3.1161 \(\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=447 \[ -\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (-141 a^2 b^2+8 a^4+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 \left (-149 a^4 b^2-516 a^2 b^4+8 a^6-36 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left (-147 a^2 b^2+8 a^4+444 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d} \]

[Out]

(-2*(8*a^4 - 141*a^2*b^2 + 36*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(693*b^2*d) - (2*a*(8*a^2 - 131*b^2)
*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(693*b^2*d) - (2*(8*a^2 - 117*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])
^(5/2))/(693*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*
(a + b*Sin[c + d*x])^(7/2))/(11*b*d) + (2*a*(8*a^4 - 147*a^2*b^2 + 444*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b
)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^6 - 149*a^4*b^2
- 516*a^2*b^4 - 36*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(693*
b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a^3*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c +
d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.42183, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345, Rules used = {2895, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (-141 a^2 b^2+8 a^4+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 \left (-149 a^4 b^2-516 a^2 b^4+8 a^6-36 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left (-147 a^2 b^2+8 a^4+444 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{a+b \sin (c+d x)}}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(8*a^4 - 141*a^2*b^2 + 36*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(693*b^2*d) - (2*a*(8*a^2 - 131*b^2)
*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(693*b^2*d) - (2*(8*a^2 - 117*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])
^(5/2))/(693*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*
(a + b*Sin[c + d*x])^(7/2))/(11*b*d) + (2*a*(8*a^4 - 147*a^2*b^2 + 444*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b
)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^6 - 149*a^4*b^2
- 516*a^2*b^4 - 36*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(693*
b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a^3*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c +
d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])

Rule 2895

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a*(n + 3)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d*f*(m
+ n + 3)*(m + n + 4)), x] + (-Dist[1/(b^2*(m + n + 3)*(m + n + 4)), Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x
])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*
(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e
+ f*x])^(m + 1))/(b*d^2*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[
m, 0] || IntegersQ[2*m, 2*n]) &&  !m < -1 &&  !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]

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 \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac{4 \int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (-\frac{99 b^2}{4}+\frac{5}{2} a b \sin (c+d x)-\frac{1}{4} \left (8 a^2-117 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{99 b^2}\\ &=-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac{8 \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac{693 a b^2}{8}+\frac{3}{4} b \left (5 a^2-18 b^2\right ) \sin (c+d x)-\frac{5}{8} a \left (8 a^2-131 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{693 b^2}\\ &=-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac{16 \int \csc (c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{3465}{16} a^2 b^2+\frac{15}{8} a b \left (a^2-68 b^2\right ) \sin (c+d x)-\frac{15}{16} \left (8 a^4-141 a^2 b^2+36 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3465 b^2}\\ &=-\frac{2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}-\frac{32 \int \frac{\csc (c+d x) \left (-\frac{10395}{32} a^3 b^2-\frac{15}{16} b \left (a^4+480 a^2 b^2+18 b^4\right ) \sin (c+d x)-\frac{15}{32} a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{10395 b^2}\\ &=-\frac{2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac{32 \int \frac{\csc (c+d x) \left (\frac{10395 a^3 b^3}{32}-\frac{15}{32} \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{10395 b^3}+\frac{\left (a \left (8 a^4-147 a^2 b^2+444 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{693 b^3}\\ &=-\frac{2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+a^3 \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{693 b^3}+\frac{\left (a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{693 b^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac{2 a \left (8 a^4-147 a^2 b^2+444 b^4\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)}}{693 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (a^3 \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}{\sqrt{a+b \sin (c+d x)}}-\frac{\left (\left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\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}{693 b^3 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^2 d}-\frac{2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac{2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac{8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac{2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac{2 a \left (8 a^4-147 a^2 b^2+444 b^4\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)}}{693 b^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\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}}}{693 b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{2 a^3 \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}}}{d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 3.79175, size = 521, normalized size = 1.17 \[ \frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)} \left (4 \left (113 a^2 b^2-54 b^4\right ) \cos (2 (c+d x))+2660 a^2 b^2-24 a^3 b \sin (c+d x)+32 a^4+1954 a b^3 \sin (c+d x)+322 a b^3 \sin (3 (c+d x))-63 b^4 \cos (4 (c+d x))-9 b^4\right )-2 \left (\frac{8 b \left (480 a^2 b^2+a^4+18 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{2 a \left (1239 a^2 b^2+8 a^4+444 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{2 i \left (-147 a^2 b^2+8 a^4+444 b^4\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{b^2 \sqrt{-\frac{1}{a+b}}}\right )}{2772 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(((2*I)*(8*a^4 - 147*a^2*b^2 + 444*b^4)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*S
in[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)])/(b^2*Sqrt
[-(a + b)^(-1)]) + (8*b*(a^4 + 480*a^2*b^2 + 18*b^4)*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*a*(8*a^4 + 1239*a^2*b^2 + 444*b^4)*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]]) + Cos[c + d*x]
*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 + 2660*a^2*b^2 - 9*b^4 + 4*(113*a^2*b^2 - 54*b^4)*Cos[2*(c + d*x)] - 63*b^4*
Cos[4*(c + d*x)] - 24*a^3*b*Sin[c + d*x] + 1954*a*b^3*Sin[c + d*x] + 322*a*b^3*Sin[3*(c + d*x)]))/(2772*b^2*d)

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Maple [B]  time = 1.829, size = 1573, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c))^(5/2),x)

[Out]

2/693*(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)*Elli
pticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6-36*((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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/
2))*b^7-8*((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^7+224*a*b^6*sin(d*x+c)^6+274*a^2*b^5*sin(d*x+c)^5+1
16*a^3*b^4*sin(d*x+c)^4-706*a*b^6*sin(d*x+c)^4-a^4*b^3*sin(d*x+c)^3-1028*a^2*b^5*sin(d*x+c)^3-4*a^5*b^2*sin(d*
x+c)^2-505*a^3*b^4*sin(d*x+c)^2+518*a*b^6*sin(d*x+c)^2+a^4*b^3*sin(d*x+c)+754*a^2*b^5*sin(d*x+c)-693*a^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)*b^4*EllipticPi(((a+b
*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))+693*a^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)*b^5*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(
a+b))^(1/2))+389*a^3*b^4-36*a*b^6+4*a^5*b^2-591*((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^4+8*((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^6*b-6*((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*b^2-149*(
(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^3+1107*((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^4-516*((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)*Ellipti
cF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^5-408*((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^6+155*((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)*E
llipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2+63*b^7*sin(d*x+c)^7-180*b^7*sin(d*x+c)^5+
153*b^7*sin(d*x+c)^3-36*b^7*sin(d*x+c))/b^4/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{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^(5/2)*cos(d*x + c)^3*cot(d*x + c), 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(cos(d*x+c)^3*cot(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(cos(d*x+c)**3*cot(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(cos(d*x+c)^3*cot(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out