3.1182 \(\int \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=416 \[ \frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (32 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left (36 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}} \]

[Out]

((6*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^
2)/(3*a*d*Sqrt[a + b*Sin[c + d*x]]) + (5*(16*a^2 - 21*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d) -
((24*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^3*b*d) + (5*(16*a^2 - 21*b^2)*El
lipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a +
b)]) - ((32*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24
*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(36*a^2 - 35*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^4*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.15481, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2724, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (32 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{24 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left (36 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{8 a^4 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^
2)/(3*a*d*Sqrt[a + b*Sin[c + d*x]]) + (5*(16*a^2 - 21*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d) -
((24*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^3*b*d) + (5*(16*a^2 - 21*b^2)*El
lipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a +
b)]) - ((32*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24
*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(36*a^2 - 35*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^4*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2724

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/(3*a^2*b*(m + 1)), Int[((a + b*Sin[e + f*x])
^(m + 1)*Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2,
x])/Sin[e + f*x]^3, x], x] - Simp[((3*a^2 + b^2*(m - 2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(3*a^2*b*
f*(m + 1)*Sin[e + f*x]^2), x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 3055

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[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

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 \frac{\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}+\frac{2 \int \frac{\csc ^3(c+d x) \left (\frac{1}{4} \left (24 a^2-35 b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{3}{4} \left (4 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a^2 b}\\ &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{5}{8} b \left (16 a^2-21 b^2\right )+\frac{7}{4} a b^2 \sin (c+d x)+\frac{1}{8} b \left (24 a^2-35 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a^3 b}\\ &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{\int \frac{\csc (c+d x) \left (\frac{3}{16} b^2 \left (36 a^2-35 b^2\right )+\frac{1}{8} a b \left (24 a^2-35 b^2\right ) \sin (c+d x)+\frac{5}{16} b^2 \left (16 a^2-21 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a^4 b}\\ &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}-\frac{\int \frac{\csc (c+d x) \left (-\frac{3}{16} b^3 \left (36 a^2-35 b^2\right )+\frac{1}{16} a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a^4 b^2}+\frac{\left (5 \left (16 a^2-21 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{48 a^4}\\ &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}-\frac{\left (32 a^2-35 b^2\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{48 a^3}+\frac{\left (b \left (36 a^2-35 b^2\right )\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{16 a^4}+\frac{\left (5 \left (16 a^2-21 b^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 a^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{5 \left (16 a^2-21 b^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 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (\left (32 a^2-35 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^3 \sqrt{a+b \sin (c+d x)}}+\frac{\left (b \left (36 a^2-35 b^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 a^4 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt{a+b \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^4 d}-\frac{\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{12 a^3 b d}+\frac{5 \left (16 a^2-21 b^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 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (32 a^2-35 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^3 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left (36 a^2-35 b^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 a^4 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 5.56461, size = 468, normalized size = 1.12 \[ \frac{-\frac{4 \left (\left (105 b^3-80 a^2 b\right ) \cos (c+d x)+a \cot (c+d x) \left (8 a^2 \csc ^2(c+d x)-32 a^2-14 a b \csc (c+d x)+35 b^2\right )\right )}{a^4 \sqrt{a+b \sin (c+d x)}}+\frac{-\frac{8 a \left (24 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{2 b \left (315 b^2-296 a^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{10 i \left (21 b^2-16 a^2\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{a b \sqrt{-\frac{1}{a+b}}}}{a^4}}{96 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*((-80*a^2*b + 105*b^3)*Cos[c + d*x] + a*Cot[c + d*x]*(-32*a^2 + 35*b^2 - 14*a*b*Csc[c + d*x] + 8*a^2*Csc[
c + d*x]^2)))/(a^4*Sqrt[a + b*Sin[c + d*x]]) + (((10*I)*(-16*a^2 + 21*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[S
qrt[-(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 + Si
n[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - (8*a*(24*a^2 - 35*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*(-296*a^2 + 315*b^2)*Elliptic
Pi[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^4)
/(96*d)

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Maple [B]  time = 1.809, size = 1496, normalized size = 3.6 \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))^(3/2),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)^3-185*((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+105*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+si
n(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+150*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+35*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)^3-105*((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+108*((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^3*b^2*sin(d*x+c)^3-108*((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)^3-105*((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+105*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+si
n(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+80*a^3*b^2*sin(d*x+c)^5-105*a*b^4*sin(d*x+c)^5+32*a^4*b*sin(d*x+c)^4-35*a^2*b^3*sin(d*x+c)^4-66*a^3*b^2*sin
(d*x+c)^3+105*a*b^4*sin(d*x+c)^3-40*a^4*b*sin(d*x+c)^2+35*a^2*b^3*sin(d*x+c)^2-14*a^3*b^2*sin(d*x+c)+8*a^4*b)/
b/a^5/sin(d*x+c)^3/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

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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(d*x+c)^4/(a+b*sin(d*x+c))^(3/2),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 \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \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))^(3/2),x, algorithm="fricas")

[Out]

integral(-sqrt(b*sin(d*x + c) + a)*cot(d*x + c)^4/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2), x)

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

[Out]

Integral(cot(c + d*x)**4/(a + b*sin(c + d*x))**(3/2), x)

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

[Out]

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