3.1151 \(\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=528 \[ -\frac{2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{8 \left (-375 a^2 b^2+160 a^4+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{16 a \left (-47 a^2 b^2+32 a^4-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 \left (-174 a^4 b^2+81 a^2 b^4+64 a^6-195 b^6\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{8 \left (-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+64 a^8+195 b^8\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left (-111 a^4 b^2+102 a^2 b^4+32 a^6-471 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d} \]

[Out]

(8*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) + (16*a*
(32*a^4 - 47*a^2*b^2 - 27*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) - (8*(160*a^4 - 375*a^2*
b^2 + 117*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(45045*b^5*d) + (8*a*(8*a^2 - 21*b^2)*Cos[c + d*x]*Sin
[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(1287*b^4*d) - (2*(80*a^2 - 221*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*
Sin[c + d*x])^(5/2))/(2145*b^3*d) + (4*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(39*b^2*d) -
(2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(15*b*d) - (16*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^
4 - 471*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*S
in[c + d*x])/(a + b)]) + (8*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 276*a^2*b^6 + 195*b^8)*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]])

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Rubi [A]  time = 1.20816, antiderivative size = 528, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{2 \left (80 a^2-221 b^2\right ) \sin ^2(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{8 \left (-375 a^2 b^2+160 a^4+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{16 a \left (-47 a^2 b^2+32 a^4-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}+\frac{8 \left (-174 a^4 b^2+81 a^2 b^4+64 a^6-195 b^6\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{8 \left (-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+64 a^8+195 b^8\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 a \left (-111 a^4 b^2+102 a^2 b^4+32 a^6-471 b^6\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{45045 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]

[Out]

(8*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) + (16*a*
(32*a^4 - 47*a^2*b^2 - 27*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) - (8*(160*a^4 - 375*a^2*
b^2 + 117*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(45045*b^5*d) + (8*a*(8*a^2 - 21*b^2)*Cos[c + d*x]*Sin
[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(1287*b^4*d) - (2*(80*a^2 - 221*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*
Sin[c + d*x])^(5/2))/(2145*b^3*d) + (4*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(39*b^2*d) -
(2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(15*b*d) - (16*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^
4 - 471*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*S
in[c + d*x])/(a + b)]) + (8*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 276*a^2*b^6 + 195*b^8)*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) (a+b \sin (c+d x))^{3/2} \, dx &=\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{4 \int \sin ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac{15}{4} \left (4 a^2-13 b^2\right )+\frac{3}{2} a b \sin (c+d x)-\frac{1}{4} \left (80 a^2-221 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{195 b^2}\\ &=-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{8 \int \sin (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac{1}{2} a \left (80 a^2-221 b^2\right )-\frac{3}{2} b \left (5 a^2+13 b^2\right ) \sin (c+d x)+\frac{15}{2} a \left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2145 b^3}\\ &=\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{16 \int (a+b \sin (c+d x))^{3/2} \left (\frac{15}{2} a^2 \left (8 a^2-21 b^2\right )+6 a b \left (5 a^2-9 b^2\right ) \sin (c+d x)-\frac{3}{4} \left (160 a^4-375 a^2 b^2+117 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{19305 b^4}\\ &=-\frac{8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{32 \int (a+b \sin (c+d x))^{3/2} \left (-\frac{45}{8} b \left (16 a^4-27 a^2 b^2+39 b^4\right )+\frac{15}{4} a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \sin (c+d x)\right ) \, dx}{135135 b^5}\\ &=\frac{16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac{8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{64 \int \sqrt{a+b \sin (c+d x)} \left (-\frac{45}{16} a b \left (16 a^4-41 a^2 b^2+249 b^4\right )+\frac{45}{16} \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \sin (c+d x)\right ) \, dx}{675675 b^5}\\ &=\frac{8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac{8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{128 \int \frac{\frac{45}{32} b \left (16 a^6-51 a^4 b^2-666 a^2 b^4-195 b^6\right )+\frac{45}{16} a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{2027025 b^5}\\ &=\frac{8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac{8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{\left (8 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 b^6\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{45045 b^6}+\frac{\left (4 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{45045 b^6}\\ &=\frac{8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac{8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{\left (8 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 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 (4 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\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{8 \left (64 a^6-174 a^4 b^2+81 a^2 b^4-195 b^6\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{45045 b^5 d}+\frac{16 a \left (32 a^4-47 a^2 b^2-27 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{45045 b^5 d}-\frac{8 \left (160 a^4-375 a^2 b^2+117 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{45045 b^5 d}+\frac{8 a \left (8 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2}}{1287 b^4 d}-\frac{2 \left (80 a^2-221 b^2\right ) \cos (c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{2145 b^3 d}+\frac{4 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{39 b^2 d}-\frac{2 \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^{5/2}}{15 b d}-\frac{16 a \left (32 a^6-111 a^4 b^2+102 a^2 b^4-471 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{8 \left (64 a^8-238 a^6 b^2+255 a^4 b^4-276 a^2 b^6+195 b^8\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 = 14.7226, size = 382, normalized size = 0.72 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (-256 \left (-174 a^5 b^2+174 a^4 b^3+81 a^3 b^4-81 a^2 b^5-64 a^6 b+64 a^7-195 a b^6+195 b^7\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+512 \left (-111 a^5 b^2+102 a^3 b^4+32 a^7-471 a b^6\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-2 b \cos (c+d x) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left (8432 a^3 b^3 \sin (c+d x)+560 a^3 b^3 \sin (3 (c+d x))+\left (3168 a^2 b^4-1280 a^4 b^2+21723 b^6\right ) \cos (2 (c+d x))+42 \left (6 a^2 b^4-13 b^6\right ) \cos (4 (c+d x))-12416 a^4 b^2+8100 a^2 b^4-3072 a^5 b \sin (c+d x)+4096 a^6-41424 a b^5 \sin (c+d x)+13776 a b^5 \sin (3 (c+d x))+7392 a b^5 \sin (5 (c+d x))-3003 b^6 \cos (6 (c+d x))+6786 b^6\right )\right )}{1441440 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]

