c*tan(e*x+d)^2)/(a-b+c)/(-4*a*c+b^2)/e/(a+b*tan(e*
x+d)^2+c*tan(e*x+d)^4)^(1/2)
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Rubi [A]
time = 0.38, antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3781, 1265,
974, 754, 820, 738, 212, 12} \begin {gather*} \frac {3 b \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 a^{5/2} e}+\frac {\tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 a^{3/2} e}-\frac {\left (3 b^2-8 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a^2 e \left (b^2-4 a c\right )}-\frac {-2 a c+b^2+b c \tan ^2(d+e x)}{a e \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {-2 a c+b^2+c (b-2 c) \tan ^2(d+e x)-b c}{e (a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\cot ^2(d+e x) \left (-2 a c+b^2+b c \tan ^2(d+e x)\right )}{a e \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e (a-b+c)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cot[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]
[Out]
ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*a^(3/2)*e) + (3
*b*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*a^(5/2)*e)
- ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4
])]/(2*(a - b + c)^(3/2)*e) - (b^2 - 2*a*c + b*c*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2
+ c*Tan[d + e*x]^4]) + (Cot[d + e*x]^2*(b^2 - 2*a*c + b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d
+ e*x]^2 + c*Tan[d + e*x]^4]) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e
*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((3*b^2 - 8*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 +
c*Tan[d + e*x]^4])/(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 3781
Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(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*Tan[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*} \int \frac {\cot ^3(d+e x)}{\left (a+b \tan ^2(d+e x)+c \tan ^4(d+e x)\right )^{3/2}} \, dx &=\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,\tan (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,\tan ^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,\tan ^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,\tan ^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,\tan ^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,\tan ^2(d+e x)\right )}{2 e}\\ &=-\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\cot ^2(d+e x) \left (b^2-2 a c+b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^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,\tan ^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,\tan ^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,\tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) e}\\ &=-\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\cot ^2(d+e x) \left (b^2-2 a c+b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(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,\tan ^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,\tan ^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,\tan ^2(d+e x)\right )}{2 (a-b+c) e}\\ &=-\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\cot ^2(d+e x) \left (b^2-2 a c+b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(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 \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^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 \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^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) \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{(a-b+c) e}\\ &=\frac {\tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 a^{3/2} e}+\frac {3 b \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 a^{5/2} e}-\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 (a-b+c)^{3/2} e}-\frac {b^2-2 a c+b c \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\cot ^2(d+e x) \left (b^2-2 a c+b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {b^2-2 a c-b c+(b-2 c) c \tan ^2(d+e x)}{(a-b+c) \left (b^2-4 a c\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) e}\\ \end {align*}
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Mathematica [A]
time = 6.08, size = 555, normalized size = 1.16 \begin {gather*} \frac {-\frac {2 \left (-\frac {b^2}{2}+2 a c\right ) \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{a^{3/2} \left (b^2-4 a c\right )}+\frac {8 \left (-\frac {b^2}{2}+2 a c\right ) \tanh ^{-1}\left (\frac {2 a-b-(-b+2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{\sqrt {a-b+c} (4 a-4 b+4 c) \left (b^2-4 a c\right )}+\frac {2 \left (-b^2+2 a c-b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {2 \cot ^2(d+e x) \left (-b^2+2 a c-b c \tan ^2(d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {2 \left (-b^2+2 a c+b c+c (-b+2 c) \tan ^2(d+e x)\right )}{(a-b+c) \left (b^2-4 a c\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}-\frac {2 \left (\frac {\left (2 a b c+\frac {1}{2} b \left (-3 b^2+8 a c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 a^{3/2}}+\frac {\left (3 b^2-8 a c\right ) \cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 a}\right )}{a \left (b^2-4 a c\right )}}{2 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Cot[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2),x]
[Out]
((-2*(-1/2*b^2 + 2*a*c)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]
^4])])/(a^(3/2)*(b^2 - 4*a*c)) + (8*(-1/2*b^2 + 2*a*c)*ArcTanh[(2*a - b - (-b + 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a
- b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(Sqrt[a - b + c]*(4*a - 4*b + 4*c)*(b^2 - 4*a*c)) +
(2*(-b^2 + 2*a*c - b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (2*
Cot[d + e*x]^2*(-b^2 + 2*a*c - b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x
]^4]) - (2*(-b^2 + 2*a*c + b*c + c*(-b + 2*c)*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e
*x]^2 + c*Tan[d + e*x]^4]) - (2*(((2*a*b*c + (b*(-3*b^2 + 8*a*c))/2)*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[
a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*a^(3/2)) + ((3*b^2 - 8*a*c)*Cot[d + e*x]^2*Sqrt[a + b*T
an[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*a)))/(a*(b^2 - 4*a*c)))/(2*e)
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Maple [F]
time = 0.63, size = 0, normalized size = 0.00 \[\int \frac {\cot ^{3}\left (e x +d \right )}{\left (a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x)
[Out]
int(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x)
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1299 vs.
