3.1135 \(\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=331 \[ \frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 a \left (-11 a^2 b^2+10 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d \sqrt{a^2-b^2}}+\frac{\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 a^2 b^3 d}+\frac{\left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{2 a b^4 d}-\frac{3 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^5 d}+\frac{3 x \left (-24 a^2 b^2+40 a^4+b^4\right )}{8 b^7} \]

[Out]

(3*(40*a^4 - 24*a^2*b^2 + b^4)*x)/(8*b^7) - (3*a*(10*a^4 - 11*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])
/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (a*(30*a^2 - 13*b^2)*Cos[c + d*x])/(2*b^6*d) - (3*(20*a^2 - 7*b^2
)*Cos[c + d*x]*Sin[c + d*x])/(8*b^5*d) + ((10*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*a*b^4*d) - ((15*a^2
- 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a*b^2*d*(a
+ b*Sin[c + d*x])^2) + ((7*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))

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Rubi [A]  time = 1.00846, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2891, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 a \left (-11 a^2 b^2+10 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d \sqrt{a^2-b^2}}+\frac{\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}-\frac{\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 a^2 b^3 d}+\frac{\left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{2 a b^4 d}-\frac{3 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^5 d}+\frac{3 x \left (-24 a^2 b^2+40 a^4+b^4\right )}{8 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(40*a^4 - 24*a^2*b^2 + b^4)*x)/(8*b^7) - (3*a*(10*a^4 - 11*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])
/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (a*(30*a^2 - 13*b^2)*Cos[c + d*x])/(2*b^6*d) - (3*(20*a^2 - 7*b^2
)*Cos[c + d*x]*Sin[c + d*x])/(8*b^5*d) + ((10*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*a*b^4*d) - ((15*a^2
- 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a*b^2*d*(a
+ b*Sin[c + d*x])^2) + ((7*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))

Rule 2891

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[((a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*b^2*d*
f*(m + 1)), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[
a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m +
n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[((a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a +
b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1))/(a^2*b^2*d*f*(m + 1)*(m + 2)), x]) /; FreeQ[{a, b, d, e, f,
n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m, -2] || EqQ[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 2735

