3.1175 \(\int \frac{\cot ^4(c+d x) \csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx\)

Optimal. Leaf size=412 \[ -\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{b \left (68 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left (188 a^2-105 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (-72 a^2 b^2+48 a^4+35 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d} \]

[Out]

-(b*(188*a^2 - 105*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d) + (5*(12*a^2 - 7*b^2)*Cot[c + d*x]*
Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(96*a^3*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])
/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(4*a*d) - (b*(188*a^2 - 105*b^2)*Elliptic
E[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])
+ (b*(68*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)])/(192*a
^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 - 72*a^2*b^2 + 35*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a +
b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^4*d*Sqrt[a + b*Sin[c + d*x]])

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Rubi [A]  time = 1.24434, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2893, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{b \left (68 a^2-35 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^3 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left (188 a^2-105 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{192 a^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (-72 a^2 b^2+48 a^4+35 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a^4 d \sqrt{a+b \sin (c+d x)}}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d} \]

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*(188*a^2 - 105*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d) + (5*(12*a^2 - 7*b^2)*Cot[c + d*x]*
Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(96*a^3*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])
/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(4*a*d) - (b*(188*a^2 - 105*b^2)*Elliptic
E[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])
+ (b*(68*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)])/(192*a
^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 - 72*a^2*b^2 + 35*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a +
b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^4*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 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) \csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}-\frac{\int \frac{\csc ^3(c+d x) \left (\frac{5}{4} \left (12 a^2-7 b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{3}{4} \left (16 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{12 a^2}\\ &=\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}-\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{1}{8} b \left (188 a^2-105 b^2\right )-\frac{1}{4} a \left (36 a^2-7 b^2\right ) \sin (c+d x)+\frac{5}{8} b \left (12 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{24 a^3}\\ &=-\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}-\frac{\int \frac{\csc (c+d x) \left (-\frac{3}{16} \left (48 a^4-72 a^2 b^2+35 b^4\right )+\frac{5}{8} a b \left (12 a^2-7 b^2\right ) \sin (c+d x)+\frac{1}{16} b^2 \left (188 a^2-105 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{24 a^4}\\ &=-\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}+\frac{\int \frac{\csc (c+d x) \left (\frac{3}{16} b \left (48 a^4-72 a^2 b^2+35 b^4\right )+\frac{1}{16} a b^2 \left (68 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{24 a^4 b}-\frac{\left (b \left (188 a^2-105 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{384 a^4}\\ &=-\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}+\frac{\left (b \left (68 a^2-35 b^2\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{384 a^3}+\frac{\left (48 a^4-72 a^2 b^2+35 b^4\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{128 a^4}-\frac{\left (b \left (188 a^2-105 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^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}-\frac{b \left (188 a^2-105 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^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (b \left (68 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}{384 a^3 \sqrt{a+b \sin (c+d x)}}+\frac{\left (\left (48 a^4-72 a^2 b^2+35 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^4 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{b \left (188 a^2-105 b^2\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{192 a^4 d}+\frac{5 \left (12 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{96 a^3 d}+\frac{7 b \cot (c+d x) \csc ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 a d}-\frac{b \left (188 a^2-105 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^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left (68 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}}}{192 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (48 a^4-72 a^2 b^2+35 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^4 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.55887, size = 647, normalized size = 1.57 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (\frac{5 \csc ^2(c+d x) \left (12 a^2 \cos (c+d x)-7 b^2 \cos (c+d x)\right )}{96 a^3}+\frac{\csc (c+d x) \left (105 b^3 \cos (c+d x)-188 a^2 b \cos (c+d x)\right )}{192 a^4}+\frac{7 b \cot (c+d x) \csc ^2(c+d x)}{24 a^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a}\right )}{d}+\frac{-\frac{2 \left (140 a b^3-240 a^3 b\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left (-620 a^2 b^2+288 a^4+315 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left (188 a^2 b^2-105 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{768 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((((-188*a^2*b*Cos[c + d*x] + 105*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a^4) + (5*(12*a^2*Cos[c + d*x] - 7*b^2*
Cos[c + d*x])*Csc[c + d*x]^2)/(96*a^3) + (7*b*Cot[c + d*x]*Csc[c + d*x]^2)/(24*a^2) - (Cot[c + d*x]*Csc[c + d*
x]^3)/(4*a))*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(-240*a^3*b + 140*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 - 620*a^2*b^2 + 315*b^4)*E
llipticPi[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)*(188*a^2*b^2 - 105*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^4*d)

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Maple [B]  time = 1.953, 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*(35*a^2*b^3*sin(d*x+c)^5+174*a^3*b^2*sin(d*x+c)^4-8*a^4*b*sin(d*x+c)+76*a^4*b*sin(d*x+c)^3-188*a^3*b^2*
sin(d*x+c)^6+105*a*b^4*sin(d*x+c)^6-68*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)^4-105*((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))*b^5*sin(d*x+c)^4-188*((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^5*sin(d*x+c)^4+14*a^3*b^2*sin(d*x+c)^2-105*a*b^4*sin(d*x+c)^4-35*
a^2*b^3*sin(d*x+c)^3+120*((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)^4-105
*((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)*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+216*((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+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)^4+293*((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)*Elli
pticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^4-35*((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+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)^4-
216*((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)^4-258*((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+68*((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^5/sin(d*x+c)^4/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*csc(d*x+c)/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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

[Out]

Timed out

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

[Out]

Timed out

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

[Out]

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