\(\int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [1169]

Optimal result

Integrand size = 31, antiderivative size = 405 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {16 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3465 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[Out]

-8/3465*(160*a^4-247*a^2*b^2+45*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+8/3465*a*(120*a^2-179*b^2)*cos(d*
x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^4/d-2/693*(80*a^2-117*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(
1/2)/b^3/d+20/99*a*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/b^2/d-2/11*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(
d*x+c))^(1/2)/b/d+16/3465*a*(160*a^4-267*a^2*b^2+69*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^6/d/((a+b*sin(d
*x+c))/(a+b))^(1/2)-8/3465*(320*a^6-614*a^4*b^2+249*a^2*b^4+45*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^6/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2974, 3128, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {8 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}-\frac {16 a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{3465 b^6 d \sqrt {a+b \sin (c+d x)}}+\frac {20 a \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d} \]

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(-8*(160*a^4 - 247*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^5*d) + (8*a*(120*a^2 - 179
*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^4*d) - (2*(80*a^2 - 117*b^2)*Cos[c + d*x]*Si
n[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]
])/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b*d) - (16*a*(160*a^4 - 267*a^2*b
^2 + 69*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^6*d*Sqrt[(a + b*Si
n[c + d*x])/(a + b)]) + (8*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/
(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3465*b^6*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 2831

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

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 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((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 + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

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])))

Rubi steps \begin{align*} \text {integral}& = \frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {4 \int \frac {\sin ^2(c+d x) \left (\frac {3}{4} \left (20 a^2-33 b^2\right )-\frac {1}{2} a b \sin (c+d x)-\frac {1}{4} \left (80 a^2-117 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{99 b^2} \\ & = -\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {8 \int \frac {\sin (c+d x) \left (-\frac {1}{2} a \left (80 a^2-117 b^2\right )+\frac {1}{2} b \left (5 a^2-27 b^2\right ) \sin (c+d x)+\frac {1}{2} a \left (120 a^2-179 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{693 b^3} \\ & = \frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {16 \int \frac {\frac {1}{2} a^2 \left (120 a^2-179 b^2\right )-2 a b \left (5 a^2-6 b^2\right ) \sin (c+d x)-\frac {3}{4} \left (160 a^4-247 a^2 b^2+45 b^4\right ) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^4} \\ & = -\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {32 \int \frac {\frac {3}{8} b \left (80 a^4-111 a^2 b^2-45 b^4\right )+\frac {3}{4} a \left (160 a^4-267 a^2 b^2+69 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{10395 b^5} \\ & = -\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {\left (8 a \left (160 a^4-267 a^2 b^2+69 b^4\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{3465 b^6}+\frac {\left (4 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{3465 b^6} \\ & = -\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {\left (8 a \left (160 a^4-267 a^2 b^2+69 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}{3465 b^6 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 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}{3465 b^6 \sqrt {a+b \sin (c+d x)}} \\ & = -\frac {8 \left (160 a^4-247 a^2 b^2+45 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d}+\frac {8 a \left (120 a^2-179 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{3465 b^4 d}-\frac {2 \left (80 a^2-117 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{693 b^3 d}+\frac {20 a \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{99 b^2 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b d}-\frac {16 a \left (160 a^4-267 a^2 b^2+69 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)}}{3465 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 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}}}{3465 b^6 d \sqrt {a+b \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.30 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {128 a \left (160 a^5+160 a^4 b-267 a^3 b^2-267 a^2 b^3+69 a b^4+69 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-64 \left (320 a^6-614 a^4 b^2+249 a^2 b^4+45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-10240 a^5+16448 a^3 b^2-3718 a b^4-128 \left (5 a^3 b^2-6 a b^4\right ) \cos (2 (c+d x))+70 a b^4 \cos (4 (c+d x))-2560 a^4 b \sin (c+d x)+3752 a^2 b^3 \sin (c+d x)+990 b^5 \sin (c+d x)+200 a^2 b^3 \sin (3 (c+d x))-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )}{27720 b^6 d \sqrt {a+b \sin (c+d x)}} \]

