\(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1135]

Optimal result

Integrand size = 29, antiderivative size = 331 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 \left (40 a^4-24 a^2 b^2+b^4\right ) x}{8 b^7}-\frac {3 a \left (10 a^4-11 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 \sqrt {a^2-b^2} d}+\frac {a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac {3 \left (20 a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^5 d}+\frac {\left (10 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 a b^4 d}-\frac {\left (15 a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{4 a^2 b^3 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {\left (7 a^2-2 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))} \]

[Out]

3/8*(40*a^4-24*a^2*b^2+b^4)*x/b^7+1/2*a*(30*a^2-13*b^2)*cos(d*x+c)/b^6/d-3/8*(20*a^2-7*b^2)*cos(d*x+c)*sin(d*x
+c)/b^5/d+1/2*(10*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^2/a/b^4/d-1/4*(15*a^2-4*b^2)*cos(d*x+c)*sin(d*x+c)^3/a^2/b^
3/d-1/2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))^2+1/2*(7*a^2-2*b^2)*cos(d*x+c)*sin(d*x+c)^4
/a^2/b^2/d/(a+b*sin(d*x+c))-3*a*(10*a^4-11*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^7
/d/(a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2970, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (7 a^2-2 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a^2 b^2 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{2 a b^2 d (a+b \sin (c+d x))^2}+\frac {a \left (30 a^2-13 b^2\right ) \cos (c+d x)}{2 b^6 d}-\frac {3 \left (20 a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^5 d}+\frac {\left (10 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{2 a b^4 d}-\frac {\left (15 a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 a^2 b^3 d}-\frac {3 a \left (10 a^4-11 a^2 b^2+2 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d \sqrt {a^2-b^2}}+\frac {3 x \left (40 a^4-24 a^2 b^2+b^4\right )}{8 b^7} \]

[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 210

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

Rule 632

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 2739

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 2814

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 2970

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 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 {\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 ) \text {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 ) \text {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 ) \arctan \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1250\) vs. \(2(331)=662\).

Time = 7.53 (sec) , antiderivative size = 1250, normalized size of antiderivative = 3.78 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {-\frac {6 \left (-8 (c+d x)+\frac {2 a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \arctan \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 {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}-\frac {3 b \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}\right )}{b^3}+\frac {6 \left (\frac {6 a b \arctan \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 (-24 \left (-8 a^2+b^2\right ) (c+d x)-\frac {6 a \left (64 a^6-168 a^4 b^2+140 a^2 b^4-35 b^6\right ) \arctan \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}}+96 a b \cos (c+d x)+\frac {a b \left (-16 a^4+20 a^2 b^2-5 b^4\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}+\frac {b \left (112 a^6-220 a^4 b^2+115 a^2 b^4-10 b^6\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-8 b^2 \sin (2 (c+d x))\right )}{b^5}+\frac {\frac {12 a \left (640 a^8-1920 a^6 b^2+2016 a^4 b^4-840 a^2 b^6+105 b^8\right ) \arctan \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 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) \sin (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))}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}}{b^7}}{256 d} \]

[In]

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

[Out]

-1/256*((-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*Tan[(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*Cos[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)*Sin[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)/d

