3.1148 \(\int \cot ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=351 \[ \frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\left (32 a^2+b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left (80 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left (12 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d} \]

[Out]

((32*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[
c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(3*a*d) + ((80*a^2 + 3*b
^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d*Sqrt[(a + b*Sin[c + d*x])
/(a + b)]) - ((32*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/
(24*a*d*Sqrt[a + b*Sin[c + d*x]]) - (b*(12*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a
+ b*Sin[c + d*x])/(a + b)])/(8*a^2*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.861873, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2725, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\left (32 a^2+b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left (80 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left (12 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]

[Out]

((32*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[
c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(3*a*d) + ((80*a^2 + 3*b
^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d*Sqrt[(a + b*Sin[c + d*x])
/(a + b)]) - ((32*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/
(24*a*d*Sqrt[a + b*Sin[c + d*x]]) - (b*(12*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a
+ b*Sin[c + d*x])/(a + b)])/(8*a^2*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2725

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(3*a*f*Sin[e + f*x]^3), x] + (-Dist[1/(6*a^2), Int[((a + b*Sin[e + f*x])^m*Simp[8*
a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x])/Sin[e + f*x]^2, x
], x] - Simp[(b*(m - 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(6*a^2*f*Sin[e + f*x]^2), x]) /; FreeQ[{a,
b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1] && IntegerQ[2*m]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3059

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2655

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

Rule 2653

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

Rule 3002

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2663

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

Rule 2661

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

Rule 2807

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx &=\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}-\frac{\int \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (\frac{1}{4} \left (32 a^2-3 b^2\right )+\frac{1}{2} a b \sin (c+d x)-\frac{3}{4} \left (8 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2}\\ &=\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}-\frac{\int \frac{\csc (c+d x) \left (\frac{3}{8} b \left (12 a^2-b^2\right )-\frac{1}{4} a \left (24 a^2+b^2\right ) \sin (c+d x)-\frac{1}{8} b \left (80 a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{6 a^2}\\ &=\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}+\frac{\int \frac{\csc (c+d x) \left (-\frac{3}{8} b^2 \left (12 a^2-b^2\right )-\frac{1}{8} a b \left (32 a^2+b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{6 a^2 b}-\frac{1}{48} \left (-80-\frac{3 b^2}{a^2}\right ) \int \sqrt{a+b \sin (c+d x)} \, dx\\ &=\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}-\frac{\left (32 a^2+b^2\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{48 a}-\frac{1}{16} \left (b \left (12-\frac{b^2}{a^2}\right )\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{\left (\left (-80-\frac{3 b^2}{a^2}\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{48 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}+\frac{\left (80+\frac{3 b^2}{a^2}\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{24 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (\left (32 a^2+b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{48 a \sqrt{a+b \sin (c+d x)}}-\frac{\left (b \left (12-\frac{b^2}{a^2}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{\csc (c+d x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{16 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\left (32 a^2-3 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}+\frac{b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{3 a d}+\frac{\left (80+\frac{3 b^2}{a^2}\right ) E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{24 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (32 a^2+b^2\right ) F\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{24 a d \sqrt{a+b \sin (c+d x)}}-\frac{b \left (12-\frac{b^2}{a^2}\right ) \Pi \left (2;\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{8 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 5.50867, size = 473, normalized size = 1.35 \[ \frac{-\frac{4 \cot (c+d x) \sqrt{a+b \sin (c+d x)} \left (8 a^2 \csc ^2(c+d x)-32 a^2+2 a b \csc (c+d x)-3 b^2\right )}{a^2}+\frac{-\frac{8 a \left (24 a^2+b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 b \left (8 a^2+9 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left (80 a^2+3 b^2\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a b \sqrt{-\frac{1}{a+b}} \left (\csc ^2(c+d x)-2\right )}}{a^2}}{96 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]

[Out]

((-4*Cot[c + d*x]*(-32*a^2 - 3*b^2 + 2*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2)*Sqrt[a + b*Sin[c + d*x]])/a^2
+ (((2*I)*(80*a^2 + 3*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)
]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[
c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]]
, (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a -
b))])/(a*b*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (8*a*(24*a^2 + b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (
2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*b*(8*a^2 + 9*b^2)*EllipticPi[2
, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/a^2)/(96
*d)

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Maple [B]  time = 1.838, size = 1495, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(80*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellip
ticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*sin(d*x+c)^3-77*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-
(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/
(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^3-3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d
*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+c)^3-48*a^5*
((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b
*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*sin(d*x+c)^3-32*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)
*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*
a^4*b*sin(d*x+c)^3+78*b^2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a
-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*sin(d*x+c)^3-b^3*((a+b*sin(d*x+c)
)/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-
b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*sin(d*x+c)^3+3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/
2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x
+c)^3-36*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellipt
icPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^3+36*((a+b*sin(d*x+c))/(a-
b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^
(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^2*b^3*sin(d*x+c)^3+3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a
+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/
2))*a*b^4*sin(d*x+c)^3-3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-
b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*b^5*sin(d*x+c)^3+32*a^3*b^2*s
in(d*x+c)^5+3*a*b^4*sin(d*x+c)^5+32*a^4*b*sin(d*x+c)^4+a^2*b^3*sin(d*x+c)^4-42*a^3*b^2*sin(d*x+c)^3-3*a*b^4*si
n(d*x+c)^3-40*a^4*b*sin(d*x+c)^2-a^2*b^3*sin(d*x+c)^2+10*a^3*b^2*sin(d*x+c)+8*a^4*b)/a^3/b/sin(d*x+c)^3/cos(d*
x+c)/(a+b*sin(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sin(d*x + c) + a)*cot(d*x + c)^4, x)

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Fricas [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(d*x+c)^4*(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin{\left (c + d x \right )}} \cot ^{4}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*sin(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a + b*sin(c + d*x))*cot(c + d*x)**4, x)

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Giac [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(d*x+c)^4*(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out