3.444 \(\int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=187 \[ \frac {10 a \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^6 d}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {5 x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{8 b^6}+\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))} \]

[Out]

-5/8*(8*a^4-12*a^2*b^2+3*b^4)*x/b^6+10*a*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^6/
d+5/12*cos(d*x+c)^3*(4*a-3*b*sin(d*x+c))/b^3/d-cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))-5/8*cos(d*x+c)*(8*a*(a^2-b^2)
-b*(4*a^2-3*b^2)*sin(d*x+c))/b^5/d

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Rubi [A]  time = 0.37, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2693, 2865, 2735, 2660, 618, 204} \[ \frac {10 a \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^6 d}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {5 x \left (-12 a^2 b^2+8 a^4+3 b^4\right )}{8 b^6}+\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]

[Out]

(-5*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6) + (10*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2
- b^2]])/(b^6*d) + (5*Cos[c + d*x]^3*(4*a - 3*b*Sin[c + d*x]))/(12*b^3*d) - Cos[c + d*x]^5/(b*d*(a + b*Sin[c
+ d*x])) - (5*Cos[c + d*x]*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*Sin[c + d*x]))/(8*b^5*d)

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])

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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b}\\ &=\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \int \frac {\cos ^2(c+d x) \left (-a b-\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3}\\ &=\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {5 \int \frac {a b \left (4 a^2-5 b^2\right )+\left (8 a^4-12 a^2 b^2+3 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8 b^5}\\ &=-\frac {5 \left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {\left (5 a \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^6}\\ &=-\frac {5 \left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {\left (10 a \left (a^2-b^2\right )^2\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 \left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\left (20 a \left (a^2-b^2\right )^2\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 \left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac {10 a \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^6 d}+\frac {5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac {\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}-\frac {5 \cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}\\ \end {align*}

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Mathematica [B]  time = 6.52, size = 3679, normalized size = 19.67 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^2,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))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x]))) - ((48*Sqrt[2]*(a -
b)*b^3*(-(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)^2*(a^2 - b^2)*Sqrt
[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) + (5*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*Si
n[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*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)^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*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))^(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*Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sqr
t[-(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]) - (2*Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*ArcTanh[(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)])))/b) + (2*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))^(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*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(S
qrt[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))/(a^2 - 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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fricas [A]  time = 0.54, size = 599, normalized size = 3.20 \[ \left [\frac {6 \, b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{5} - 12 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x - 60 \, {\left (a^{4} - a^{2} b^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 15 \, {\left (8 \, a^{4} b - 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right ) + 5 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - 3 \, {\left (8 \, a^{4} b - 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} d x - 3 \, {\left (4 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}}, \frac {6 \, b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{5} - 12 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x - 120 \, {\left (a^{4} - a^{2} b^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 15 \, {\left (8 \, a^{4} b - 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right ) + 5 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - 3 \, {\left (8 \, a^{4} b - 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} d x - 3 \, {\left (4 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/24*(6*b^5*cos(d*x + c)^5 - 5*(4*a^2*b^3 - 3*b^5)*cos(d*x + c)^3 - 15*(8*a^5 - 12*a^3*b^2 + 3*a*b^4)*d*x - 6
0*(a^4 - a^2*b^2 + (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*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)) - 15*(8*a^4*b - 12*a^2*b^3 + 3*b^5)*cos(d*x + c) + 5*(2*a*b^4*cos(d*x + c
)^3 - 3*(8*a^4*b - 12*a^2*b^3 + 3*b^5)*d*x - 3*(4*a^3*b^2 - 5*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^7*d*sin(d*
x + c) + a*b^6*d), 1/24*(6*b^5*cos(d*x + c)^5 - 5*(4*a^2*b^3 - 3*b^5)*cos(d*x + c)^3 - 15*(8*a^5 - 12*a^3*b^2
+ 3*a*b^4)*d*x - 120*(a^4 - a^2*b^2 + (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*(8*a^4*b - 12*a^2*b^3 + 3*b^5)*cos(d*x + c) + 5*(2*a*b^4*cos(d*x + c)^
3 - 3*(8*a^4*b - 12*a^2*b^3 + 3*b^5)*d*x - 3*(4*a^3*b^2 - 5*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^7*d*sin(d*x
+ c) + a*b^6*d)]

