3.1.18 \(\int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) [18]
Optimal. Leaf size=141 \[ -\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}-\frac {\tanh ^{-1}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {c} e} \]
[Out]
-1/2*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*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))/e/(a-b+c)^(1/2)
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Rubi [A]
time = 0.14, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3782, 1265,
857, 635, 212, 738} \begin {gather*} -\frac {\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 \sqrt {a-b+c}}-\frac {\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 )}{2 \sqrt {c} e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
[Out]
-1/2*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])]/(Sqrt[a - b + c]*e) - 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])]/(2*Sqrt[c]*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 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 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*} \int \frac {\cot ^3(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^3}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \frac {x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 e}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{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 e}\\ &=-\frac {\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 )}{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 )}{e}\\ &=-\frac {\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 \sqrt {a-b+c} e}-\frac {\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 )}{2 \sqrt {c} 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 = 125.40, size = 64578, normalized size = 458.00 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[d + e*x]^3/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 = 153, normalized size = 1.09
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{2 \sqrt {c}}-\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2 \sqrt {a -b +c}}}{e}\) |
\(153\) |
| default |
\(\frac {-\frac {\ln \left (\frac {\frac {b}{2}+c \left (\cot ^{2}\left (e x +d \right )\right )}{\sqrt {c}}+\sqrt {a +b \left (\cot ^{2}\left (e x +d \right )\right )+c \left (\cot ^{4}\left (e x +d \right )\right )}\right )}{2 \sqrt {c}}-\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+2 \sqrt {a -b +c}\, \sqrt {c \left (\cot ^{2}\left (e x +d \right )+1\right )^{2}+\left (b -2 c \right ) \left (\cot ^{2}\left (e x +d \right )+1\right )+a -b +c}}{\cot ^{2}\left (e x +d \right )+1}\right )}{2 \sqrt {a -b +c}}}{e}\) |
\(153\) |
| | |
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|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/e*(-1/2*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(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)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="maxima")
[Out]
integrate(cot(x*e + d)^3/sqrt(c*cot(x*e + d)^4 + b*cot(x*e + d)^2 + a), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs.
\(2 (125) = 250\).
time = 4.55, size = 1759, normalized size = 12.48 \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*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")
[Out]
[1/4*(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)*c
os(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)) + (a - b + 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)
))*e^(-1)/((a - b)*c + c^2), -1/4*(2*(a - b + c)*sqrt(-c)*arctan(-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))) - 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)))*e^(-1)/((a - b)*c + c^2), 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))) + (a - b + 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)*sqr
t(((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)))*e^(-1
)/((a - b)*c + c^2), 1/2*(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)/(c
os(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))) - (a - b + c)*sqrt(-c)*arctan(-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))))*e^(-1)/((a - b)*c + c^2)]
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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 )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(e*x+d)**3/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)
[Out]
Integral(cot(d + e*x)**3/sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4), 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*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 \frac {{\mathrm {cot}\left (d+e\,x\right )}^3}{\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)^3/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)
[Out]
int(cot(d + e*x)^3/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)
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