\(\int \frac {\cot ^2(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [541]
Optimal result
Integrand size = 25, antiderivative size = 287 \[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f}-\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}}
\]
[Out]
1/3*cot(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(3/2)+1/3*(3*a+4*b)*cot(f*x+e)/a^2/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*
(7*a+8*b)*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^3/(a+b)/f-1/3*(7*a+8*b)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec
(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*sin(f*x+e)^2)^(1/2)/a^3/(a+b)/f/(1+b*sin(f*x+e)^2/a)^(1/2)+4/3*EllipticF(sin
(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(1+b*sin(f*x+e)^2/a)^(1/2)/a^2/f/(a+b*sin(f*x+e)^2)^(1/2
)
Rubi [A] (verified)
Time = 0.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3275, 480, 593, 597, 538, 437,
435, 432, 430} \[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^3 f (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)}+\frac {4 \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 f (a+b) \sqrt {a+b \sin ^2(e+f x)}}+\frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}
\]
[In]
Int[Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]
[Out]
Cot[e + f*x]/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((3*a + 4*b)*Cot[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin
[e + f*x]^2]) - ((7*a + 8*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)*f) - ((7*a + 8*b)*Sqrt[Co
s[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)*
f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*
x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])
Rule 430
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Rule 432
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Rule 435
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]
Rule 437
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]
, Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ
[a, 0]
Rule 480
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(e*x)^
(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*e*n*(p + 1))), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^
n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b,
c, d, e, m, n, p, q, x]
Rule 538
Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))
Rule 593
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Rule 597
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]
Rule 3275
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff^(m + 1)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[x^m*((a + b*ff^2*
x^2)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2]
&& !IntegerQ[p]
Rubi steps \begin{align*}
\text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2}}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-4+3 x^2}{x^2 \sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a f} \\ & = \frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {7 a+8 b+(-3 a-4 b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 (a+b) f} \\ & = \frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {a (3 a+4 b)+b (7 a+8 b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b) f} \\ & = \frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f}+\frac {\left (4 \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 f}-\frac {\left ((7 a+8 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b) f} \\ & = \frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f}-\frac {\left ((7 a+8 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (4 \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f}-\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.24 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.73
\[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-\frac {\left (24 a^3+68 a^2 b+69 a b^2+24 b^3-4 b \left (11 a^2+19 a b+8 b^2\right ) \cos (2 (e+f x))+b^2 (7 a+8 b) \cos (4 (e+f x))\right ) \cot (e+f x)}{\sqrt {2}}-2 a^2 (7 a+8 b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+8 a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{6 a^3 (a+b) f (2 a+b-b \cos (2 (e+f x)))^{3/2}}
\]
[In]
Integrate[Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]
[Out]
(-(((24*a^3 + 68*a^2*b + 69*a*b^2 + 24*b^3 - 4*b*(11*a^2 + 19*a*b + 8*b^2)*Cos[2*(e + f*x)] + b^2*(7*a + 8*b)*
Cos[4*(e + f*x)])*Cot[e + f*x])/Sqrt[2]) - 2*a^2*(7*a + 8*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*Elliptic
E[e + f*x, -(b/a)] + 8*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticF[e + f*x, -(b/a)])/(6*a^3
*(a + b)*f*(2*a + b - b*Cos[2*(e + f*x)])^(3/2))
Maple [A] (verified)
Time = 5.29 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.43
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method | result | size |
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default |