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giac [B]  time = 2.50, size = 469, normalized size = 2.51 \[ -\frac {\frac {15 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {240 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \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 {48 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - 2 \, a^{3} b^{2} + a 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 )} a b^{5}} + \frac {2 \, {\left (36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 144 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, 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} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 36 \, 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} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 304 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, a^{3} - 112 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{5}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/24*(15*(8*a^4 - 12*a^2*b^2 + 3*b^4)*(d*x + c)/b^6 - 240*(a^5 - 2*a^3*b^2 + a*b^4)*(pi*floor(1/2*(d*x + c)/p
i + 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) + 48*(a^4*b*tan(
1/2*d*x + 1/2*c) - 2*a^2*b^3*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c) + a^5 - 2*a^3*b^2 + a*b^4)/((a*ta
n(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a*b^5) + 2*(36*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 27*b^3*tan(
1/2*d*x + 1/2*c)^7 + 96*a^3*tan(1/2*d*x + 1/2*c)^6 - 144*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 36*a^2*b*tan(1/2*d*x +
1/2*c)^5 - 3*b^3*tan(1/2*d*x + 1/2*c)^5 + 288*a^3*tan(1/2*d*x + 1/2*c)^4 - 336*a*b^2*tan(1/2*d*x + 1/2*c)^4 -
36*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 3*b^3*tan(1/2*d*x + 1/2*c)^3 + 288*a^3*tan(1/2*d*x + 1/2*c)^2 - 304*a*b^2*t
an(1/2*d*x + 1/2*c)^2 - 36*a^2*b*tan(1/2*d*x + 1/2*c) + 27*b^3*tan(1/2*d*x + 1/2*c) + 96*a^3 - 112*a*b^2)/((ta
n(1/2*d*x + 1/2*c)^2 + 1)^4*b^5))/d

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maple [B]  time = 0.25, size = 1021, normalized size = 5.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a+b*sin(d*x+c))^2,x)

[Out]

4/d/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a^2-2/d/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)
*b+a)/a*tan(1/2*d*x+1/2*c)+9/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7+1/4/d/b^2/(1+tan(1/2*d*x+
1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5-1/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3-9/4/d/b^2/(1+tan(1/
2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)-8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*a^3+28/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2
)^4*a-10/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^4+15/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a^2-2/d/b^5/(tan(1/2*d*x+1/2
*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a^4-15/4/d/b^2*arctan(tan(1/2*d*x+1/2*c))-2/d/b/(tan(1/2*d*x+1/2*c)^2*a+2*ta
n(1/2*d*x+1/2*c)*b+a)+4/d/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a*tan(1/2*d*x+1/2*c)+10/d/b^6*
a^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))-20/d/b^4*a^3/(a^2-b^2)^(1/2)*arct
an(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+10/d/b^2*a/(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))-3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*a^2-8/d/b^5/(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^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^6*a-3/d/b^4/(1+tan
(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a^2-24/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^4*a^3+28/
d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^4*a+3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^
3*a^2-24/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^2*a^3+76/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1
/2*d*x+1/2*c)^2*a+3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*a^2-2/d/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*
tan(1/2*d*x+1/2*c)*b+a)*a^3*tan(1/2*d*x+1/2*c)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^2,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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 7.58, size = 2530, normalized size = 13.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(a + b*sin(c + d*x))^2,x)

[Out]

