\(\int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) [1161]

Optimal result

Integrand size = 29, antiderivative size = 447 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{693 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {2 a^3 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-2/693*a*(8*a^2-131*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/b^2/d-2/693*(8*a^2-117*b^2)*cos(d*x+c)*(a+b*sin(d*x
+c))^(5/2)/b^2/d+8/99*a*cos(d*x+c)*(a+b*sin(d*x+c))^(7/2)/b^2/d-2/11*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(7
/2)/b/d-2/693*(8*a^4-141*a^2*b^2+36*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^2/d-2/693*a*(8*a^4-147*a^2*b^2+44
4*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/
2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+2/693*(8*a^6-149*a^4*b^2-516*a
^2*b^4-36*b^6)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*
x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^3/d/(a+b*sin(d*x+c))^(1/2)-2*a^3*(sin(1/2*c+1/4*P
i+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2,2^(1/2)*(b/(a+b))^(1/2))*
((a+b*sin(d*x+c))/(a+b))^(1/2)/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2974, 3128, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {2 a^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (8 a^2-117 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{693 b^2 d}-\frac {2 a \left (8 a^2-131 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{693 b^2 d}-\frac {2 \left (8 a^4-141 a^2 b^2+36 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^2 d}+\frac {2 a \left (8 a^4-147 a^2 b^2+444 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (8 a^6-149 a^4 b^2-516 a^2 b^4-36 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 a \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{99 b^2 d}-\frac {2 \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{11 b d} \]

[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 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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]

Rule 2886

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/
(c + d))*Sin[e + f*x]]), 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 2974

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 3081

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 3128

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 3138

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \operatorname {EllipticF}\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 \operatorname {EllipticPi}\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] (verified)

Result contains complex when optimal does not.

Time = 3.18 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.17 \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {-2 \left (\frac {2 i \left (8 a^4-147 a^2 b^2+444 b^4\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{b^2 \sqrt {-\frac {1}{a+b}}}+\frac {8 b \left (a^4+480 a^2 b^2+18 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (8 a^4+1239 a^2 b^2+444 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )+\cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4+2660 a^2 b^2-9 b^4+4 \left (113 a^2 b^2-54 b^4\right ) \cos (2 (c+d x))-63 b^4 \cos (4 (c+d x))-24 a^3 b \sin (c+d x)+1954 a b^3 \sin (c+d x)+322 a b^3 \sin (3 (c+d x))\right )}{2772 b^2 d} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1572\) vs. \(2(510)=1020\).

Time = 89.30 (sec) , antiderivative size = 1573, normalized size of antiderivative = 3.52

method	result	size
default	\(\text {Expression too large to display}\)	\(1573\)

[In]

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

[Out]

-2/693*(-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)*Elli
pticF(((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)*(-(s
in(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)*EllipticF(((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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*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^4-444*((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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6+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))+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)*Ellip
ticF(((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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*
a^7-389*a^3*b^4-4*a^5*b^2+36*a*b^6-224*a*b^6*sin(d*x+c)^6-274*a^2*b^5*sin(d*x+c)^5-116*a^3*b^4*sin(d*x+c)^4+70
6*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)-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

Fricas [F]

\[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right ) \,d x } \]

[In]

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

[Out]

integral((2*a*b*cos(d*x + c)^3*cot(d*x + c)*sin(d*x + c) - (b^2*cos(d*x + c)^5 - (a^2 + b^2)*cos(d*x + c)^3)*c
ot(d*x + c))*sqrt(b*sin(d*x + c) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right ) \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^3\,\mathrm {cot}\left (c+d\,x\right )\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

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

[Out]

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