3.456 \(\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=197 \[ -\frac{5 \left (-5 a^2 b^2+4 a^4+b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^6 d \sqrt{a^2-b^2}}+\frac{5 \cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{2 b^5 d}+\frac{5 a x \left (4 a^2-3 b^2\right )}{2 b^6}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2} \]

[Out]

(5*a*(4*a^2 - 3*b^2)*x)/(2*b^6) - (5*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]
])/(b^6*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]^5/(2*b*d*(a + b*Sin[c + d*x])^2) - (5*Cos[c + d*x]^3*(4*a + b*Sin[c
+ d*x]))/(6*b^3*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(2*b^5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.366881, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2693, 2863, 2865, 2735, 2660, 618, 204} \[ -\frac{5 \left (-5 a^2 b^2+4 a^4+b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^6 d \sqrt{a^2-b^2}}+\frac{5 \cos (c+d x) \left (4 a^2-2 a b \sin (c+d x)-b^2\right )}{2 b^5 d}+\frac{5 a x \left (4 a^2-3 b^2\right )}{2 b^6}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]

[Out]

(5*a*(4*a^2 - 3*b^2)*x)/(2*b^6) - (5*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]
])/(b^6*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]^5/(2*b*d*(a + b*Sin[c + d*x])^2) - (5*Cos[c + d*x]^3*(4*a + b*Sin[c
+ d*x]))/(6*b^3*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(2*b^5*d)

Rule 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2863

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

Rule 2865

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

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 ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 b}\\ &=-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac{5 \int \frac{\cos ^2(c+d x) (-b-4 a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{2 b^3}\\ &=-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac{5 \cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{2 b^5 d}+\frac{5 \int \frac{2 b \left (2 a^2-b^2\right )+2 a \left (4 a^2-3 b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{4 b^5}\\ &=\frac{5 a \left (4 a^2-3 b^2\right ) x}{2 b^6}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac{5 \cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{2 b^5 d}-\frac{\left (5 \left (4 a^4-5 a^2 b^2+b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 b^6}\\ &=\frac{5 a \left (4 a^2-3 b^2\right ) x}{2 b^6}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac{5 \cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{2 b^5 d}-\frac{\left (5 \left (4 a^4-5 a^2 b^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^6 d}\\ &=\frac{5 a \left (4 a^2-3 b^2\right ) x}{2 b^6}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac{5 \cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{2 b^5 d}+\frac{\left (10 \left (4 a^4-5 a^2 b^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^6 d}\\ &=\frac{5 a \left (4 a^2-3 b^2\right ) x}{2 b^6}-\frac{5 \left (4 a^4-5 a^2 b^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^6 \sqrt{a^2-b^2} d}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}+\frac{5 \cos (c+d x) \left (4 a^2-b^2-2 a b \sin (c+d x)\right )}{2 b^5 d}\\ \end{align*}

Mathematica [B]  time = 6.58301, size = 3905, normalized size = 19.82 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]

[Out]

(Cos[c + d*x]^5*(-(b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(7
/2))/(2*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])^2) - ((-3*a*b^3*(-(b/
(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(7/2))/((a^2 - b^2)*((a*b)/(
a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])) - ((144*Sqrt[2]*a*b^5*(-(b/(a - b))
- (b*Sin[c + d*x])/(a - b))^(7/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*
Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((7*(3/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)
)^3) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) -
(b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/12 + (35*b^4*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b
- ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) + (2*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*
x])/(a - b))^3)/(15*b^3) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a -
b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b
)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(128*(a - b)^4*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^4*(1 + ((a
- b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(7*(a - b)*(a + b)^4*Sqrt[((a + b)*(b/(a + b) - (b
*Sin[c + d*x])/(a + b)))/b]) + (((18*a^2*b^5)/((a - b)^2*(a + b)^2) + (b^5*(2*a^2 - 5*b^2))/((a - b)^2*(a + b)
^2))*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*
(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5/(16*(1 + ((a - b)*(-(b/(a - b)) - (b
*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) +
(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/2 - (15*b^3*(((a - b)*(-(b/(a - b)) - (
b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) - (Sqrt[2]*Sqrt[
a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)
) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))
/(64*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a -
b)))/(2*b))^3)))/(5*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b))
+ b^2/(a - b))*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(3/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])
/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((3*(5/(8*(1 + ((a - b)*(-(b/(
a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(6*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))
/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/8 + (15*b^2*(((a - b)*(-(b
/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Si
n[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a
- b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a -
b))^2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(3*(a + b)^2*Sqrt[((a + b)*(b/(a +
b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*Sqrt[-(b/(a - b)) - (b*S
in[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x]
)/(a - b)))/(2*b))^(7/2)*((5*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt
[2]*Sqrt[b])])/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)
) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)) + (15/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))
/(2*b))^3) + 5/(4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a -
b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/6))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a +
b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*(-(((-((a*b)/(a + b)) - b^2
/(a + b))*((-2*(-((a*b)/(a + b)) - b^2/(a + b))*ArcTan[(Sqrt[(a*b)/(a + b) + b^2/(a + b)]*Sqrt[-(b/(a - b)) -
(b*Sin[c + d*x])/(a - b)])/(Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])
/(b*Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[(a*b)/(a + b) + b^2/(a + b)]) + (2*Sqrt[a - b]*ArcTanh[(Sqrt[a -
b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[a + b]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/
(b*Sqrt[a + b])))/b) + (2*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Si
n[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)*((Sqrt[b]*ArcSinh[(
Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(
a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)) + 1
/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)))))/(b*(a + b)*Sqrt[((a + b)*(b/(a + b) - (
b*Sin[c + d*x])/(a + b)))/b])))/b) + (4*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(
a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((3*S
qrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(4*Sqrt[2]*Sqrt
[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))
/(2*b))^(5/2)) + (3/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b
/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/4))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])
/(a + b)))/b])))/b))/b))/b))/b)/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))))/(2*((a*b)/(a -
b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b)))))/(d*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/2)*(1 - (a + b*Sin
[c + d*x])/(a + b))^(5/2))