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(128*a*(160*a^5 + 160*a^4*b - 267*a^3*b^2 - 267*a^2*b^3 + 69*a*b^4 + 69*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4,
(2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 64*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*Elliptic
F[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-10240*a^5 + 1644
8*a^3*b^2 - 3718*a*b^4 - 128*(5*a^3*b^2 - 6*a*b^4)*Cos[2*(c + d*x)] + 70*a*b^4*Cos[4*(c + d*x)] - 2560*a^4*b*S
in[c + d*x] + 3752*a^2*b^3*Sin[c + d*x] + 990*b^5*Sin[c + d*x] + 200*a^2*b^3*Sin[3*(c + d*x)] - 765*b^5*Sin[3*
(c + d*x)] - 315*b^5*Sin[5*(c + d*x)]))/(27720*b^6*d*Sqrt[a + b*Sin[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1355\) vs. \(2(439)=878\).

Time = 1.42 (sec) , antiderivative size = 1356, normalized size of antiderivative = 3.35

method	result	size
default	\(\text {Expression too large to display}\)	\(1356\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*(1280*((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
llipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b-960*((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^2-2456*((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+1692*((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+996*((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^5-732*((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+3416*((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^2-2688*((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+552*((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+180*((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)*EllipticF(((a+b*sin(d*
x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^7-1280*((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^7-988*a^3*b^
4+640*a^5*b^2+180*a*b^6+35*a*b^6*sin(d*x+c)^6-50*a^2*b^5*sin(d*x+c)^5+80*a^3*b^4*sin(d*x+c)^4-166*a*b^6*sin(d*
x+c)^4-160*a^4*b^3*sin(d*x+c)^3+322*a^2*b^5*sin(d*x+c)^3-640*a^5*b^2*sin(d*x+c)^2+908*a^3*b^4*sin(d*x+c)^2-49*
a*b^6*sin(d*x+c)^2+160*a^4*b^3*sin(d*x+c)-272*a^2*b^5*sin(d*x+c)-315*b^7*sin(d*x+c)^7+900*b^7*sin(d*x+c)^5-765
*b^7*sin(d*x+c)^3+180*b^7*sin(d*x+c))/b^7/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} {\left (640 \, a^{6} - 1308 \, a^{4} b^{2} + 609 \, a^{2} b^{4} + 135 \, b^{6}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (640 \, a^{6} - 1308 \, a^{4} b^{2} + 609 \, a^{2} b^{4} + 135 \, b^{6}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 12 \, \sqrt {2} {\left (-160 i \, a^{5} b + 267 i \, a^{3} b^{3} - 69 i \, a b^{5}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 12 \, \sqrt {2} {\left (160 i \, a^{5} b - 267 i \, a^{3} b^{3} + 69 i \, a b^{5}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (315 \, b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (80 \, a^{2} b^{4} + 9 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (320 \, a^{4} b^{2} - 294 \, a^{2} b^{4} - 45 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (175 \, a b^{5} \cos \left (d x + c\right )^{3} - 3 \, {\left (80 \, a^{3} b^{3} - 61 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{10395 \, b^{7} d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/10395*(2*sqrt(2)*(640*a^6 - 1308*a^4*b^2 + 609*a^2*b^4 + 135*b^6)*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2
- 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + 2*sqrt
(2)*(640*a^6 - 1308*a^4*b^2 + 609*a^2*b^4 + 135*b^6)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2,
-8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) - 12*sqrt(2)*(-160*I*
a^5*b + 267*I*a^3*b^3 - 69*I*a*b^5)*sqrt(I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a
*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c
) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) - 12*sqrt(2)*(160*I*a^5*b - 267*I*a^3*b^3 + 69*I*a*b^5)*sqrt(-I*b)*weierst
rassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/
b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 3*(315*b^6*co
s(d*x + c)^5 - 5*(80*a^2*b^4 + 9*b^6)*cos(d*x + c)^3 + 2*(320*a^4*b^2 - 294*a^2*b^4 - 45*b^6)*cos(d*x + c) + 2
*(175*a*b^5*cos(d*x + c)^3 - 3*(80*a^3*b^3 - 61*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(
b^7*d)

Sympy [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

Integral(sin(c + d*x)**2*cos(c + d*x)**4/sqrt(a + b*sin(c + d*x)), x)

Maxima [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^2/sqrt(b*sin(d*x + c) + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^2}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]

[In]

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^(1/2),x)

[Out]

int((cos(c + d*x)^4*sin(c + d*x)^2)/(a + b*sin(c + d*x))^(1/2), x)