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {-\frac {2 a \left (\frac {\left (-\frac {9}{2} a^{3} b^{2}+2 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \,b^{2} \left (31 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5 a^{2} b \left (2 a^{2}-b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (10 a^{4}-11 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}+\frac {\frac {2 \left (\left (3 a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{3} b -6 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -12 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -10 a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10 a^{3} b -4 a \,b^{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (40 a^{4}-24 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{7}}}{d}\)	\(450\)
default	\(\frac {-\frac {2 a \left (\frac {\left (-\frac {9}{2} a^{3} b^{2}+2 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \,b^{2} \left (31 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5 a^{2} b \left (2 a^{2}-b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 \left (10 a^{4}-11 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}+\frac {\frac {2 \left (\left (3 a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{3} b -6 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -12 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 a^{3} b -10 a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10 a^{3} b -4 a \,b^{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (40 a^{4}-24 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{7}}}{d}\)	\(450\)
risch	\(\frac {15 x \,a^{4}}{b^{7}}-\frac {9 x \,a^{2}}{b^{5}}+\frac {3 x}{8 b^{3}}-\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}}{8 b^{4} d}-\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {5 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{b^{6} d}-\frac {15 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{4} d}+\frac {5 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{b^{6} d}-\frac {15 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{4} d}+\frac {15 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{7}}+\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{8 b^{4} d}-\frac {i a^{2} \left (-12 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+32 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-17 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+22 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-11 a^{2} b^{2}+6 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{7}}-\frac {15 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{7}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {33 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,b^{5}}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{4 b^{5} d}+\frac {33 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,b^{5}}+\frac {\sin \left (4 d x +4 c \right )}{32 b^{3} d}\)	\(834\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*a/b^7*(((-9/2*a^3*b^2+2*a*b^4)*tan(1/2*d*x+1/2*c)^3-5/2*b*(2*a^4+3*a^2*b^2-2*b^4)*tan(1/2*d*x+1/2*c)^2
-1/2*a*b^2*(31*a^2-16*b^2)*tan(1/2*d*x+1/2*c)-5/2*a^2*b*(2*a^2-b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1
/2*c)+a)^2+3/2*(10*a^4-11*a^2*b^2+2*b^4)/(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)))+2/b^7*(((3*a^2*b^2-5/8*b^4)*tan(1/2*d*x+1/2*c)^7+(10*a^3*b-6*a*b^3)*tan(1/2*d*x+1/2*c)^6+(3*a^2*b^2+3/8*b
^4)*tan(1/2*d*x+1/2*c)^5+(30*a^3*b-12*a*b^3)*tan(1/2*d*x+1/2*c)^4+(-3*a^2*b^2-3/8*b^4)*tan(1/2*d*x+1/2*c)^3+(3
0*a^3*b-10*a*b^3)*tan(1/2*d*x+1/2*c)^2+(-3*a^2*b^2+5/8*b^4)*tan(1/2*d*x+1/2*c)+10*a^3*b-4*a*b^3)/(1+tan(1/2*d*
x+1/2*c)^2)^4+3/8*(40*a^4-24*a^2*b^2+b^4)*arctan(tan(1/2*d*x+1/2*c))))

