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

Optimal. Leaf size=188 \[ -\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^6 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {a x \left (8 a^4-20 a^2 b^2+15 b^4\right )}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d} \]

[Out]

1/8*a*(8*a^4-20*a^2*b^2+15*b^4)*x/b^6-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^6/d
+1/5*cos(d*x+c)^5/b/d-1/12*cos(d*x+c)^3*(4*a^2-4*b^2-3*a*b*sin(d*x+c))/b^3/d+1/8*cos(d*x+c)*(8*(a^2-b^2)^2-a*b
*(4*a^2-7*b^2)*sin(d*x+c))/b^5/d

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Rubi [A]  time = 0.46, antiderivative size = 188, 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 = {2695, 2865, 2735, 2660, 618, 204} \[ -\frac {2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^6 d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {a x \left (-20 a^2 b^2+8 a^4+15 b^4\right )}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 -
b^2]])/(b^6*d) + Cos[c + d*x]^5/(5*b*d) - (Cos[c + d*x]^3*(4*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(12*b^3*d) + (
Cos[c + d*x]*(8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*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 2695

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 + p)), x] + Dist[(g^2*(p - 1))/(b*(m + p)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && 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)} \, dx &=\frac {\cos ^5(c+d x)}{5 b d}+\frac {\int \frac {\cos ^4(c+d x) (b+a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b}\\ &=\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\int \frac {\cos ^2(c+d x) \left (-b \left (a^2-4 b^2\right )-a \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3}\\ &=\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {\int \frac {b \left (4 a^4-9 a^2 b^2+8 b^4\right )+a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8 b^5}\\ &=\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^6}\\ &=\frac {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}-\frac {\left (2 \left (a^2-b^2\right )^3\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 {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}+\frac {\left (4 \left (a^2-b^2\right )^3\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 {a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2-b^2\right )^{5/2} \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 {\cos ^5(c+d x)}{5 b d}-\frac {\cos ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 a b \sin (c+d x)\right )}{12 b^3 d}+\frac {\cos (c+d x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{8 b^5 d}\\ \end {align*}

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Mathematica [B]  time = 6.31, size = 2827, normalized size = 15.04 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x]^5*((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) - (Sq
rt[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*Si
n[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]*Sq
rt[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*S
in[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]*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]) - (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)])/(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*S
in[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))/(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 = 0.50, size = 483, normalized size = 2.57 \[ \left [\frac {24 \, b^{5} \cos \left (d x + c\right )^{5} - 40 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 60 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 120 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) + 15 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, b^{6} d}, \frac {24 \, b^{5} \cos \left (d x + c\right )^{5} - 40 \, {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 120 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 120 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right ) + 15 \, {\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, b^{6} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/120*(24*b^5*cos(d*x + c)^5 - 40*(a^2*b^3 - b^5)*cos(d*x + c)^3 + 15*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*d*x + 6
0*(a^4 - 2*a^2*b^2 + b^4)*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)) + 120*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c) + 15*(2*a*b^4*cos(d*x + c)^3 - (4*a^3*b^2 - 7*a*b^4
)*cos(d*x + c))*sin(d*x + c))/(b^6*d), 1/120*(24*b^5*cos(d*x + c)^5 - 40*(a^2*b^3 - b^5)*cos(d*x + c)^3 + 15*(
8*a^5 - 20*a^3*b^2 + 15*a*b^4)*d*x + 120*(a^4 - 2*a^2*b^2 + b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/
(sqrt(a^2 - b^2)*cos(d*x + c))) + 120*(a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c) + 15*(2*a*b^4*cos(d*x + c)^3 - (4
*a^3*b^2 - 7*a*b^4)*cos(d*x + c))*sin(d*x + c))/(b^6*d)]