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

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^3(c+d x) \left (6 \left (4 a^2-b^2\right )-a b \sin (c+d x)-2 \left (15 a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 b^2}\\ &=-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\int \frac{\sin ^2(c+d x) \left (-6 a \left (15 a^2-4 b^2\right )+6 a^2 b \sin (c+d x)+12 a \left (10 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8 a^2 b^3}\\ &=\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\int \frac{\sin (c+d x) \left (24 a^2 \left (10 a^2-3 b^2\right )-30 a^3 b \sin (c+d x)-18 a^2 \left (20 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^2 b^4}\\ &=-\frac{3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\int \frac{-18 a^3 \left (20 a^2-7 b^2\right )+6 a^2 b \left (20 a^2-3 b^2\right ) \sin (c+d x)+24 a^3 \left (30 a^2-13 b^2\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{48 a^2 b^5}\\ &=\frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\int \frac{-18 a^3 b \left (20 a^2-7 b^2\right )-18 a^2 \left (40 a^4-24 a^2 b^2+b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{48 a^2 b^6}\\ &=\frac{3 \left (40 a^4-24 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left (3 a \left (10 a^4-11 a^2 b^2+2 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 b^7}\\ &=\frac{3 \left (40 a^4-24 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac{\left (3 a \left (10 a^4-11 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^7 d}\\ &=\frac{3 \left (40 a^4-24 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}+\frac{\left (6 a \left (10 a^4-11 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^7 d}\\ &=\frac{3 \left (40 a^4-24 a^2 b^2+b^4\right ) x}{8 b^7}-\frac{3 a \left (10 a^4-11 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^7 \sqrt{a^2-b^2} d}+\frac{a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac{3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac{\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac{\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 10.2277, size = 1250, normalized size = 3.78 \[ -\frac{-\frac{6 \left (-8 (c+d x)+\frac{2 a \left (8 a^4-20 b^2 a^2+15 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac{3 b \left (4 a^4-7 b^2 a^2+2 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a b \left (4 a^2-3 b^2\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}\right )}{b^3}+\frac{6 \left (\frac{6 a b \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+\frac{\cos (c+d x) \left (a \left (2 a^2+b^2\right )+b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}+\frac{2 \left (-8 \sin (2 (c+d x)) b^2+96 a \cos (c+d x) b+\frac{\left (112 a^6-220 b^2 a^4+115 b^4 a^2-10 b^6\right ) \cos (c+d x) b}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a \left (-16 a^4+20 b^2 a^2-5 b^4\right ) \cos (c+d x) b}{(a-b) (a+b) (a+b \sin (c+d x))^2}-24 \left (b^2-8 a^2\right ) (c+d x)-\frac{6 a \left (64 a^6-168 b^2 a^4+140 b^4 a^2-35 b^6\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}\right )}{b^5}+\frac{\frac{12 a \left (640 a^8-1920 b^2 a^6+2016 b^4 a^4-840 b^6 a^2+105 b^8\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{-3840 (c+d x) a^{10}-3840 b \cos (c+d x) a^9-7680 b (c+d x) \sin (c+d x) a^9+7680 b^2 (c+d x) a^8+1920 b^2 (c+d x) \cos (2 (c+d x)) a^8-2880 b^2 \sin (2 (c+d x)) a^8+8640 b^3 \cos (c+d x) a^7+320 b^3 \cos (3 (c+d x)) a^7+19200 b^3 (c+d x) \sin (c+d x) a^7-2976 b^4 (c+d x) a^6-4800 b^4 (c+d x) \cos (2 (c+d x)) a^6+6880 b^4 \sin (2 (c+d x)) a^6-40 b^4 \sin (4 (c+d x)) a^6-5696 b^5 \cos (c+d x) a^5-760 b^5 \cos (3 (c+d x)) a^5-8 b^5 \cos (5 (c+d x)) a^5-15552 b^5 (c+d x) \sin (c+d x) a^5-1776 b^6 (c+d x) a^4+3888 b^6 (c+d x) \cos (2 (c+d x)) a^4-5182 b^6 \sin (2 (c+d x)) a^4+88 b^6 \sin (4 (c+d x)) a^4+2 b^6 \sin (6 (c+d x)) a^4+788 b^7 \cos (c+d x) a^3+560 b^7 \cos (3 (c+d x)) a^3+16 b^7 \cos (5 (c+d x)) a^3+4224 b^7 (c+d x) \sin (c+d x) a^3+960 b^8 (c+d x) a^2-1056 b^8 (c+d x) \cos (2 (c+d x)) a^2+1221 b^8 \sin (2 (c+d x)) a^2-56 b^8 \sin (4 (c+d x)) a^2-4 b^8 \sin (6 (c+d x)) a^2+114 b^9 \cos (c+d x) a-120 b^9 \cos (3 (c+d x)) a-8 b^9 \cos (5 (c+d x)) a-192 b^9 (c+d x) \sin (c+d x) a-48 b^{10} (c+d x)+48 b^{10} (c+d x) \cos (2 (c+d x))-36 b^{10} \sin (2 (c+d x))+8 b^{10} \sin (4 (c+d x))+2 b^{10} \sin (6 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}}{b^7}}{256 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-6*(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^
2 - b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7
*a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^3 + (6*((6*a*b*ArcTan[(b + a*Ta
n[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d
*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) + (2*(-24*(-8*a^2 + b^2)*(c + d*x) - (6*a*(64*a^6 - 168*a
^4*b^2 + 140*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 96*a*b*Co
s[c + d*x] + (a*b*(-16*a^4 + 20*a^2*b^2 - 5*b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (b*(
112*a^6 - 220*a^4*b^2 + 115*a^2*b^4 - 10*b^6)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) - 8*b^2
*Sin[2*(c + d*x)]))/b^5 + ((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 840*a^2*b^6 + 105*b^8)*ArcTan[(b + a
*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 2976
*a^6*b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10*(c + d*x) - 3840*a^9*b*Cos[c + d
*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5*b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*x
] + 1920*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x)
*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x)*Cos[2*(c + d*x)] + 48*b^10*(c + d*x)*Cos[2*(c + d*x)] + 320*a^7*b^3
*Cos[3*(c + d*x)] - 760*a^5*b^5*Cos[3*(c + d*x)] + 560*a^3*b^7*Cos[3*(c + d*x)] - 120*a*b^9*Cos[3*(c + d*x)] -
8*a^5*b^5*Cos[5*(c + d*x)] + 16*a^3*b^7*Cos[5*(c + d*x)] - 8*a*b^9*Cos[5*(c + d*x)] - 7680*a^9*b*(c + d*x)*Si
n[c + d*x] + 19200*a^7*b^3*(c + d*x)*Sin[c + d*x] - 15552*a^5*b^5*(c + d*x)*Sin[c + d*x] + 4224*a^3*b^7*(c + d
*x)*Sin[c + d*x] - 192*a*b^9*(c + d*x)*Sin[c + d*x] - 2880*a^8*b^2*Sin[2*(c + d*x)] + 6880*a^6*b^4*Sin[2*(c +
d*x)] - 5182*a^4*b^6*Sin[2*(c + d*x)] + 1221*a^2*b^8*Sin[2*(c + d*x)] - 36*b^10*Sin[2*(c + d*x)] - 40*a^6*b^4*
Sin[4*(c + d*x)] + 88*a^4*b^6*Sin[4*(c + d*x)] - 56*a^2*b^8*Sin[4*(c + d*x)] + 8*b^10*Sin[4*(c + d*x)] + 2*a^4
*b^6*Sin[6*(c + d*x)] - 4*a^2*b^8*Sin[6*(c + d*x)] + 2*b^10*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d*
x])^2))/b^7)/(256*d)