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 1110, normalized size of antiderivative = 3.35 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.82 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (40 \, a^{4} - 24 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{b^{7}} - \frac {24 \, {\left (10 \, a^{5} - 11 \, a^{3} b^{2} + 2 \, a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{7}} + \frac {8 \, {\left (9 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 31 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 16 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, a^{5} - 5 \, a^{3} b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} b^{6}} + \frac {2 \, {\left (24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 80 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 96 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 80 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 80 \, a^{3} - 32 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{6}}}{8 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 17.95 (sec) , antiderivative size = 3581, normalized size of antiderivative = 10.82 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((30*a^5 - 13*a^3*b^2)/b^6 + (15*tan(c/2 + (d*x)/2)^8*(10*a^5 - 6*a*b^4 + 9*a^3*b^2))/b^6 + (3*tan(c/2 + (d*x)
/2)^10*(10*a^5 - 5*a*b^4 + 9*a^3*b^2))/b^6 + (tan(c/2 + (d*x)/2)^2*(150*a^5 - 37*a*b^4 + 15*a^3*b^2))/b^6 + (2
*tan(c/2 + (d*x)/2)^4*(150*a^5 - 59*a*b^4 + 75*a^3*b^2))/b^6 + (tan(c/2 + (d*x)/2)*(420*a^4 - 187*a^2*b^2))/(4
*b^5) + (3*tan(c/2 + (d*x)/2)^11*(20*a^4 - 7*a^2*b^2))/(4*b^5) + (3*tan(c/2 + (d*x)/2)^7*(340*a^4 + 2*b^4 - 13
9*a^2*b^2))/(2*b^5) - (tan(c/2 + (d*x)/2)^9*(20*b^4 - 660*a^4 + 231*a^2*b^2))/(4*b^5) - (tan(c/2 + (d*x)/2)^5*
(6*b^4 - 1380*a^4 + 623*a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2)^3*(1740*a^4 + 20*b^4 - 809*a^2*b^2))/(4*b^5) -
(2*tan(c/2 + (d*x)/2)^6*(13*a*b^2 - 30*a^3)*(5*a^2 + 6*b^2))/b^6)/(d*(tan(c/2 + (d*x)/2)^2*(6*a^2 + 4*b^2) +
tan(c/2 + (d*x)/2)^10*(6*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(15*a^2 + 16*b^2) + tan(c/2 + (d*x)/2)^8*(15*a^2
+ 16*b^2) + tan(c/2 + (d*x)/2)^6*(20*a^2 + 24*b^2) + a^2*tan(c/2 + (d*x)/2)^12 + a^2 + 20*a*b*tan(c/2 + (d*x)/
2)^3 + 40*a*b*tan(c/2 + (d*x)/2)^5 + 40*a*b*tan(c/2 + (d*x)/2)^7 + 20*a*b*tan(c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2
+ (d*x)/2)^11 + 4*a*b*tan(c/2 + (d*x)/2))) + (atan((((a^4*40i + b^4*1i - a^2*b^2*24i)*(((9*a^2*b^14)/2 - 216*
a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16 - 1449*a^3*b^14
+ 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) - (3*(a^4*40i + b^4*1i - a^2*b^
2*24i)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 - (3*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 12
8*a^3*b^20))/(2*b^18))*(a^4*40i + b^4*1i - a^2*b^2*24i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(384*a^2*b^18 - 2112*a^
4*b^16 + 1920*a^6*b^14))/(2*b^18)))/(8*b^7))*3i)/(8*b^7) + ((a^4*40i + b^4*1i - a^2*b^2*24i)*(((9*a^2*b^14)/2
- 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16 - 1449*a^
3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) + (3*(a^4*40i + b^4*1i -
a^2*b^2*24i)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 + (3*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^2
2 - 128*a^3*b^20))/(2*b^18))*(a^4*40i + b^4*1i - a^2*b^2*24i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(384*a^2*b^18 - 2
112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18)))/(8*b^7))*3i)/(8*b^7))/((108000*a^13 - 189*a^3*b^10 + (12231*a^5*b^8)
/2 - 49383*a^7*b^6 + 159840*a^9*b^4 - 221400*a^11*b^2)/b^17 - (3*(a^4*40i + b^4*1i - a^2*b^2*24i)*(((9*a^2*b^1
4)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16 - 14
49*a^3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) - (3*(a^4*40i + b^4*
1i - a^2*b^2*24i)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 - (3*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*