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giac [B]  time = 0.49, size = 496, normalized size = 2.64 \[ \frac {\frac {15 \, {\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{b^{6}} - \frac {240 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\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 {2 \, {\left (60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1200 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 720 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 720 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1120 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1040 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 560 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, a^{4} - 280 \, a^{2} b^{2} + 184 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} b^{5}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*(d*x + c)/b^6 - 240*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(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
*(60*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 135*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*a^4*tan(1/2*d*x + 1/2*c)^8 - 360*a^
2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 150*a*b^3*t
an(1/2*d*x + 1/2*c)^7 + 480*a^4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 720*b^4*tan(1/2
*d*x + 1/2*c)^6 + 720*a^4*tan(1/2*d*x + 1/2*c)^4 - 1600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1120*b^4*tan(1/2*d*x
+ 1/2*c)^4 - 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 150*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 480*a^4*tan(1/2*d*x + 1/2*c
)^2 - 1040*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 560*b^4*tan(1/2*d*x + 1/2*c)^2 - 60*a^3*b*tan(1/2*d*x + 1/2*c) + 1
35*a*b^3*tan(1/2*d*x + 1/2*c) + 120*a^4 - 280*a^2*b^2 + 184*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^5))/d

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maple [B]  time = 0.16, size = 1055, normalized size = 5.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/4/d/b^2*a*arctan(tan(1/2*d*x+1/2*c))+2/d/(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))+12/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4*a^4-80/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(
1/2*d*x+1/2*c)^4*a^2-2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^3*a^3+5/2/d/b^2/(1+tan(1/2*d*x+1/2*
c)^2)^5*tan(1/2*d*x+1/2*c)^3*a+8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2*a^4-52/3/d/b^3/(1+tan(1
/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2*a^2-1/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)*a^3+9/4/d/b^
2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)*a+1/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9*a^3-
9/4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9*a+2/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2
*c)^8*a^4+8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6*a^4-6/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1
/2*d*x+1/2*c)^8*a^2+2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7*a^3-5/2/d/b^2/(1+tan(1/2*d*x+1/2*c
)^2)^5*tan(1/2*d*x+1/2*c)^7*a-20/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6*a^2-2/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^6+6/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^4+12/d/b/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6+2/d/b^5/(1+tan
(1/2*d*x+1/2*c)^2)^5*a^4+2/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^5+28/3/d/b/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*
x+1/2*c)^2-5/d/b^4*arctan(tan(1/2*d*x+1/2*c))*a^3-14/3/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*a^2+56/3/d/b/(1+tan(1/
2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4-6/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))*a^2+46/15/d/b/(1+tan(1/2*d*x+1/2*c)^2)^5+6/d/b/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8

