3.1169 \(\int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx\)

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

[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]])

________________________________________________________________________________________

Rubi [A]  time = 0.885941, antiderivative size = 405, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {2895, 3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (80 a^2-117 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{693 b^3 d}+\frac{8 a \left (120 a^2-179 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^4 d}-\frac{8 \left (-247 a^2 b^2+160 a^4+45 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3465 b^5 d}+\frac{8 \left (-614 a^4 b^2+249 a^2 b^4+320 a^6+45 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3465 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left (-267 a^2 b^2+160 a^4+69 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3465 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{99 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{11 b d} \]

Antiderivative was successfully verified.

[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 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 3023

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 2752

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 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 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]

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\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 ) 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}}}{3465 b^6 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 4.11822, size = 326, normalized size = 0.8 \[ \frac{b \cos (c+d x) \left (3752 a^2 b^3 \sin (c+d x)+200 a^2 b^3 \sin (3 (c+d x))-128 \left (5 a^3 b^2-6 a b^4\right ) \cos (2 (c+d x))+16448 a^3 b^2-2560 a^4 b \sin (c+d x)-10240 a^5+70 a b^4 \cos (4 (c+d x))-3718 a b^4+990 b^5 \sin (c+d x)-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )-64 \left (-614 a^4 b^2+249 a^2 b^4+320 a^6+45 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+128 a \left (-267 a^3 b^2-267 a^2 b^3+160 a^4 b+160 a^5+69 a b^4+69 b^5\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{27720 b^6 d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[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]])

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Maple [B]  time = 1.609, size = 1356, normalized size = 3.4 \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)^4*sin(d*x+c)^2/(a+b*sin(d*x+c))^(1/2),x)

[Out]

-2/3465*(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)*El
lipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^6+180*((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^7-1280*((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+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)-98
8*a^3*b^4+180*a*b^6+640*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+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)*EllipticF(((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+9
96*((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)*Ellip
ticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2-315*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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}}{\sqrt{b \sin \left (d x + c\right ) + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(cos(d*x + c)^6 - cos(d*x + c)^4)/sqrt(b*sin(d*x + c) + a), x)

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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)**4*sin(d*x+c)**2/(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{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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