3.6.20 \(\int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx\) [520]

Optimal. Leaf size=240 \[ \frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

2/3*(2*a+b)*cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^2/f-1/3*cot(f*x+e)*csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)/a/f
+2/3*(2*a+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^2/f
/(1+b*sin(f*x+e)^2/a)^(1/2)-1/3*(a+b)*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/f/(a+b*sin(f*x+e)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3275, 485, 597, 538, 437, 435, 432, 430} \begin {gather*} \frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\text {ArcSin}(\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\text {ArcSin}(\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

(2*(2*a + b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[
e + f*x]^2])/(3*a*f) + (2*(2*a + b)*Sqrt[Cos[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^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((a + b)*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*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 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -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 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*} \int \frac {\cot ^4(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{x^4 \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-2 (2 a+b)+(3 a+b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a f}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a (3 a+b)-2 b (2 a+b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 f}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}-\frac {\left ((a+b) \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 f}+\frac {\left (2 (2 a+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^2 f}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {\left (2 (2 a+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^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left ((a+b) \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 f \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {2 (2 a+b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a f}+\frac {2 (2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(a+b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.70, size = 186, normalized size = 0.78 \begin {gather*} \frac {\frac {\left (-2 \left (4 a^2+5 a b+2 b^2\right ) \cos (2 (e+f x))+(2 a+b) (2 a+3 b+b \cos (4 (e+f x)))\right ) \cot (e+f x) \csc ^2(e+f x)}{\sqrt {2}}+4 a (2 a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-2 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} F\left (e+f x\left |-\frac {b}{a}\right .\right )}{6 a^2 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

(((-2*(4*a^2 + 5*a*b + 2*b^2)*Cos[2*(e + f*x)] + (2*a + b)*(2*a + 3*b + b*Cos[4*(e + f*x)]))*Cot[e + f*x]*Csc[
e + f*x]^2)/Sqrt[2] + 4*a*(2*a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 2*a*(a
+ b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a^2*f*Sqrt[2*a + b - b*Cos[2*(e +
f*x)]])

________________________________________________________________________________________

Maple [A]
time = 10.26, size = 351, normalized size = 1.46

method	result	size
default	\(-\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )+b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \left (\sin ^{3}\left (f x +e \right )\right )-4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \left (\sin ^{3}\left (f x +e \right )\right )+4 a b \left (\sin ^{6}\left (f x +e \right )\right )+2 b^{2} \left (\sin ^{6}\left (f x +e \right )\right )+4 a^{2} \left (\sin ^{4}\left (f x +e \right )\right )-3 a b \left (\sin ^{4}\left (f x +e \right )\right )-2 b^{2} \left (\sin ^{4}\left (f x +e \right )\right )-5 a^{2} \left (\sin ^{2}\left (f x +e \right )\right )-a b \left (\sin ^{2}\left (f x +e \right )\right )+a^{2}}{3 a^{2} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\)	\(351\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*((cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2*sin(f*x+e)^3+
b*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a*sin(f*x+e)^3-4*(cos
(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*a^2*sin(f*x+e)^3*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))-2*(cos(f*x
+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b*sin(f*x+e)^3+4*a*b*sin(f*x+
e)^6+2*b^2*sin(f*x+e)^6+4*a^2*sin(f*x+e)^4-3*a*b*sin(f*x+e)^4-2*b^2*sin(f*x+e)^4-5*a^2*sin(f*x+e)^2-a*b*sin(f*
x+e)^2+a^2)/a^2/sin(f*x+e)^3/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f

________________________________________________________________________________________

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(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 0.17, size = 1045, normalized size = 4.35 \begin {gather*} \frac {{\left (2 \, {\left (-2 i \, a b^{2} - i \, b^{3} + {\left (2 i \, a b^{2} + i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) - {\left (4 i \, a^{2} b + 4 i \, a b^{2} + i \, b^{3} + {\left (-4 i \, a^{2} b - 4 i \, a b^{2} - i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (2 i \, a b^{2} + i \, b^{3} + {\left (-2 i \, a b^{2} - i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) - {\left (-4 i \, a^{2} b - 4 i \, a b^{2} - i \, b^{3} + {\left (4 i \, a^{2} b + 4 i \, a b^{2} + i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (3 i \, a^{2} b + 5 i \, a b^{2} + 2 i \, b^{3} + {\left (-3 i \, a^{2} b - 5 i \, a b^{2} - 2 i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) - {\left (6 i \, a^{3} + 5 i \, a^{2} b + i \, a b^{2} + {\left (-6 i \, a^{3} - 5 i \, a^{2} b - i \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (-3 i \, a^{2} b - 5 i \, a b^{2} - 2 i \, b^{3} + {\left (3 i \, a^{2} b + 5 i \, a b^{2} + 2 i \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \sin \left (f x + e\right ) - {\left (-6 i \, a^{3} - 5 i \, a^{2} b - i \, a b^{2} + {\left (6 i \, a^{3} + 5 i \, a^{2} b + i \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \sin \left (f x + e\right )\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{3 \, {\left (a^{2} b^{2} f \cos \left (f x + e\right )^{2} - a^{2} b^{2} f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*((2*(-2*I*a*b^2 - I*b^3 + (2*I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e)
- (4*I*a^2*b + 4*I*a*b^2 + I*b^3 + (-4*I*a^2*b - 4*I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqr
t((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)*(co
s(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*I*a*
b^2 + I*b^3 + (-2*I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (-4*I*a^2*b -
4*I*a*b^2 - I*b^3 + (4*I*a^2*b + 4*I*a*b^2 + I*b^3)*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*
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*(3*I*a^2*b + 5*I*a*b^2
+ 2*I*b^3 + (-3*I*a^2*b - 5*I*a*b^2 - 2*I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) -
(6*I*a^3 + 5*I*a^2*b + I*a*b^2 + (-6*I*a^3 - 5*I*a^2*b - I*a*b^2)*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*(-3*I*a^2
*b - 5*I*a*b^2 - 2*I*b^3 + (3*I*a^2*b + 5*I*a*b^2 + 2*I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*si
n(f*x + e) - (-6*I*a^3 - 5*I*a^2*b - I*a*b^2 + (6*I*a^3 + 5*I*a^2*b + I*a*b^2)*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*a*b^2 + b^3)*cos(f*x + e)^3 - (3*a*b^2 + 2*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2*b^
2*f*cos(f*x + e)^2 - a^2*b^2*f)*sin(f*x + e))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4/(a+b*sin(f*x+e)**2)**(1/2),x)

[Out]

Integral(cot(e + f*x)**4/sqrt(a + b*sin(e + f*x)**2), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(cot(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(1/2),x)

[Out]

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(1/2), x)

________________________________________________________________________________________