3.1149 \(\int \cot ^4(c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=412 \[ \frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{b \left (196 a^2+5 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left (68 a^2-15 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (24 a^2 b^2+48 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d} \]

[Out]

(b*(68*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d) + (5*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c
+ d*x]*Sqrt[a + b*Sin[c + d*x]])/(32*a^2*d) + (5*b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(24
*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(4*a*d) + (b*(68*a^2 - 15*b^2)*EllipticE[(c
- Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b
*(196*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(192*a^2*d
*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 + 24*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S
qrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^3*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.23542, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{b \left (196 a^2+5 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left (68 a^2-15 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (24 a^2 b^2+48 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(68*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d) + (5*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c
+ d*x]*Sqrt[a + b*Sin[c + d*x]])/(32*a^2*d) + (5*b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(24
*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(4*a*d) + (b*(68*a^2 - 15*b^2)*EllipticE[(c
- Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b
*(196*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(192*a^2*d
*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 + 24*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S
qrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^3*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2893

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*d*f*(n + 1)), x] +
(-Dist[1/(a^2*d^2*(n + 1)*(n + 2)), Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) -
b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4))*Sin[e +
f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 2))/
(a^2*d^2*f*(n + 1)*(n + 2)), x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Intege
rsQ[2*m, 2*n]) &&  !m < -1 && LtQ[n, -1] && (LtQ[n, -2] || EqQ[m + n + 4, 0])

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 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 \cot ^4(c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)} \, dx &=\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}-\frac{\int \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (\frac{15}{4} \left (4 a^2-b^2\right )+\frac{1}{2} a b \sin (c+d x)-\frac{1}{4} \left (48 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2}\\ &=\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}-\frac{\int \frac{\csc ^2(c+d x) \left (\frac{1}{8} b \left (68 a^2-15 b^2\right )-\frac{1}{4} a \left (36 a^2+b^2\right ) \sin (c+d x)-\frac{1}{8} b \left (132 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{24 a^2}\\ &=\frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}-\frac{\int \frac{\csc (c+d x) \left (-\frac{3}{16} \left (48 a^4+24 a^2 b^2-5 b^4\right )-\frac{1}{8} a b \left (132 a^2-5 b^2\right ) \sin (c+d x)-\frac{1}{16} b^2 \left (68 a^2-15 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{24 a^3}\\ &=\frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}+\frac{\int \frac{\csc (c+d x) \left (\frac{3}{16} b \left (48 a^4+24 a^2 b^2-5 b^4\right )+\frac{1}{16} a b^2 \left (196 a^2+5 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{24 a^3 b}+\frac{\left (b \left (68 a^2-15 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{384 a^3}\\ &=\frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}+\frac{1}{384} \left (b \left (196+\frac{5 b^2}{a^2}\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx+\frac{\left (48 a^4+24 a^2 b^2-5 b^4\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{128 a^3}+\frac{\left (b \left (68 a^2-15 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}{384 a^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}+\frac{b \left (68 a^2-15 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)}}{192 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (b \left (196+\frac{5 b^2}{a^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}{384 \sqrt{a+b \sin (c+d x)}}+\frac{\left (\left (48 a^4+24 a^2 b^2-5 b^4\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}{128 a^3 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{b \left (68 a^2-15 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^3 d}+\frac{5 \left (4 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{32 a^2 d}+\frac{5 b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{4 a d}+\frac{b \left (68 a^2-15 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)}}{192 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left (196+\frac{5 b^2}{a^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}}}{192 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (48 a^4+24 a^2 b^2-5 b^4\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}}}{64 a^3 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.59123, size = 643, normalized size = 1.56 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (\frac{5 \csc ^2(c+d x) \left (12 a^2 \cos (c+d x)+b^2 \cos (c+d x)\right )}{96 a^2}+\frac{\csc (c+d x) \left (68 a^2 b \cos (c+d x)-15 b^3 \cos (c+d x)\right )}{192 a^3}-\frac{b \cot (c+d x) \csc ^2(c+d x)}{24 a}-\frac{1}{4} \cot (c+d x) \csc ^3(c+d x)\right )}{d}+\frac{-\frac{2 \left (528 a^3 b-20 a b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left (212 a^2 b^2+288 a^4-45 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left (15 b^4-68 a^2 b^2\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{768 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((((68*a^2*b*Cos[c + d*x] - 15*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a^3) + (5*(12*a^2*Cos[c + d*x] + b^2*Cos[c
+ d*x])*Csc[c + d*x]^2)/(96*a^2) - (b*Cot[c + d*x]*Csc[c + d*x]^2)/(24*a) - (Cot[c + d*x]*Csc[c + d*x]^3)/4)*
Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(528*a^3*b - 20*a*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(
a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(288*a^4 + 212*a^2*b^2 - 45*b^4)*EllipticPi[2, (-c
+ Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((2*I)*(-68*a^
2*b^2 + 15*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(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)]))*Sqrt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[
1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2
*a*(a + b*Sin[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(768*a^3*d)

________________________________________________________________________________________

Maple [B]  time = 1.859, size = 1761, 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*csc(d*x+c)*(a+b*sin(d*x+c))^(1/2),x)

[Out]

1/192*(5*a^2*b^3*sin(d*x+c)^5+66*a^3*b^2*sin(d*x+c)^4-56*a^4*b*sin(d*x+c)+244*a^4*b*sin(d*x+c)^3-68*a^3*b^2*si
n(d*x+c)^6+15*a*b^4*sin(d*x+c)^6-188*a^4*b*sin(d*x+c)^5-144*((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^5*sin(d*x+c)^4-15*((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)^4-68*((a+b*si
n(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^5*sin(d*x+c)^4+2*a^3*b^2*sin(d*x+c)^2-15*a*b^4*sin(d*x+c)^4-5*a^2*b^3
*sin(d*x+c)^3+264*((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^5*sin(d*x+c)^4-48*a^5-120*a^5*sin(d*x+c)^4+
168*a^5*sin(d*x+c)^2+144*((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^4*b*sin(d*x+c)^4-15*((a+b*s
in(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)^4+72*((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)^4+15*((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)^4+83*
((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)^4-5*((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))*a^2*b^3*sin(d*x+c)^4+15*((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)^4-72*((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)^4-78*((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*b^2*sin(d*x+c)^4-196*((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)^4)/a^4/sin(d*x
+c)^4/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

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} \csc \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*csc(d*x+c)*(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*csc(d*x + c), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out