[Out]

(Sqrt[a + b*Sin[c + d*x]]*(512*(32*a^7 - 111*a^5*b^2 + 102*a^3*b^4 - 471*a*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/
4, (2*b)/(a + b)] - 256*(64*a^7 - 64*a^6*b - 174*a^5*b^2 + 174*a^4*b^3 + 81*a^3*b^4 - 81*a^2*b^5 - 195*a*b^6 +
195*b^7)*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)
]*(4096*a^6 - 12416*a^4*b^2 + 8100*a^2*b^4 + 6786*b^6 + (-1280*a^4*b^2 + 3168*a^2*b^4 + 21723*b^6)*Cos[2*(c +
d*x)] + 42*(6*a^2*b^4 - 13*b^6)*Cos[4*(c + d*x)] - 3003*b^6*Cos[6*(c + d*x)] - 3072*a^5*b*Sin[c + d*x] + 8432*
a^3*b^3*Sin[c + d*x] - 41424*a*b^5*Sin[c + d*x] + 560*a^3*b^3*Sin[3*(c + d*x)] + 13776*a*b^5*Sin[3*(c + d*x)]
+ 7392*a*b^5*Sin[5*(c + d*x)])))/(1441440*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])

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Maple [B]  time = 1.694, size = 1801, 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))^(3/2),x)

[Out]

-2/45045*(-10*a^4*b^5*sin(d*x+c)^5+10362*a^2*b^7*sin(d*x+c)^5+16*a^5*b^4*sin(d*x+c)^4-62*a^3*b^6*sin(d*x+c)^4-
12603*a*b^8*sin(d*x+c)^4-32*a^6*b^3*sin(d*x+c)^3+122*a^4*b^5*sin(d*x+c)^3-8115*a^2*b^7*sin(d*x+c)^3-128*a^7*b^
2*sin(d*x+c)^2+412*a^5*b^4*sin(d*x+c)^2-305*a^3*b^6*sin(d*x+c)^2+840*a*b^8*sin(d*x+c)^2+2988*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-
(sin(d*x+c)-1)*b/(a+b))^(1/2)*a*b^8+17682*a*b^8*sin(d*x+c)^6+128*a^7*b^2-428*a^5*b^4+360*a^3*b^6+780*a*b^8+780
*(-(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*sin(d*x+c
))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*b^9+32*a^6*b^3*sin(d*x+c)-112*a^4*b^5*sin(d*x+c)+1512*a^2*b^7*
sin(d*x+c)-6699*a*b^8*sin(d*x+c)^8-3759*a^2*b^7*sin(d*x+c)^7+7*a^3*b^6*sin(d*x+c)^6-3003*b^9*sin(d*x+c)^9+7644
*b^9*sin(d*x+c)^7-5109*b^9*sin(d*x+c)^5-312*b^9*sin(d*x+c)^3+780*b^9*sin(d*x+c)-256*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+
c)-1)*b/(a+b))^(1/2)*a^9+1144*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^7*b^2-1704*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(si
n(d*x+c)-1)*b/(a+b))^(1/2)*a^5*b^4+4584*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^3*b^6-3768*(-(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*sin(d*x+c))/(a-b))^(
1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a*b^8-192*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^7*b^2-952*(-(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*sin(d*x+c))/(a
-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^6*b^3+684*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^5*b^4+
1020*(-(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*sin(d
*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^4*b^5-3480*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2
)*a^3*b^6+256*(-(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*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*a^8*b-1104*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellipt
icF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b
))^(1/2)*a^2*b^7)/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{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \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))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^(3/2)*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 (a \cos \left (d x + c\right )^{6} - a \cos \left (d x + c\right )^{4} +{\left (b \cos \left (d x + c\right )^{6} - b \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\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))^(3/2),x, algorithm="fricas")

[Out]

integral(-(a*cos(d*x + c)^6 - a*cos(d*x + c)^4 + (b*cos(d*x + c)^6 - b*cos(d*x + c)^4)*sin(d*x + c))*sqrt(b*si
n(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))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \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))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^4*sin(d*x + c)^2, x)