\(2 (452) = 904\).
time = 18.19, size = 5299, normalized size = 11.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="fricas")
[Out]
[-1/8*(2*((a^3*b^2*c - 4*a^4*c^2)*tan(x*e + d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(x*e + d)^4 + (a^4*b^2 - 4*a^5*c)*
tan(x*e + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(x*e + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a -
b)*c)*tan(x*e + d)^2 - 4*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)*((b - 2*c)*tan(x*e + d)^2 + 2*a - b)*sq
rt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(x*e + d)^4 + 2*tan(x*e + d)^2 + 1)) - ((4*(2*a^2 + 3*a*b)*c^
4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*b^2 + 5*a*b^3 + 3*b^4)*c^2 - (2*a^
3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(x*e + d)^6 - (2*a^3*b^3 - a^2*b^4 - 4*a*b^5 + 3*b^6 - 4*(2*a^2*b + 3
*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 +
3*b^5)*c)*tan(x*e + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2*a^3 + 3*a^2*b)*c^3 - (16*a^4 + 8*
a^3*b - 26*a^2*b^2 - 3*a*b^3)*c^2 - 2*(4*a^5 - 2*a^4*b - 10*a^3*b^2 + 5*a^2*b^3 + 3*a*b^4)*c)*tan(x*e + d)^2)*
sqrt(a)*log(((b^2 + 4*a*c)*tan(x*e + d)^4 + 8*a*b*tan(x*e + d)^2 + 4*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2
+ a)*(b*tan(x*e + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(x*e + d)^4) - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 4*a^3*c^3
- (8*a^2*c^4 + 3*(4*a^3 - 6*a^2*b - a*b^2)*c^3 + 2*(2*a^4 - 7*a^3*b + 3*a^2*b^2 + 3*a*b^3)*c^2 - (a^3*b^2 - 4*
a^2*b^3 + 3*a*b^4)*c)*tan(x*e + d)^4 - (8*a^4 - 8*a^3*b - a^2*b^2)*c^2 + (a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 2*(2
*a^3 + 5*a^2*b)*c^3 - (4*a^4 + 10*a^3*b - 22*a^2*b^2 - 3*a*b^3)*c^2 - 2*(2*a^4*b - 8*a^3*b^2 + 4*a^2*b^3 + 3*a
*b^4)*c)*tan(x*e + d)^2 - 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c)*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2
+ a))/((4*a^4*c^4 + (8*a^5 - 8*a^4*b - a^3*b^2)*c^3 + 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c^2 - (a^5*b^2 -
2*a^4*b^3 + a^3*b^4)*c)*e*tan(x*e + d)^6 - (a^5*b^3 - 2*a^4*b^4 + a^3*b^5 - 4*a^4*b*c^3 - (8*a^5*b - 8*a^4*b^
2 - a^3*b^3)*c^2 - 2*(2*a^6*b - 4*a^5*b^2 + a^4*b^3 + a^3*b^4)*c)*e*tan(x*e + d)^4 - (a^6*b^2 - 2*a^5*b^3 + a^
4*b^4 - 4*a^5*c^3 - (8*a^6 - 8*a^5*b - a^4*b^2)*c^2 - 2*(2*a^7 - 4*a^6*b + a^5*b^2 + a^4*b^3)*c)*e*tan(x*e + d
)^2), -1/4*(((4*(2*a^2 + 3*a*b)*c^4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*
b^2 + 5*a*b^3 + 3*b^4)*c^2 - (2*a^3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(x*e + d)^6 - (2*a^3*b^3 - a^2*b^4
- 4*a*b^5 + 3*b^6 - 4*(2*a^2*b + 3*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2
*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 + 3*b^5)*c)*tan(x*e + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2
*a^3 + 3*a^2*b)*c^3 - (16*a^4 + 8*a^3*b - 26*a^2*b^2 - 3*a*b^3)*c^2 - 2*(4*a^5 - 2*a^4*b - 10*a^3*b^2 + 5*a^2*
b^3 + 3*a*b^4)*c)*tan(x*e + d)^2)*sqrt(-a)*arctan(1/2*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)*(b*tan(x*e