a*b^22 - 128*a^3*b^20))/(2*b^18))*(a^4*40i + b^4*1i - a^2*b^2*24i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(384*a^2*b^1
8 - 2112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18)))/(8*b^7)))/(8*b^7) + (3*(a^4*40i + b^4*1i - a^2*b^2*24i)*(((9*a^
2*b^14)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16
- 1449*a^3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) + (3*(a^4*40i +
b^4*1i - a^2*b^2*24i)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 + (3*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*
(192*a*b^22 - 128*a^3*b^20))/(2*b^18))*(a^4*40i + b^4*1i - a^2*b^2*24i))/(8*b^7) + (tan(c/2 + (d*x)/2)*(384*a^
2*b^18 - 2112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18)))/(8*b^7)))/(8*b^7) + (tan(c/2 + (d*x)/2)*(432000*a^14 + 54*
a^2*b^12 - 2889*a^4*b^10 + 49950*a^6*b^8 - 311472*a^8*b^6 + 833760*a^10*b^4 - 993600*a^12*b^2))/b^18))*(a^4*40
i + b^4*1i - a^2*b^2*24i)*3i)/(4*b^7*d) + (a*atan(((a*(-(a + b)*(a - b))^(1/2)*(10*a^4 + 2*b^4 - 11*a^2*b^2)*(
((9*a^2*b^14)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*
a*b^16 - 1449*a^3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) - (3*a*(-
(a + b)*(a - b))^(1/2)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 + (tan(c/2 + (d*x)/2)*(384*a^2*b^18 - 2
112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18) - (3*a*(-(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192
*a*b^22 - 128*a^3*b^20))/(2*b^18))*(10*a^4 + 2*b^4 - 11*a^2*b^2))/(2*(b^9 - a^2*b^7)))*(10*a^4 + 2*b^4 - 11*a^
2*b^2))/(2*(b^9 - a^2*b^7)))*3i)/(2*(b^9 - a^2*b^7)) + (a*(-(a + b)*(a - b))^(1/2)*(10*a^4 + 2*b^4 - 11*a^2*b^
2)*(((9*a^2*b^14)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*
(18*a*b^16 - 1449*a^3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8 - 28800*a^11*b^6))/(2*b^18) + (3*
a*(-(a + b)*(a - b))^(1/2)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 + (tan(c/2 + (d*x)/2)*(384*a^2*b^18
- 2112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18) + (3*a*(-(a + b)*(a - b))^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*
(192*a*b^22 - 128*a^3*b^20))/(2*b^18))*(10*a^4 + 2*b^4 - 11*a^2*b^2))/(2*(b^9 - a^2*b^7)))*(10*a^4 + 2*b^4 - 1
1*a^2*b^2))/(2*(b^9 - a^2*b^7)))*3i)/(2*(b^9 - a^2*b^7)))/((108000*a^13 - 189*a^3*b^10 + (12231*a^5*b^8)/2 - 4
9383*a^7*b^6 + 159840*a^9*b^4 - 221400*a^11*b^2)/b^17 + (tan(c/2 + (d*x)/2)*(432000*a^14 + 54*a^2*b^12 - 2889*
a^4*b^10 + 49950*a^6*b^8 - 311472*a^8*b^6 + 833760*a^10*b^4 - 993600*a^12*b^2))/b^18 - (3*a*(-(a + b)*(a - b))
^(1/2)*(10*a^4 + 2*b^4 - 11*a^2*b^2)*(((9*a^2*b^14)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*a^1
0*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16 - 1449*a^3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*b^8
- 28800*a^11*b^6))/(2*b^18) - (3*a*(-(a + b)*(a - b))^(1/2)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^17 +
(tan(c/2 + (d*x)/2)*(384*a^2*b^18 - 2112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18) - (3*a*(-(a + b)*(a - b))^(1/2)*
(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 128*a^3*b^20))/(2*b^18))*(10*a^4 + 2*b^4 - 11*a^2*b^2))/(2*(b^
9 - a^2*b^7)))*(10*a^4 + 2*b^4 - 11*a^2*b^2))/(2*(b^9 - a^2*b^7))))/(2*(b^9 - a^2*b^7)) + (3*a*(-(a + b)*(a -
b))^(1/2)*(10*a^4 + 2*b^4 - 11*a^2*b^2)*(((9*a^2*b^14)/2 - 216*a^4*b^12 + 2952*a^6*b^10 - 8640*a^8*b^8 + 7200*
a^10*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(18*a*b^16 - 1449*a^3*b^14 + 18576*a^5*b^12 - 63648*a^7*b^10 + 77760*a^9*
b^8 - 28800*a^11*b^6))/(2*b^18) + (3*a*(-(a + b)*(a - b))^(1/2)*((12*a*b^18 - 204*a^3*b^16 + 240*a^5*b^14)/b^1
7 + (tan(c/2 + (d*x)/2)*(384*a^2*b^18 - 2112*a^4*b^16 + 1920*a^6*b^14))/(2*b^18) + (3*a*(-(a + b)*(a - b))^(1/
2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^22 - 128*a^3*b^20))/(2*b^18))*(10*a^4 + 2*b^4 - 11*a^2*b^2))/(2*
(b^9 - a^2*b^7)))*(10*a^4 + 2*b^4 - 11*a^2*b^2))/(2*(b^9 - a^2*b^7))))/(2*(b^9 - a^2*b^7))))*(-(a + b)*(a - b)
)^(1/2)*(10*a^4 + 2*b^4 - 11*a^2*b^2)*3i)/(d*(b^9 - a^2*b^7))