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Maple [B]  time = 0.16, size = 1193, normalized size = 3.6 \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)^3/(a+b*sin(d*x+c))^3,x)

[Out]

-6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*a^2+9/d*a^4/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1
/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3-4/d*a^2/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+
1/2*c)^3+10/d*a^5/b^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2+15/d*a^3/b^4/(t
an(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2-10/d*a/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*ta
n(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2-24/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^4*a-6/d/b^
5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*a^2+60/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^2
*a^3+60/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^4*a^3+31/d*a^4/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1
/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)-16/d*a^2/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1
/2*d*x+1/2*c)+3/4/d/b^3*arctan(tan(1/2*d*x+1/2*c))-20/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^2*a+
6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*a^2+20/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/
2*c)^6*a^3-12/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^6*a+6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1
/2*d*x+1/2*c)^5*a^2-18/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^2+10/d*a^5/b^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x
+1/2*c)*b+a)^2-5/d*a^3/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2-5/4/d/b^3/(1+tan(1/2*d*x+1/2*c)
^2)^4*tan(1/2*d*x+1/2*c)^7+3/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5-3/4/d/b^3/(1+tan(1/2*d*x+
1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3+5/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)+20/d/b^6/(1+tan(1/2*d
*x+1/2*c)^2)^4*a^3-8/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*a+30/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a^4-30/d*a^5/b^7/(
a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+33/d*a^3/b^5/(a^2-b^2)^(1/2)*arctan(1/
2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-6/d*a/b^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2
*b)/(a^2-b^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.88746, size = 2415, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/8*(4*(a^3*b^5 - a*b^7)*cos(d*x + c)^5 - 3*(40*a^6*b^2 - 64*a^4*b^4 + 25*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2
- 2*(20*a^5*b^3 - 27*a^3*b^5 + 7*a*b^7)*cos(d*x + c)^3 + 3*(40*a^8 - 24*a^6*b^2 - 39*a^4*b^4 + 24*a^2*b^6 - b^
8)*d*x - 6*(10*a^7 - a^5*b^2 - 9*a^3*b^4 + 2*a*b^6 - (10*a^5*b^2 - 11*a^3*b^4 + 2*a*b^6)*cos(d*x + c)^2 + 2*(1
0*a^6*b - 11*a^4*b^3 + 2*a^2*b^5)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*si
n(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^
2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 6*(20*a^7*b - 22*a^5*b^3 - a^3*b^5 + 3*a*b^7)*cos(d*x + c) - (2*(a^2*b^
6 - b^8)*cos(d*x + c)^5 - (10*a^4*b^4 - 11*a^2*b^6 + b^8)*cos(d*x + c)^3 - 6*(40*a^7*b - 64*a^5*b^3 + 25*a^3*b
^5 - a*b^7)*d*x - 3*(60*a^6*b^2 - 91*a^4*b^4 + 32*a^2*b^6 - b^8)*cos(d*x + c))*sin(d*x + c))/((a^2*b^9 - b^11)
*d*cos(d*x + c)^2 - 2*(a^3*b^8 - a*b^10)*d*sin(d*x + c) - (a^4*b^7 - b^11)*d), -1/8*(4*(a^3*b^5 - a*b^7)*cos(d
*x + c)^5 - 3*(40*a^6*b^2 - 64*a^4*b^4 + 25*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 - 2*(20*a^5*b^3 - 27*a^3*b^5 + 7
*a*b^7)*cos(d*x + c)^3 + 3*(40*a^8 - 24*a^6*b^2 - 39*a^4*b^4 + 24*a^2*b^6 - b^8)*d*x + 12*(10*a^7 - a^5*b^2 -
9*a^3*b^4 + 2*a*b^6 - (10*a^5*b^2 - 11*a^3*b^4 + 2*a*b^6)*cos(d*x + c)^2 + 2*(10*a^6*b - 11*a^4*b^3 + 2*a^2*b^
5)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 6*(20*a^7*b -
22*a^5*b^3 - a^3*b^5 + 3*a*b^7)*cos(d*x + c) - (2*(a^2*b^6 - b^8)*cos(d*x + c)^5 - (10*a^4*b^4 - 11*a^2*b^6 +
b^8)*cos(d*x + c)^3 - 6*(40*a^7*b - 64*a^5*b^3 + 25*a^3*b^5 - a*b^7)*d*x - 3*(60*a^6*b^2 - 91*a^4*b^4 + 32*a^2
*b^6 - b^8)*cos(d*x + c))*sin(d*x + c))/((a^2*b^9 - b^11)*d*cos(d*x + c)^2 - 2*(a^3*b^8 - a*b^10)*d*sin(d*x +
c) - (a^4*b^7 - b^11)*d)]

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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)**3/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.36459, size = 729, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(3*(40*a^4 - 24*a^2*b^2 + b^4)*(d*x + c)/b^7 - 24*(10*a^5 - 11*a^3*b^2 + 2*a*b^4)*(pi*floor(1/2*(d*x + c)/
pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^7) + 8*(9*a^4*b*ta
n(1/2*d*x + 1/2*c)^3 - 4*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 10*a^5*tan(1/2*d*x + 1/2*c)^2 + 15*a^3*b^2*tan(1/2*d
*x + 1/2*c)^2 - 10*a*b^4*tan(1/2*d*x + 1/2*c)^2 + 31*a^4*b*tan(1/2*d*x + 1/2*c) - 16*a^2*b^3*tan(1/2*d*x + 1/2
*c) + 10*a^5 - 5*a^3*b^2)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*b^6) + 2*(24*a^2*b*tan(
1/2*d*x + 1/2*c)^7 - 5*b^3*tan(1/2*d*x + 1/2*c)^7 + 80*a^3*tan(1/2*d*x + 1/2*c)^6 - 48*a*b^2*tan(1/2*d*x + 1/2
*c)^6 + 24*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 3*b^3*tan(1/2*d*x + 1/2*c)^5 + 240*a^3*tan(1/2*d*x + 1/2*c)^4 - 96*a
*b^2*tan(1/2*d*x + 1/2*c)^4 - 24*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 3*b^3*tan(1/2*d*x + 1/2*c)^3 + 240*a^3*tan(1/2
*d*x + 1/2*c)^2 - 80*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 24*a^2*b*tan(1/2*d*x + 1/2*c) + 5*b^3*tan(1/2*d*x + 1/2*c)
+ 80*a^3 - 32*a*b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*b^6))/d