3.509 \(\int \cot ^4(e+f x) (a+b \sin ^2(e+f x))^{3/2} \, dx\)

Optimal. Leaf size=276 \[ \frac{(3 a-5 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{(5 a-3 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]

[Out]

((a - b)*Cos[e + f*x]^2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f + ((3*a - 5*b)*Cos[e + f*x]*Sin[e + f*x]*Sq
rt[a + b*Sin[e + f*x]^2])/(3*f) - (Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2))/(3*f) + (8*(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*f*Sqrt[1 + (b*Si
n[e + f*x]^2)/a]) - ((5*a - 3*b)*(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*f*Sqrt[a + b*Sin[e + f*x]^2])

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Rubi [A]  time = 0.350888, antiderivative size = 276, 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, 473, 580, 528, 524, 426, 424, 421, 419} \[ \frac{(3 a-5 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{(5 a-3 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*Cos[e + f*x]^2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f + ((3*a - 5*b)*Cos[e + f*x]*Sin[e + f*x]*Sq
rt[a + b*Sin[e + f*x]^2])/(3*f) - (Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2))/(3*f) + (8*(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*f*Sqrt[1 + (b*Si
n[e + f*x]^2)/a]) - ((5*a - 3*b)*(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*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 473

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*(c + d*x^n)^q)/(e*(m + 1)), x] - Dist[n/(e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^(p -
1)*(c + d*x^n)^(q - 1)*Simp[b*c*p + a*d*q + b*d*(p + q)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x
]

Rule 580

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)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

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 \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 ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{x^4} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{\left (2 \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2} \left (-\frac{3}{2} (a-b)-3 b x^2\right ) \sqrt{a+b x^2}}{x^2} \, dx,x,\sin (e+f x)\right )}{3 f}\\ &=\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{\left (2 \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2} \left (\frac{3}{2} \left (2 a^2-5 a b+b^2\right )+\frac{3}{2} (3 a-5 b) b x^2\right )}{\sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 f}\\ &=\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(3 a-5 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{\left (2 \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\frac{3}{2} (a-3 b) (3 a-b) b+12 (a-b) b^2 x^2}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{9 b f}\\ &=\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(3 a-5 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{\left (8 (a-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 f}-\frac{\left ((5 a-3 b) (a+b) \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 f}\\ &=\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(3 a-5 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{\left (8 (a-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 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{\left ((5 a-3 b) (a+b) \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 f \sqrt{a+b \sin ^2(e+f x)}}\\ &=\frac{(a-b) \cos ^2(e+f x) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{f}+\frac{(3 a-5 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\cot ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac{8 (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 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{(5 a-3 b) (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 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 5.05421, size = 218, normalized size = 0.79 \[ \frac{-4 \left (5 a^2+2 a b-3 b^2\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )-\frac{\cot (e+f x) \csc ^2(e+f x) \left (\left (64 a^2-32 a b-79 b^2\right ) \cos (2 (e+f x))-32 a^2-2 b (6 a-11 b) \cos (4 (e+f x))+44 a b-b^2 \cos (6 (e+f x))+58 b^2\right )}{4 \sqrt{2}}+32 a (a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{12 f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((-32*a^2 + 44*a*b + 58*b^2 + (64*a^2 - 32*a*b - 79*b^2)*Cos[2*(e + f*x)] - 2*(6*a - 11*b)*b*Cos[4*(e + f*x)
] - b^2*Cos[6*(e + f*x)])*Cot[e + f*x]*Csc[e + f*x]^2)/(4*Sqrt[2]) + 32*a*(a - b)*Sqrt[(2*a + b - b*Cos[2*(e +
f*x)])/a]*EllipticE[e + f*x, -(b/a)] - 4*(5*a^2 + 2*a*b - 3*b^2)*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*Ellip
ticF[e + f*x, -(b/a)])/(12*f*Sqrt[2*a + b - b*Cos[2*(e + f*x)]])

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Maple [A]  time = 1.303, size = 419, normalized size = 1.5 \begin{align*} -{\frac{1}{3\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f} \left ( -{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{8}+5\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+2\,b\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) a \left ( \sin \left ( fx+e \right ) \right ) ^{3}-3\,{b}^{2}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{3}-8\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+8\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}ab \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{6}-3\,{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{6}+4\,{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}-8\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{4}+4\,{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}-5\,{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+5\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{2}+{a}^{2} \right ){\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}

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)

[Out]

-1/3*(-b^2*sin(f*x+e)^8+5*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/
2)*a^2*sin(f*x+e)^3+2*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)^3-3*b^2*(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))*s
in(f*x+e)^3-8*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*a^2*sin(f
*x+e)^3+8*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*a*b*sin(f*x+e
)^3+3*a*b*sin(f*x+e)^6-3*b^2*sin(f*x+e)^6+4*a^2*sin(f*x+e)^4-8*a*b*sin(f*x+e)^4+4*b^2*sin(f*x+e)^4-5*a^2*sin(f
*x+e)^2+5*a*b*sin(f*x+e)^2+a^2)/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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}

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]

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

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

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]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}

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