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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)),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 = 7.66, size = 3075, normalized size = 16.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(15*a^4 + 23*b^4 - 35*a^2*b^2))/(15*b^5) + (tan(c/2 + (d*x)/2)^3*(5*a*b^2 - 4*a^3))/(2*b^4) - (tan(c/2 + (
d*x)/2)^7*(5*a*b^2 - 4*a^3))/(2*b^4) - (tan(c/2 + (d*x)/2)^9*(9*a*b^2 - 4*a^3))/(4*b^4) + (2*tan(c/2 + (d*x)/2
)^8*(a^4 + 3*b^4 - 3*a^2*b^2))/b^5 + (4*tan(c/2 + (d*x)/2)^6*(2*a^4 + 3*b^4 - 5*a^2*b^2))/b^5 + (4*tan(c/2 + (
d*x)/2)^2*(6*a^4 + 7*b^4 - 13*a^2*b^2))/(3*b^5) + (4*tan(c/2 + (d*x)/2)^4*(9*a^4 + 14*b^4 - 20*a^2*b^2))/(3*b^
5) + (tan(c/2 + (d*x)/2)*(9*a*b^2 - 4*a^3))/(4*b^4))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10
*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (atan((((-(a + b)^5*(a - b)^5)^
(1/2)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)
*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b
^15) + ((-(a + b)^5*(a - b)^5)^(1/2)*((28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((-(a + b)^5*(a - b)^5)
^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15)))/b^6 - (tan(c/2 + (d*x)/2)*(12
8*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6)*1i)/b^6 + ((-(a + b)^5*(a - b)^5)^(1/2
)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64
*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15)
+ ((-(a + b)^5*(a - b)^5)^(1/2)*(((-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19
- 128*a^3*b^17))/(2*b^15)))/b^6 - (28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (tan(c/2 + (d*x)/2)*(128*a*
b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6)*1i)/b^6)/((32*a^16 - 120*a^2*b^14 + 655*a^
4*b^12 - 1549*a^6*b^10 + 2069*a^8*b^8 - 1695*a^10*b^6 + 856*a^12*b^4 - 248*a^14*b^2)/b^14 + ((-(a + b)^5*(a -
b)^5)^(1/2)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d
*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5)
)/(2*b^15) + ((-(a + b)^5*(a - b)^5)^(1/2)*((28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((-(a + b)^5*(a -
b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15)))/b^6 - (tan(c/2 + (d*x)/
2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6))/b^6 - ((-(a + b)^5*(a - b)^5)^(
1/2)*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*
(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^
15) + ((-(a + b)^5*(a - b)^5)^(1/2)*(((-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^
19 - 128*a^3*b^17))/(2*b^15)))/b^6 - (28*a^2*b^16 - 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (tan(c/2 + (d*x)/2)*(128
*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15)))/b^6))/b^6 + (tan(c/2 + (d*x)/2)*(128*a^17 -
450*a^3*b^14 + 2550*a^5*b^12 - 6230*a^7*b^10 + 8530*a^9*b^8 - 7088*a^11*b^6 + 3584*a^13*b^4 - 1024*a^15*b^2))/
b^15))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(b^6*d) + (a*atan(((a*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9
- 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*
b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2*b^2)*((28*a^2*b^16
- 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^
12))/(2*b^15) + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*(8*a^4 + 15*b^4 -
20*a^2*b^2)*1i)/(8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2))/(8*b^6) + (a*(((225*a^4*b^13)/2 - 300*a^6
*b^11 + 320*a^8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*
a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2
*b^2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16
- 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15)
)*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2))/(8*b^6))/((32*a^16 -
120*a^2*b^14 + 655*a^4*b^12 - 1549*a^6*b^10 + 2069*a^8*b^8 - 1695*a^10*b^6 + 856*a^12*b^4 - 248*a^14*b^2)/b^14
+ (tan(c/2 + (d*x)/2)*(128*a^17 - 450*a^3*b^14 + 2550*a^5*b^12 - 6230*a^7*b^10 + 8530*a^9*b^8 - 7088*a^11*b^6
+ 3584*a^13*b^4 - 1024*a^15*b^2))/b^15 + (a*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^8*b^9 - 160*a^10*b^7 +
32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 3160*a^7*b^11 + 2240*a^9*b
^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2*b^2)*((28*a^2*b^16 - 44*a^4*b^14 + 1
6*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15) + (
a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(
8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(8*b^6) - (a*(((225*a^4*b^13)/2 - 300*a^6*b^11 + 320*a^
8*b^9 - 160*a^10*b^7 + 32*a^12*b^5)/b^14 - (tan(c/2 + (d*x)/2)*(64*a*b^17 - 834*a^3*b^15 + 2385*a^5*b^13 - 316
0*a^7*b^11 + 2240*a^9*b^9 - 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) + (a*(8*a^4 + 15*b^4 - 20*a^2*b^2)*((tan(c/
2 + (d*x)/2)*(128*a*b^18 - 384*a^3*b^16 + 384*a^5*b^14 - 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 - 44*a^4*b^14
+ 16*a^6*b^12)/b^14 + (a*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 - 128*a^3*b^17))/(2*b^15))*(8*a^4 + 15*
b^4 - 20*a^2*b^2)*1i)/(8*b^6))*1i)/(8*b^6))*(8*a^4 + 15*b^4 - 20*a^2*b^2)*1i)/(8*b^6)))*(8*a^4 + 15*b^4 - 20*a
^2*b^2))/(4*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)),x)

[Out]

Timed out

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