3.1165 \(\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=449 \[ -\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{b \left (148 a^2+169 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left (492 a^2-5 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (-360 a^2 b^2+48 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d} \]

[Out]

-(b^2*(196*a^2 + 5*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d) + (5*b*(68*a^2 + b^2)*Cot[c + d*x]*(
a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((60*a^2 + b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))
/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*
x]^3*(a + b*Sin[c + d*x])^(7/2))/(4*a*d) + (b*(492*a^2 - 5*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S
qrt[a + b*Sin[c + d*x]])/(64*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(148*a^2 + 169*b^2)*EllipticF[(c - P
i/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 -
360*a^2*b^2 - 5*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*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.51137, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {2893, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{b \left (148 a^2+169 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 d \sqrt{a+b \sin (c+d x)}}+\frac{b \left (492 a^2-5 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (-360 a^2 b^2+48 a^4-5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{64 a d \sqrt{a+b \sin (c+d x)}}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

-(b^2*(196*a^2 + 5*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d) + (5*b*(68*a^2 + b^2)*Cot[c + d*x]*(
a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((60*a^2 + b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))
/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*
x]^3*(a + b*Sin[c + d*x])^(7/2))/(4*a*d) + (b*(492*a^2 - 5*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S
qrt[a + b*Sin[c + d*x]])/(64*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(148*a^2 + 169*b^2)*EllipticF[(c - P
i/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 -
360*a^2*b^2 - 5*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*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 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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) (a+b \sin (c+d x))^{5/2} \, dx &=\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac{\int \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac{1}{4} \left (60 a^2+b^2\right )+\frac{5}{2} a b \sin (c+d x)-\frac{3}{4} \left (16 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2}\\ &=\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac{\int \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac{5}{8} b \left (68 a^2+b^2\right )-\frac{3}{4} a \left (12 a^2-5 b^2\right ) \sin (c+d x)-\frac{3}{8} b \left (124 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac{\int \csc (c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{3}{16} \left (48 a^4-360 a^2 b^2-5 b^4\right )-\frac{3}{8} a b \left (148 a^2-5 b^2\right ) \sin (c+d x)-\frac{9}{16} b^2 \left (196 a^2+5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=-\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac{\int \frac{\csc (c+d x) \left (-\frac{9}{32} a \left (48 a^4-360 a^2 b^2-5 b^4\right )-\frac{9}{16} a^2 b \left (172 a^2-87 b^2\right ) \sin (c+d x)-\frac{9}{32} a b^2 \left (492 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{36 a^2}\\ &=-\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac{\int \frac{\csc (c+d x) \left (\frac{9}{32} a b \left (48 a^4-360 a^2 b^2-5 b^4\right )-\frac{9}{32} a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{36 a^2 b}+\frac{\left (b \left (492 a^2-5 b^2\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{128 a}\\ &=-\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}-\frac{1}{128} \left (b \left (148 a^2+169 b^2\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx+\frac{\left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{128 a}+\frac{\left (b \left (492 a^2-5 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}{128 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac{b \left (492 a^2-5 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)}}{64 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (b \left (148 a^2+169 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}{128 \sqrt{a+b \sin (c+d x)}}+\frac{\left (\left (48 a^4-360 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 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{64 a^2 d}+\frac{5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac{b \left (492 a^2-5 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)}}{64 a d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{b \left (148 a^2+169 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}}}{64 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (48 a^4-360 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 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.59333, size = 655, normalized size = 1.46 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (\frac{1}{96} \csc ^2(c+d x) \left (60 a^2 \cos (c+d x)-59 b^2 \cos (c+d x)\right )+\frac{5 \csc (c+d x) \left (116 a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right )}{192 a}-\frac{1}{4} a^2 \cot (c+d x) \csc ^3(c+d x)-\frac{17}{24} a b \cot (c+d x) \csc ^2(c+d x)-\frac{2}{3} b^2 \cos (c+d x)\right )}{d}+\frac{-\frac{2 \left (688 a^3 b-348 a b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left (-228 a^2 b^2+96 a^4-15 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left (5 b^4-492 a^2 b^2\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{256 a d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(((-2*b^2*Cos[c + d*x])/3 + (5*(116*a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x])*Csc[c + d*x])/(192*a) + ((60*a^2*
Cos[c + d*x] - 59*b^2*Cos[c + d*x])*Csc[c + d*x]^2)/96 - (17*a*b*Cot[c + d*x]*Csc[c + d*x]^2)/24 - (a^2*Cot[c
+ d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(688*a^3*b - 348*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*(96*a^4 - 228*a^2*b^2 -
15*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)*(-492*a^2*b^2 + 5*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)]*Sq
rt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*S
in[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)]))/(256*a*d)

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Maple [B]  time = 1.879, size = 1777, normalized size = 4. \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))^(5/2),x)

[Out]

1/192*(128*a^2*b^3*sin(d*x+c)^7+5*a^2*b^3*sin(d*x+c)^5+706*a^3*b^2*sin(d*x+c)^4-184*a^4*b*sin(d*x+c)+884*a^4*b
*sin(d*x+c)^3-452*a^3*b^2*sin(d*x+c)^6+15*a*b^4*sin(d*x+c)^6-700*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)^4-1476*((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)^4-254*a^3*b^2*sin(d*x+c)^2
-15*a*b^4*sin(d*x+c)^4-133*a^2*b^3*sin(d*x+c)^3+1032*((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*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)^4-1080*((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+1491*((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+507
*((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+1080*((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-
1998*((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+444*((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^2/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))^(5/2),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))^(5/2),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))**(5/2),x)

[Out]

Timed out

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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*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out