- ((2*(15*a^4 + 3*b^4 - 20*a^2*b^2))/(3*b^5) - (5*tan(c/2 + (d*x)/2)^8*(b^4 - 4*a^4 + 4*a^2*b^2))/(2*b^5) + (5
*tan(c/2 + (d*x)/2)^6*(16*a^4 + 3*b^4 - 20*a^2*b^2))/(2*b^5) + (5*tan(c/2 + (d*x)/2)^2*(48*a^4 + 15*b^4 - 68*a
^2*b^2))/(6*b^5) + (5*tan(c/2 + (d*x)/2)^4*(72*a^4 + 15*b^4 - 100*a^2*b^2))/(6*b^5) + (tan(c/2 + (d*x)/2)*(180
*a^4 + 24*b^4 - 245*a^2*b^2))/(12*a*b^4) + (4*tan(c/2 + (d*x)/2)^5*(15*a^4 + 3*b^4 - 20*a^2*b^2))/(a*b^4) + (t
an(c/2 + (d*x)/2)^9*(20*a^4 + 8*b^4 - 25*a^2*b^2))/(4*a*b^4) + (tan(c/2 + (d*x)/2)^7*(60*a^4 + 16*b^4 - 85*a^2
*b^2))/(2*a*b^4) + (tan(c/2 + (d*x)/2)^3*(300*a^4 + 48*b^4 - 385*a^2*b^2))/(6*a*b^4))/(d*(a + 2*b*tan(c/2 + (d
*x)/2) + 5*a*tan(c/2 + (d*x)/2)^2 + 10*a*tan(c/2 + (d*x)/2)^4 + 10*a*tan(c/2 + (d*x)/2)^6 + 5*a*tan(c/2 + (d*x
)/2)^8 + a*tan(c/2 + (d*x)/2)^10 + 8*b*tan(c/2 + (d*x)/2)^3 + 12*b*tan(c/2 + (d*x)/2)^5 + 8*b*tan(c/2 + (d*x)/
2)^7 + 2*b*tan(c/2 + (d*x)/2)^9)) - (atan((((a^4*8i + b^4*3i - a^2*b^2*12i)*(((225*a^2*b^13)/2 - 900*a^4*b^11
+ 2400*a^6*b^9 - 2400*a^8*b^7 + 800*a^10*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(450*a*b^15 - 5425*a^3*b^13 + 17800*a
^5*b^11 - 24000*a^7*b^9 + 14400*a^9*b^7 - 3200*a^11*b^5))/(2*b^15) - (5*(a^4*8i + b^4*3i - a^2*b^2*12i)*((60*a
*b^16 - 140*a^3*b^14 + 80*a^5*b^12)/b^14 - (5*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(
2*b^15))*(a^4*8i + b^4*3i - a^2*b^2*12i))/(8*b^6) + (tan(c/2 + (d*x)/2)*(640*a^2*b^16 - 1280*a^4*b^14 + 640*a^
6*b^12))/(2*b^15)))/(8*b^6))*5i)/(8*b^6) + ((a^4*8i + b^4*3i - a^2*b^2*12i)*(((225*a^2*b^13)/2 - 900*a^4*b^11
+ 2400*a^6*b^9 - 2400*a^8*b^7 + 800*a^10*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(450*a*b^15 - 5425*a^3*b^13 + 17800*a
^5*b^11 - 24000*a^7*b^9 + 14400*a^9*b^7 - 3200*a^11*b^5))/(2*b^15) + (5*(a^4*8i + b^4*3i - a^2*b^2*12i)*((60*a
*b^16 - 140*a^3*b^14 + 80*a^5*b^12)/b^14 + (5*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(
2*b^15))*(a^4*8i + b^4*3i - a^2*b^2*12i))/(8*b^6) + (tan(c/2 + (d*x)/2)*(640*a^2*b^16 - 1280*a^4*b^14 + 640*a^
6*b^12))/(2*b^15)))/(8*b^6))*5i)/(8*b^6))/((4000*a^13 - 1875*a^3*b^10 + 12750*a^5*b^8 - 30875*a^7*b^6 + 35000*
a^9*b^4 - 19000*a^11*b^2)/b^14 - (5*(a^4*8i + b^4*3i - a^2*b^2*12i)*(((225*a^2*b^13)/2 - 900*a^4*b^11 + 2400*a
^6*b^9 - 2400*a^8*b^7 + 800*a^10*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(450*a*b^15 - 5425*a^3*b^13 + 17800*a^5*b^11
- 24000*a^7*b^9 + 14400*a^9*b^7 - 3200*a^11*b^5))/(2*b^15) - (5*(a^4*8i + b^4*3i - a^2*b^2*12i)*((60*a*b^16 -