+ d)^2 + 2*a)*sqrt(-a)/(a*c*tan(x*e + d)^4 + a*b*tan(x*e + d)^2 + a^2)) + ((a^3*b^2*c - 4*a^4*c^2)*tan(x*e +
d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(x*e + d)^4 + (a^4*b^2 - 4*a^5*c)*tan(x*e + d)^2)*sqrt(a - b + c)*log(((b^2 +
4*(a - 2*b)*c + 8*c^2)*tan(x*e + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(x*e + d)^2 - 4*sqrt(c*tan(x*e + d)
^4 + b*tan(x*e + d)^2 + a)*((b - 2*c)*tan(x*e + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)
/(tan(x*e + d)^4 + 2*tan(x*e + d)^2 + 1)) - 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 4*a^3*c^3 - (8*a^2*c^4 + 3*(4*a
^3 - 6*a^2*b - a*b^2)*c^3 + 2*(2*a^4 - 7*a^3*b + 3*a^2*b^2 + 3*a*b^3)*c^2 - (a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*c)
*tan(x*e + d)^4 - (8*a^4 - 8*a^3*b - a^2*b^2)*c^2 + (a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 2*(2*a^3 + 5*a^2*b)*c^3 -
(4*a^4 + 10*a^3*b - 22*a^2*b^2 - 3*a*b^3)*c^2 - 2*(2*a^4*b - 8*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*c)*tan(x*e + d)
^2 - 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c)*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a))/((4*a^4*c^4 + (
8*a^5 - 8*a^4*b - a^3*b^2)*c^3 + 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)
*c)*e*tan(x*e + d)^6 - (a^5*b^3 - 2*a^4*b^4 + a^3*b^5 - 4*a^4*b*c^3 - (8*a^5*b - 8*a^4*b^2 - a^3*b^3)*c^2 - 2*
(2*a^6*b - 4*a^5*b^2 + a^4*b^3 + a^3*b^4)*c)*e*tan(x*e + d)^4 - (a^6*b^2 - 2*a^5*b^3 + a^4*b^4 - 4*a^5*c^3 - (
8*a^6 - 8*a^5*b - a^4*b^2)*c^2 - 2*(2*a^7 - 4*a^6*b + a^5*b^2 + a^4*b^3)*c)*e*tan(x*e + d)^2), 1/8*(4*((a^3*b^
2*c - 4*a^4*c^2)*tan(x*e + d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(x*e + d)^4 + (a^4*b^2 - 4*a^5*c)*tan(x*e + d)^2)*s
qrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(x*e + d)^4 + b*tan(x*e + d)^2 + a)*((b - 2*c)*tan(x*e + d)^2 + 2*a - b)
*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(x*e + d)^4 + (a*b - b^2 + b*c)*tan(x*e + d)^2 + a^2 - a*b + a*c)) + (
(4*(2*a^2 + 3*a*b)*c^4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*b^2 + 5*a*b^3
+ 3*b^4)*c^2 - (2*a^3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(x*e + d)^6 - (2*a^3*b^3 - a^2*b^4 - 4*a*b^5 + 3
*b^6 - 4*(2*a^2*b + 3*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2*a^3*b^2 - 10
*a^2*b^3 + 5*a*b^4 + 3*b^5)*c)*tan(x*e + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2*a^3 + 3*a^2*
b)*c^3 - (16*a^4 + 8*a^3*b - 26*a^2*b^2 - 3*a*b...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{3}{\left (d + e x \right )}}{\left (a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(e*x+d)**3/(a+b*tan(e*x+d)**2+c*tan(e*x+d)**4)**(3/2),x)
[Out]
Integral(cot(d + e*x)**3/(a + b*tan(d + e*x)**2 + c*tan(d + e*x)**4)**(3/2), x)
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm="giac")
[Out]
Timed out
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d + e*x)^3/(a + b*tan(d + e*x)^2 + c*tan(d + e*x)^4)^(3/2),x)
[Out]
\text{Hanged}
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