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Maple [B]  time = 0.121, size = 1060, normalized size = 5.4 \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)^6/(a+b*sin(d*x+c))^3,x)

[Out]

-14/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3-1/d/b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2+12/d/b^5/(1+t
an(1/2*d*x+1/2*c)^2)^3*a^2+20/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^3-15/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a-15/d/
b/(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+8/d/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*t
an(1/2*d*x+1/2*c)*b+a)^2*a^4-7/d/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a^2-5/d/b^2/(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/d/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*
c)*b+a)^2/a*tan(1/2*d*x+1/2*c)^3-2/d/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/a*tan(1/2*d*x+1/2*c)-
6/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^4-8/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^
2+3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*a*tan(1/2*d*x+1/2*c)^5+12/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/
2*c)^4*a^2+24/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^2*a^2-3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*a*t
an(1/2*d*x+1/2*c)+7/d/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a^3*tan(1/2*d*x+1/2*c)^3-5/d/b^2
/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a*tan(1/2*d*x+1/2*c)^3+8/d/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*
tan(1/2*d*x+1/2*c)*b+a)^2*a^4*tan(1/2*d*x+1/2*c)^2+9/d/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2
*a^2*tan(1/2*d*x+1/2*c)^2-2/d*b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/a^2*tan(1/2*d*x+1/2*c)^2+2
5/d/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a^3*tan(1/2*d*x+1/2*c)-23/d/b^2/(tan(1/2*d*x+1/2*c
)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a*tan(1/2*d*x+1/2*c)-20/d/b^6/(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))*a^4+25/d/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
))*a^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)^6/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.82276, size = 1669, normalized size = 8.47 \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)^6/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/12*(4*b^5*cos(d*x + c)^5 - 30*(4*a^3*b^2 - 3*a*b^4)*d*x*cos(d*x + c)^2 - 20*(2*a^2*b^3 - b^5)*cos(d*x + c)
^3 + 30*(4*a^5 + a^3*b^2 - 3*a*b^4)*d*x - 15*(4*a^4 + 3*a^2*b^2 - b^4 - (4*a^2*b^2 - b^4)*cos(d*x + c)^2 + 2*(
4*a^3*b - a*b^3)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(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)) + 30*(4*a^4*b - a^2*b^3 - b^5)*cos(d*x + c) + 10*(a*b^4*cos(d*x + c)^3 + 6*(4*a^4*b - 3*a^
2*b^3)*d*x + 6*(3*a^3*b^2 - 2*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^8*d*cos(d*x + c)^2 - 2*a*b^7*d*sin(d*x + c
) - (a^2*b^6 + b^8)*d), -1/6*(2*b^5*cos(d*x + c)^5 - 15*(4*a^3*b^2 - 3*a*b^4)*d*x*cos(d*x + c)^2 - 10*(2*a^2*b
^3 - b^5)*cos(d*x + c)^3 + 15*(4*a^5 + a^3*b^2 - 3*a*b^4)*d*x + 15*(4*a^4 + 3*a^2*b^2 - b^4 - (4*a^2*b^2 - b^4
)*cos(d*x + c)^2 + 2*(4*a^3*b - a*b^3)*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))) + 15*(4*a^4*b - a^2*b^3 - b^5)*cos(d*x + c) + 5*(a*b^4*cos(d*x + c)^3 + 6*(4*a^4*b - 3*a^2
*b^3)*d*x + 6*(3*a^3*b^2 - 2*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^8*d*cos(d*x + c)^2 - 2*a*b^7*d*sin(d*x + c)
- (a^2*b^6 + b^8)*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)**6/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.16336, size = 617, normalized size = 3.13 \begin{align*} \frac{\frac{15 \,{\left (4 \, a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{b^{6}} - \frac{30 \,{\left (4 \, a^{4} - 5 \, a^{2} b^{2} + 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^{6}} + \frac{2 \,{\left (9 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 36 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 18 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 72 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 36 \, a^{2} - 14 \, b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3} b^{5}} + \frac{6 \,{\left (7 \, a^{5} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5 \, a^{3} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, a b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, a^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 \, a^{4} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{2} b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, b^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 25 \, a^{5} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 23 \, a^{3} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, a b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, a^{6} - 7 \, a^{4} b^{2} - a^{2} b^{4}\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} a^{2} b^{5}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/6*(15*(4*a^3 - 3*a*b^2)*(d*x + c)/b^6 - 30*(4*a^4 - 5*a^2*b^2 + 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^6) + 2*(9*a*b*tan(1/2*d*x + 1/2*c
)^5 + 36*a^2*tan(1/2*d*x + 1/2*c)^4 - 18*b^2*tan(1/2*d*x + 1/2*c)^4 + 72*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*b^2*t
an(1/2*d*x + 1/2*c)^2 - 9*a*b*tan(1/2*d*x + 1/2*c) + 36*a^2 - 14*b^2)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^5) + 6
*(7*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 5*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 8*a^6*t
an(1/2*d*x + 1/2*c)^2 + 9*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 - 2*b^6*tan(1/2*d
*x + 1/2*c)^2 + 25*a^5*b*tan(1/2*d*x + 1/2*c) - 23*a^3*b^3*tan(1/2*d*x + 1/2*c) - 2*a*b^5*tan(1/2*d*x + 1/2*c)
+ 8*a^6 - 7*a^4*b^2 - a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^2*b^5))/d