140*a^3*b^14 + 80*a^5*b^12)/b^14 - (5*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))
*(a^4*8i + b^4*3i - a^2*b^2*12i))/(8*b^6) + (tan(c/2 + (d*x)/2)*(640*a^2*b^16 - 1280*a^4*b^14 + 640*a^6*b^12))
/(2*b^15)))/(8*b^6)))/(8*b^6) + (5*(a^4*8i + b^4*3i - a^2*b^2*12i)*(((225*a^2*b^13)/2 - 900*a^4*b^11 + 2400*a^
6*b^9 - 2400*a^8*b^7 + 800*a^10*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(450*a*b^15 - 5425*a^3*b^13 + 17800*a^5*b^11 -
24000*a^7*b^9 + 14400*a^9*b^7 - 3200*a^11*b^5))/(2*b^15) + (5*(a^4*8i + b^4*3i - a^2*b^2*12i)*((60*a*b^16 - 1
40*a^3*b^14 + 80*a^5*b^12)/b^14 + (5*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*
(a^4*8i + b^4*3i - a^2*b^2*12i))/(8*b^6) + (tan(c/2 + (d*x)/2)*(640*a^2*b^16 - 1280*a^4*b^14 + 640*a^6*b^12))/
(2*b^15)))/(8*b^6)))/(8*b^6) + (tan(c/2 + (d*x)/2)*(16000*a^14 + 2250*a^2*b^12 - 22500*a^4*b^10 + 86250*a^6*b^
8 - 162000*a^8*b^6 + 160000*a^10*b^4 - 80000*a^12*b^2))/b^15))*(a^4*8i + b^4*3i - a^2*b^2*12i)*5i)/(4*b^6*d) -
(10*a*atanh((1125*a^3*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(3250*a^5*b - 1125*a^3*b^3 - (3125*a^7)/b +
(1000*a^9)/b^3 - 6250*a^6*tan(c/2 + (d*x)/2) - 2250*a^2*b^4*tan(c/2 + (d*x)/2) + 6500*a^4*b^2*tan(c/2 + (d*x)/
2) + (2000*a^8*tan(c/2 + (d*x)/2))/b^2) + (1000*a^5*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(3125*a^7*b + 1
125*a^3*b^5 - 3250*a^5*b^3 - (1000*a^9)/b - 2000*a^8*tan(c/2 + (d*x)/2) + 2250*a^2*b^6*tan(c/2 + (d*x)/2) - 65
00*a^4*b^4*tan(c/2 + (d*x)/2) + 6250*a^6*b^2*tan(c/2 + (d*x)/2)) + (2250*a^2*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3
*a^2*b^4 + 3*a^4*b^2)^(1/2))/(3250*a^5 - 1125*a^3*b^2 - (3125*a^7)/b^2 + (1000*a^9)/b^4 + 6500*a^4*b*tan(c/2 +
(d*x)/2) - 2250*a^2*b^3*tan(c/2 + (d*x)/2) - (6250*a^6*tan(c/2 + (d*x)/2))/b + (2000*a^8*tan(c/2 + (d*x)/2))/
b^3) + (3125*a^4*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(3125*a^7 + 1125*a^3*b^4 - 3250
*a^5*b^2 - (1000*a^9)/b^2 + 6250*a^6*b*tan(c/2 + (d*x)/2) + 2250*a^2*b^5*tan(c/2 + (d*x)/2) - 6500*a^4*b^3*tan
(c/2 + (d*x)/2) - (2000*a^8*tan(c/2 + (d*x)/2))/b) + (1000*a^6*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a
^4*b^2)^(1/2))/(1000*a^9 - 1125*a^3*b^6 + 3250*a^5*b^4 - 3125*a^7*b^2 + 2000*a^8*b*tan(c/2 + (d*x)/2) - 2250*a
^2*b^7*tan(c/2 + (d*x)/2) + 6500*a^4*b^5*tan(c/2 + (d*x)/2) - 6250*a^6*b^3*tan(c/2 + (d*x)/2)))*(-(a + b)^3*(a
- b)^3)^(1/2))/(b^6*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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