\(\frac {\left (-7 a \,b^{2}-8 b^{3}\right ) \left (\cos ^{6}\left (f x +e \right )\right )+\left (11 a^{2} b +26 a \,b^{2}+16 b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right )-\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a b \left (4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -7 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-3 a^{3}-14 a^{2} b -19 a \,b^{2}-8 b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a \left (4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+8 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-7 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-15 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}\right ) \sin \left (f x +e \right )}{3 \sin \left (f x +e \right ) a^{3} \left (a +b \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \cos \left (f x +e \right ) f}\) |
\(411\) |
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[In]
int(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/3*((-7*a*b^2-8*b^3)*cos(f*x+e)^6+(11*a^2*b+26*a*b^2+16*b^3)*cos(f*x+e)^4-(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(
cos(f*x+e)^2)^(1/2)*a*b*(4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-7*E
llipticE(sin(f*x+e),(-1/a*b)^(1/2))*a-8*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*b)*cos(f*x+e)^2*sin(f*x+e)+(-3*a^
3-14*a^2*b-19*a*b^2-8*b^3)*cos(f*x+e)^2+(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*a*(4*EllipticF(
sin(f*x+e),(-1/a*b)^(1/2))*a^2+8*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a*b+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2
))*b^2-7*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a^2-15*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b-8*EllipticE(sin(
f*x+e),(-1/a*b)^(1/2))*b^2)*sin(f*x+e))/sin(f*x+e)/a^3/(a+b)/(a+b*sin(f*x+e)^2)^(3/2)/cos(f*x+e)/f
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 1595, normalized size of antiderivative = 5.56
\[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")
[Out]
1/6*((2*(-7*I*a^3*b^2 - 22*I*a^2*b^3 - 23*I*a*b^4 - 8*I*b^5 + (-7*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 - 2*(-7*I*
a^2*b^3 - 15*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (14*I*a^4*b + 51
*I*a^3*b^2 + 68*I*a^2*b^3 + 39*I*a*b^4 + 8*I*b^5 + (14*I*a^2*b^3 + 23*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^4 + 2*(-
14*I*a^3*b^2 - 37*I*a^2*b^3 - 31*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2
+ a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*s
in(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(7*I*a^3*b^2 + 22*I*a^2
*b^3 + 23*I*a*b^4 + 8*I*b^5 + (7*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^4 - 2*(7*I*a^2*b^3 + 15*I*a*b^4 + 8*I*b^5)*co
s(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (-14*I*a^4*b - 51*I*a^3*b^2 - 68*I*a^2*b^3 - 39*I*
a*b^4 - 8*I*b^5 + (-14*I*a^2*b^3 - 23*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 + 2*(14*I*a^3*b^2 + 37*I*a^2*b^3 + 31*
I*a*b^4 + 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*ellipt
ic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b
^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 2*(2*(-3*I*a^4*b - 17*I*a^3*b^2 - 33*I*a^2*b^3 - 27*I*a*b^4
- 8*I*b^5 + (-3*I*a^2*b^3 - 11*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 + 2*(3*I*a^3*b^2 + 14*I*a^2*b^3 + 19*I*a*b^4
+ 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + (6*I*a^5 + 23*I*a^4*b + 32*I*a^3*b^2
+ 19*I*a^2*b^3 + 4*I*a*b^4 + (6*I*a^3*b^2 + 11*I*a^2*b^3 + 4*I*a*b^4)*cos(f*x + e)^4 + 2*(-6*I*a^4*b - 17*I*a
^3*b^2 - 15*I*a^2*b^3 - 4*I*a*b^4)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*
a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8
*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 2*(2*(3*I*a^4*b + 17*I*a^3*b^2 + 33*I*a^2*b
^3 + 27*I*a*b^4 + 8*I*b^5 + (3*I*a^2*b^3 + 11*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^4 + 2*(-3*I*a^3*b^2 - 14*I*a^2*b
^3 - 19*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + (-6*I*a^5 - 23*I*a^4*
b - 32*I*a^3*b^2 - 19*I*a^2*b^3 - 4*I*a*b^4 + (-6*I*a^3*b^2 - 11*I*a^2*b^3 - 4*I*a*b^4)*cos(f*x + e)^4 + 2*(6*
I*a^4*b + 17*I*a^3*b^2 + 15*I*a^2*b^3 + 4*I*a*b^4)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2
+ a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*si
n(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 2*((7*a*b^4 + 8*b^5)*cos(f*
x + e)^5 - (11*a^2*b^3 + 26*a*b^4 + 16*b^5)*cos(f*x + e)^3 + (3*a^3*b^2 + 14*a^2*b^3 + 19*a*b^4 + 8*b^5)*cos(f
*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(((a^4*b^4 + a^3*b^5)*f*cos(f*x + e)^4 - 2*(a^5*b^3 + 2*a^4*b^4 + a^
3*b^5)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)*sin(f*x + e))
Sympy [F]
\[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx
\]
[In]
integrate(cot(f*x+e)**2/(a+b*sin(f*x+e)**2)**(5/2),x)
[Out]
Integral(cot(e + f*x)**2/(a + b*sin(e + f*x)**2)**(5/2), x)
Maxima [F]
\[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")
[Out]
integrate(cot(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)
Giac [F]
\[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x }
\]
[In]
integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")
[Out]
integrate(cot(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x
\]
[In]
int(cot(e + f*x)^2/(a + b*sin(e + f*x)^2)^(5/2),x)
[Out]
int(cot(e + f*x)^2/(a + b*sin(e + f*x)^2)^(5/2), x)