3.1.24 \(\int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\) [24]
Optimal. Leaf size=179 \[ \frac {\sqrt {a-b+c} \tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}-\frac {(b-2 c) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e} \]
[Out]
-1/4*(b-2*c)*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e/c^(1/2)+1/2*a
rctanh(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)^(1/2)/e
-1/2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)/e
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Rubi [A]
time = 0.15, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3782, 1261,
748, 857, 635, 212, 738} \begin {gather*} -\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}+\frac {\sqrt {a-b+c} \tanh ^{-1}\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}-\frac {(b-2 c) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
[Out]
(Sqrt[a - b + c]*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*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 +
c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*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 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 748
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) && !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Rule 857
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&& !IGtQ[m, 0]
Rule 1261
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(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]
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*} \int \cot (d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x \sqrt {a+b x^2+c x^4}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{1+x} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}+\frac {\text {Subst}\left (\int \frac {-2 a+b-(b-2 c) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{4 e}\\ &=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}-\frac {(b-2 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{4 e}-\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}-\frac {(b-2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \cot ^2(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {(a-b+c) \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 )}{e}\\ &=\frac {\sqrt {a-b+c} \tanh ^{-1}\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 e}-\frac {(b-2 c) \tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}{2 e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 34.93, size = 286262, normalized size = 1599.23 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[d + e*x]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
[Out]
Result too large to show
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Maple [A]
time = 0.24, size = 281, normalized size = 1.57
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{2}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) b}{4 \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) \sqrt {c}}{2}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2}}{e}\) |
\(281\) |
| default |
\(\frac {-\frac {\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{2}-\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) b}{4 \sqrt {c}}+\frac {\ln \left (\frac {\frac {b}{2}-c +\left (\cot ^{2}\left (e x +d \right )+1\right ) c}{\sqrt {c}}+\sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}\right ) \sqrt {c}}{2}+\frac {\sqrt {a -b +c}\, \ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2}}{e}\) |
\(281\) |
| | |
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|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/e*(-1/2*(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2)-1/4*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)/c^(1
/2)+(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/c^(1/2)*b+1/2*ln((1/2*b-c+(cot(e*x+d)^2+1)*c)
/c^(1/2)+(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))*c^(1/2)+1/2*(a-b+c)^(1/2)*ln((2*a-2*b+2*
c+(b-2*c)*(cot(e*x+d)^2+1)+2*(a-b+c)^(1/2)*(c*(cot(e*x+d)^2+1)^2+(b-2*c)*(cot(e*x+d)^2+1)+a-b+c)^(1/2))/(cot(e
*x+d)^2+1)))
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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(cot(e*x+d)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(c*cot(x*e + d)^4 + b*cot(x*e + d)^2 + a)*cot(x*e + d), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs.
\(2 (160) = 320\).
time = 6.43, size = 2012, normalized size = 11.24 \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)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")
[Out]
[1/8*(2*sqrt(a - b + c)*c*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + 2*a^2 - b^2 + 2*c
^2 + 2*((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b)*cos(2*x*e + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)
*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)) -
4*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d)) - (b - 2*c)*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*x*e
+ 2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*cos(2*x*e + 2*d) + b + 2*c)*sqr
t(c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*co
s(2*x*e + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*x*e + 2*d))/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)
) - 4*c*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2
*cos(2*x*e + 2*d) + 1)))*e^(-1)/c, -1/4*((b - 2*c)*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*co
s(2*x*e + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b +
c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1))/(((a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + (a + b)*c + c^2 - 2
*(a*c - c^2)*cos(2*x*e + 2*d))) - sqrt(a - b + c)*c*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*x*e +
2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b)*cos(2*x*e + 2*d) + a - c)*sqrt(a
- b + c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 -
2*cos(2*x*e + 2*d) + 1)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d)) + 2*c*sqrt(((a - b + c)*cos(2*x*e + 2*d
)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)))*e^(-1)/c, -1/8*(
4*sqrt(-a + b - c)*c*arctan(((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b)*cos(2*x*e + 2*d) + a - c)*sqrt(-a + b
- c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*co
s(2*x*e + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*x*e + 2*d)^2 + a^2 - b^2 + 2*a*c + c^2 - 2
*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d))) + (b - 2*c)*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*x*e +
2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 + 4*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*cos(2*x*e + 2*d) + b + 2*c)*sqrt
(c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos
(2*x*e + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*x*e + 2*d))/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1))
+ 4*c*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*
cos(2*x*e + 2*d) + 1)))*e^(-1)/c, -1/4*(2*sqrt(-a + b - c)*c*arctan(((a - b + c)*cos(2*x*e + 2*d)^2 - (2*a - b
)*cos(2*x*e + 2*d) + a - c)*sqrt(-a + b - c)*sqrt(((a - b + c)*cos(2*x*e + 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d)
+ a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*x*
e + 2*d)^2 + a^2 - b^2 + 2*a*c + c^2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*x*e + 2*d))) + (b - 2*c)*sqrt(-c)*arcta
n(-1/2*((b - 2*c)*cos(2*x*e + 2*d)^2 - 2*b*cos(2*x*e + 2*d) + b + 2*c)*sqrt(-c)*sqrt(((a - b + c)*cos(2*x*e +
2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1))/(((a - b)*c +
c^2)*cos(2*x*e + 2*d)^2 + (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*x*e + 2*d))) + 2*c*sqrt(((a - b + c)*cos(2*x*e
+ 2*d)^2 - 2*(a - c)*cos(2*x*e + 2*d) + a + b + c)/(cos(2*x*e + 2*d)^2 - 2*cos(2*x*e + 2*d) + 1)))*e^(-1)/c]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}} \cot {\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(e*x+d)*(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)
[Out]
Integral(sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)*cot(d + e*x), 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)*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="giac")
[Out]
Timed out
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {cot}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d + e*x)*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)
[Out]
int(cot(d + e*x)*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)
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