3.6.31 \(\int \frac {\cot ^4(e+f x)}{(a+b \sin ^2(e+f x))^{3/2}} \, dx\) [531]

Optimal. Leaf size=297 \[ \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {4 (a+b) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \]

[Out]

(a+b)*cot(f*x+e)*csc(f*x+e)^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/f-1/3*(3*a+4*b)*cot(f*x+e)*csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2)/a^2/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/f/(1+b*sin(f*x+e)^2/a)^(1/2)-4/3
*(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^2/f/(a+
b*sin(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3275, 479, 597, 538, 437, 435, 432, 430} \begin {gather*} \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 (\text {ArcSin}(\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^3 f \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}-\frac {4 (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^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

((a + b)*Cot[e + f*x]*Csc[e + f*x]^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*f) - ((3*a + 4*b)*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b*f
) + ((7*a + 8*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^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*(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^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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(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] && GtQ[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 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)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, 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 \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{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 {-3 a-4 b+(2 a+3 b) x^2}{x^4 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{a b f}\\ &=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-b (7 a+8 b)+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 b f}\\ &=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{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 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a b (3 a+4 b)-b^2 (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 b f}\\ &=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{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 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}-\frac {\left (4 (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^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 f}\\ &=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{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 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 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 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left (4 (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^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ &=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{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 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(7 a+8 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^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {4 (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^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 2.62, size = 199, normalized size = 0.67 \begin {gather*} \frac {\frac {\left (8 a^2+37 a b+24 b^2-4 \left (4 a^2+11 a b+8 b^2\right ) \cos (2 (e+f x))+b (7 a+8 b) \cos (4 (e+f x))\right ) \cot (e+f x) \csc ^2(e+f x)}{2 \sqrt {2}}+2 a (7 a+8 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-8 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^3 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2),x]

[Out]

(((8*a^2 + 37*a*b + 24*b^2 - 4*(4*a^2 + 11*a*b + 8*b^2)*Cos[2*(e + f*x)] + b*(7*a + 8*b)*Cos[4*(e + f*x)])*Cot
[e + f*x]*Csc[e + f*x]^2)/(2*Sqrt[2]) + 2*a*(7*a + 8*b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f
*x, -(b/a)] - 8*a*(a + b)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, -(b/a)])/(6*a^3*f*Sqrt[2*a
+ b - b*Cos[2*(e + f*x)]])

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Maple [A]
time = 9.99, size = 353, normalized size = 1.19

method	result	size
default	\(-\frac {4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )+4 b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \left (\sin ^{3}\left (f x +e \right )\right )-7 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \left (\sin ^{3}\left (f x +e \right )\right )+7 a b \left (\sin ^{6}\left (f x +e \right )\right )+8 b^{2} \left (\sin ^{6}\left (f x +e \right )\right )+4 a^{2} \left (\sin ^{4}\left (f x +e \right )\right )-3 a b \left (\sin ^{4}\left (f x +e \right )\right )-8 b^{2} \left (\sin ^{4}\left (f x +e \right )\right )-5 a^{2} \left (\sin ^{2}\left (f x +e \right )\right )-4 a b \left (\sin ^{2}\left (f x +e \right )\right )+a^{2}}{3 a^{3} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(4*(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+4*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)^3-7*
(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))-8*(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+7*a*b*sin(
f*x+e)^6+8*b^2*sin(f*x+e)^6+4*a^2*sin(f*x+e)^4-3*a*b*sin(f*x+e)^4-8*b^2*sin(f*x+e)^4-5*a^2*sin(f*x+e)^2-4*a*b*
sin(f*x+e)^2+a^2)/a^3/sin(f*x+e)^3/cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2)/f

