3.541 \(\int \frac{\cot ^2(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=287 \[ -\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)}+\frac{(3 a+4 b) \cot (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{4 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]

[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])

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Rubi [A]  time = 0.365945, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3196, 469, 579, 583, 524, 426, 424, 421, 419} \[ -\frac{(7 a+8 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 f (a+b)}+\frac{(3 a+4 b) \cot (e+f x)}{3 a^2 f (a+b) \sqrt{a+b \sin ^2(e+f x)}}+\frac{4 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(7 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^3 f (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[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 3196

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]

Rule 469

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 579

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 583

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 524

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 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 (\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 (a+b) f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}+\frac{4 \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*}

Mathematica [A]  time = 2.70433, size = 209, normalized size = 0.73 \[ \frac{-\frac{\cot (e+f x) \left (-4 b \left (11 a^2+19 a b+8 b^2\right ) \cos (2 (e+f x))+68 a^2 b+24 a^3+b^2 (7 a+8 b) \cos (4 (e+f x))+69 a b^2+24 b^3\right )}{\sqrt{2}}+8 a^2 (a+b) \left (\frac{2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} F\left (e+f x\left |-\frac{b}{a}\right .\right )-2 a^2 (7 a+8 b) \left (\frac{2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{6 a^3 f (a+b) (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 1.698, size = 411, normalized size = 1.4 \begin{align*}{\frac{1}{3\,\sin \left ( fx+e \right ){a}^{3} \left ( a+b \right ) \cos \left ( fx+e \right ) f} \left ( -\sqrt{-{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a+b}{a}}}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}ab \left ( 4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) a+4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) b-7\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) a-8\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) b \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\sqrt{-{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a+b}{a}}}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}a \left ( 4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){a}^{2}+8\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) ab+4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){b}^{2}-7\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){a}^{2}-15\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) ab-8\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){b}^{2} \right ) \sin \left ( fx+e \right ) + \left ( -7\,a{b}^{2}-8\,{b}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{6}+ \left ( 11\,{a}^{2}b+26\,a{b}^{2}+16\,{b}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -3\,{a}^{3}-14\,{a}^{2}b-19\,a{b}^{2}-8\,{b}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x)

[Out]

1/3*(-(-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*E
llipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-7*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a-8*EllipticE(sin(f*x+e),(-1/a*b)
^(1/2))*b)*sin(f*x+e)*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)+(-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+(-3*a^
3-14*a^2*b-19*a*b^2-8*b^3)*cos(f*x+e)^2)/sin(f*x+e)/a^3/(a+b)/(a+b*sin(f*x+e)^2)^(3/2)/cos(f*x+e)/f

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \cot \left (f x + e\right )^{2}}{b^{3} \cos \left (f x + e\right )^{6} - 3 \,{\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \,{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^2/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4
- a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2/(a+b*sin(f*x+e)**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)