3.1143 \(\int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=463 \[ -\frac{10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{8 \left (-937 a^2 b^2+480 a^4+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left (-279 a^2 b^2+160 a^4+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (-439 a^4 b^2+306 a^2 b^4+160 a^6-27 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d} \]

[Out]

(16*a*(160*a^4 - 279*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) - (8*(480*a^4 - 93
7*a^2*b^2 + 231*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) + (8*a*(40*a^2 - 81*b^2)*Cos[c + d
*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(3003*b^4*d) - (10*(16*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*
(a + b*Sin[c + d*x])^(3/2))/(1287*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(143*
b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(13*b*d) - (8*(320*a^6 - 798*a^4*b^2 + 435
*a^2*b^4 - 693*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(
a + b*Sin[c + d*x])/(a + b)]) + (16*a*(160*a^6 - 439*a^4*b^2 + 306*a^2*b^4 - 27*b^6)*EllipticF[(c - Pi/2 + d*x
)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^6*d*Sqrt[a + b*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.04769, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2895, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{10 \left (16 a^2-33 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{8 \left (-937 a^2 b^2+480 a^4+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{16 a \left (-279 a^2 b^2+160 a^4+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (-439 a^4 b^2+306 a^2 b^4+160 a^6-27 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left (-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d} \]

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]

(16*a*(160*a^4 - 279*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) - (8*(480*a^4 - 93
7*a^2*b^2 + 231*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) + (8*a*(40*a^2 - 81*b^2)*Cos[c + d
*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(3003*b^4*d) - (10*(16*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*
(a + b*Sin[c + d*x])^(3/2))/(1287*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(143*
b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(13*b*d) - (8*(320*a^6 - 798*a^4*b^2 + 435
*a^2*b^4 - 693*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(
a + b*Sin[c + d*x])/(a + b)]) + (16*a*(160*a^6 - 439*a^4*b^2 + 306*a^2*b^4 - 27*b^6)*EllipticF[(c - Pi/2 + d*x
)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*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 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

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 \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) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{4 \int \sin ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (\frac{1}{4} \left (60 a^2-143 b^2\right )+\frac{1}{2} a b \sin (c+d x)-\frac{5}{4} \left (16 a^2-33 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{143 b^2}\\ &=-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{8 \int \sin (c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{5}{2} a \left (16 a^2-33 b^2\right )-\frac{1}{2} b \left (5 a^2+33 b^2\right ) \sin (c+d x)+\frac{3}{2} a \left (40 a^2-81 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{1287 b^3}\\ &=\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{16 \int \sqrt{a+b \sin (c+d x)} \left (\frac{3}{2} a^2 \left (40 a^2-81 b^2\right )+5 a b \left (2 a^2-3 b^2\right ) \sin (c+d x)-\frac{1}{4} \left (480 a^4-937 a^2 b^2+231 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{9009 b^4}\\ &=-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{32 \int \sqrt{a+b \sin (c+d x)} \left (-\frac{3}{8} b \left (80 a^4-127 a^2 b^2+231 b^4\right )+\frac{3}{4} a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \sin (c+d x)\right ) \, dx}{45045 b^5}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{64 \int \frac{\frac{3}{16} a b \left (80 a^4-177 a^2 b^2-639 b^4\right )+\frac{3}{16} \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{135135 b^5}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{\left (4 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{45045 b^6}+\frac{\left (8 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 b^6\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{45045 b^6}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{\left (4 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{45045 b^6 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (8 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 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}{45045 b^6 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{16 a \left (160 a^4-279 a^2 b^2+27 b^4\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}-\frac{8 \left (480 a^4-937 a^2 b^2+231 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 a \left (40 a^2-81 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2}}{3003 b^4 d}-\frac{10 \left (16 a^2-33 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{1287 b^3 d}+\frac{20 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{143 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{3/2}}{13 b d}-\frac{8 \left (320 a^6-798 a^4 b^2+435 a^2 b^4-693 b^6\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)}}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{16 a \left (160 a^6-439 a^4 b^2+306 a^2 b^4-27 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}}}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 4.84953, size = 327, normalized size = 0.71 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (-256 a \left (-279 a^3 b^2+279 a^2 b^3-160 a^4 b+160 a^5+27 a b^4-27 b^5\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+128 \left (-798 a^4 b^2+435 a^2 b^4+320 a^6-693 b^6\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left (13592 a^2 b^3 \sin (c+d x)+1400 a^2 b^3 \sin (3 (c+d x))-1600 \left (2 a^3 b^2-3 a b^4\right ) \cos (2 (c+d x))-21056 a^3 b^2-7680 a^4 b \sin (c+d x)+10240 a^5+630 a b^4 \cos (4 (c+d x))+5898 a b^4-19866 b^5 \sin (c+d x)+5775 b^5 \sin (3 (c+d x))+3465 b^5 \sin (5 (c+d x))\right )\right )}{720720 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]

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]

(Sqrt[a + b*Sin[c + d*x]]*(128*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4
, (2*b)/(a + b)] - 256*a*(160*a^5 - 160*a^4*b - 279*a^3*b^2 + 279*a^2*b^3 + 27*a*b^4 - 27*b^5)*EllipticF[(-2*c
+ Pi - 2*d*x)/4, (2*b)/(a + b)] - 2*b*Cos[c + d*x]*Sqrt[(a + b*Sin[c + d*x])/(a + b)]*(10240*a^5 - 21056*a^3*
b^2 + 5898*a*b^4 - 1600*(2*a^3*b^2 - 3*a*b^4)*Cos[2*(c + d*x)] + 630*a*b^4*Cos[4*(c + d*x)] - 7680*a^4*b*Sin[c
+ d*x] + 13592*a^2*b^3*Sin[c + d*x] - 19866*b^5*Sin[c + d*x] + 1400*a^2*b^3*Sin[3*(c + d*x)] + 5775*b^5*Sin[3
*(c + d*x)] + 3465*b^5*Sin[5*(c + d*x)])))/(720720*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])

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Maple [B]  time = 1.677, size = 1619, 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)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x)

[Out]

-2/45045*(-50*a^3*b^5*sin(d*x+c)^5+10470*a*b^7*sin(d*x+c)^5+80*a^4*b^4*sin(d*x+c)^4-232*a^2*b^6*sin(d*x+c)^4-1
60*a^5*b^3*sin(d*x+c)^3+454*a^3*b^5*sin(d*x+c)^3-8322*a*b^7*sin(d*x+c)^3-640*a^6*b^2*sin(d*x+c)^2+1436*a^4*b^4
*sin(d*x+c)^2-511*a^2*b^6*sin(d*x+c)^2+160*a^5*b^3*sin(d*x+c)-404*a^3*b^5*sin(d*x+c)+1632*a*b^7*sin(d*x+c)-378
0*a*b^7*sin(d*x+c)^7-1516*a^4*b^4+708*a^2*b^6+640*a^6*b^2-4296*((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^6+35*a^2*b^6*sin(d*x+c)^6-3465*b^8*sin(d*x+c)^8+9240*b^8*sin(d*x+c)^6-6699*b^8*sin(d*x+c)^4+924*b^8*sin(
d*x+c)^2-216*((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
lipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^7-4932*((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^4*b^4-960*((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^6*b^2-3512*((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^3+4472*((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^6*b^2+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^7*b+4512*((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^2*b^6+2484*((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^4+2448*((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^5-1280*((
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*s
in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^8-2772*((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))*b^8+2772*
((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^8)/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 \sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}\,{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(sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^4*sin(d*x + c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \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 \sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{2}\,{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(sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^4*sin(d*x + c)^2, x)