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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(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [C] Result contains complex when optimal does not.
time = 0.20, size = 1465, normalized size = 4.93 \begin {gather*} \frac {{\left (2 \, {\left ({\left (7 i \, a b^{3} + 8 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 7 i \, a^{2} b^{2} + 15 i \, a b^{3} + 8 i \, b^{4} + {\left (-7 i \, a^{2} b^{2} - 22 i \, a b^{3} - 16 i \, b^{4}\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 ({\left (-14 i \, a^{2} b^{2} - 23 i \, a b^{3} - 8 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - 14 i \, a^{3} b - 37 i \, a^{2} b^{2} - 31 i \, a b^{3} - 8 i \, b^{4} + {\left (14 i \, a^{3} b + 51 i \, a^{2} b^{2} + 54 i \, a b^{3} + 16 i \, b^{4}\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 ({\left (-7 i \, a b^{3} - 8 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - 7 i \, a^{2} b^{2} - 15 i \, a b^{3} - 8 i \, b^{4} + {\left (7 i \, a^{2} b^{2} + 22 i \, a b^{3} + 16 i \, b^{4}\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 ({\left (14 i \, a^{2} b^{2} + 23 i \, a b^{3} + 8 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 14 i \, a^{3} b + 37 i \, a^{2} b^{2} + 31 i \, a b^{3} + 8 i \, b^{4} + {\left (-14 i \, a^{3} b - 51 i \, a^{2} b^{2} - 54 i \, a b^{3} - 16 i \, b^{4}\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}}) - 2 \, {\left (2 \, {\left ({\left (3 i \, a^{2} b^{2} + 11 i \, a b^{3} + 8 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} + 3 i \, a^{3} b + 14 i \, a^{2} b^{2} + 19 i \, a b^{3} + 8 i \, b^{4} + {\left (-3 i \, a^{3} b - 17 i \, a^{2} b^{2} - 30 i \, a b^{3} - 16 i \, b^{4}\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 ({\left (-6 i \, a^{3} b - 11 i \, a^{2} b^{2} - 4 i \, a b^{3}\right )} \cos \left (f x + e\right )^{4} - 6 i \, a^{4} - 17 i \, a^{3} b - 15 i \, a^{2} b^{2} - 4 i \, a b^{3} + {\left (6 i \, a^{4} + 23 i \, a^{3} b + 26 i \, a^{2} b^{2} + 8 i \, a 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}} 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}}) - 2 \, {\left (2 \, {\left ({\left (-3 i \, a^{2} b^{2} - 11 i \, a b^{3} - 8 i \, b^{4}\right )} \cos \left (f x + e\right )^{4} - 3 i \, a^{3} b - 14 i \, a^{2} b^{2} - 19 i \, a b^{3} - 8 i \, b^{4} + {\left (3 i \, a^{3} b + 17 i \, a^{2} b^{2} + 30 i \, a b^{3} + 16 i \, b^{4}\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 ({\left (6 i \, a^{3} b + 11 i \, a^{2} b^{2} + 4 i \, a b^{3}\right )} \cos \left (f x + e\right )^{4} + 6 i \, a^{4} + 17 i \, a^{3} b + 15 i \, a^{2} b^{2} + 4 i \, a b^{3} + {\left (-6 i \, a^{4} - 23 i \, a^{3} b - 26 i \, a^{2} b^{2} - 8 i \, a 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}} 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}}) + 2 \, {\left ({\left (7 \, a b^{3} + 8 \, b^{4}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (2 \, a^{2} b^{2} + 9 \, a b^{3} + 8 \, b^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (3 \, a^{2} b^{2} + 11 \, a b^{3} + 8 \, b^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{6 \, {\left (a^{3} b^{3} f \cos \left (f x + e\right )^{4} - {\left (a^{4} b^{2} + 2 \, a^{3} b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{2} + a^{3} b^{3}\right )} 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)^(3/2),x, algorithm="fricas")

[Out]

1/6*((2*((7*I*a*b^3 + 8*I*b^4)*cos(f*x + e)^4 + 7*I*a^2*b^2 + 15*I*a*b^3 + 8*I*b^4 + (-7*I*a^2*b^2 - 22*I*a*b^
3 - 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - ((-14*I*a^2*b^2 - 23*I*a*b^3 - 8*I
*b^4)*cos(f*x + e)^4 - 14*I*a^3*b - 37*I*a^2*b^2 - 31*I*a*b^3 - 8*I*b^4 + (14*I*a^3*b + 51*I*a^2*b^2 + 54*I*a*
b^3 + 16*I*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_
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*((-7*I*a*b^3 - 8*I*b^4)*cos(f*x + e)^4 - 7*I*a^2*b^2 - 15*I
*a*b^3 - 8*I*b^4 + (7*I*a^2*b^2 + 22*I*a*b^3 + 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*
x + e) - ((14*I*a^2*b^2 + 23*I*a*b^3 + 8*I*b^4)*cos(f*x + e)^4 + 14*I*a^3*b + 37*I*a^2*b^2 + 31*I*a*b^3 + 8*I*
b^4 + (-14*I*a^3*b - 51*I*a^2*b^2 - 54*I*a*b^3 - 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sq
rt((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*(2*((3*I*a^2*b^2
+ 11*I*a*b^3 + 8*I*b^4)*cos(f*x + e)^4 + 3*I*a^3*b + 14*I*a^2*b^2 + 19*I*a*b^3 + 8*I*b^4 + (-3*I*a^3*b - 17*I*
a^2*b^2 - 30*I*a*b^3 - 16*I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + ((-6*I*a^3*b -
11*I*a^2*b^2 - 4*I*a*b^3)*cos(f*x + e)^4 - 6*I*a^4 - 17*I*a^3*b - 15*I*a^2*b^2 - 4*I*a*b^3 + (6*I*a^4 + 23*I*a
^3*b + 26*I*a^2*b^2 + 8*I*a*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_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^2*b^2 - 11*I*a*b^3 - 8*I*b^4)*c
os(f*x + e)^4 - 3*I*a^3*b - 14*I*a^2*b^2 - 19*I*a*b^3 - 8*I*b^4 + (3*I*a^3*b + 17*I*a^2*b^2 + 30*I*a*b^3 + 16*
I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) + ((6*I*a^3*b + 11*I*a^2*b^2 + 4*I*a*b^3)*c
os(f*x + e)^4 + 6*I*a^4 + 17*I*a^3*b + 15*I*a^2*b^2 + 4*I*a*b^3 + (-6*I*a^4 - 23*I*a^3*b - 26*I*a^2*b^2 - 8*I*
a*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_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*((7*a*b^3 + 8*b^4)*cos(f*x + e)^5 - 2*(2*a^2*b^2 + 9*a*b^3 + 8*b^4)*c
os(f*x + e)^3 + (3*a^2*b^2 + 11*a*b^3 + 8*b^4)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3*b^3*f*cos(
f*x + e)^4 - (a^4*b^2 + 2*a^3*b^3)*f*cos(f*x + e)^2 + (a^4*b^2 + a^3*b^3)*f)*sin(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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)**(3/2),x)

[Out]

Integral(cot(e + f*x)**4/(a + b*sin(e + f*x)**2)**(3/2), x)

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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)^(3/2),x, algorithm="giac")

[Out]

integrate(cot(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,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)^(3/2),x)

[Out]

int(cot(e + f*x)^4/(a + b*sin(e + f*x)^2)^(3/2), x)

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