3.478 \(\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=307 \[ \frac{\left (-23 a^2 b^2+12 a^4+6 b^4\right ) \sin (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (-28 a^4 b^2+35 a^2 b^4+8 a^6-20 b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left (4 a^2-9 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{a^3 \left (-11 a^2 b^2+4 a^4+12 b^4\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{4 a x}{b^5} \]

[Out]

(-4*a*x)/b^5 + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a +
b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d
) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*Cos[c
+ d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Sin
[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

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Rubi [A]  time = 0.899682, antiderivative size = 307, 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 = {2792, 3047, 3031, 3023, 2735, 2659, 205} \[ \frac{\left (-23 a^2 b^2+12 a^4+6 b^4\right ) \sin (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}+\frac{a^2 \left (-28 a^4 b^2+35 a^2 b^4+8 a^6-20 b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left (4 a^2-9 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{a^3 \left (-11 a^2 b^2+4 a^4+12 b^4\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{4 a x}{b^5} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^4,x]

[Out]

(-4*a*x)/b^5 + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a +
b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d
) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*Cos[c
+ d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Sin
[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 2792

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

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*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 205

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

Rubi steps

\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\int \frac{\cos ^2(c+d x) \left (3 a^2-3 a b \cos (c+d x)-\left (4 a^2-3 b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a^2 \left (4 a^2-9 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{\cos (c+d x) \left (-2 a^2 \left (4 a^2-9 b^2\right )+2 a b \left (a^2-6 b^2\right ) \cos (c+d x)+\left (12 a^4-23 a^2 b^2+6 b^4\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a^2 \left (4 a^2-9 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a^3 \left (4 a^4-11 a^2 b^2+12 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{-3 a^2 b \left (4 a^4-11 a^2 b^2+12 b^4\right )-a \left (12 a^6-37 a^4 b^2+43 a^2 b^4-18 b^6\right ) \cos (c+d x)+b \left (a^2-b^2\right ) \left (12 a^4-23 a^2 b^2+6 b^4\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=\frac{\left (12 a^4-23 a^2 b^2+6 b^4\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a^2 \left (4 a^2-9 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a^3 \left (4 a^4-11 a^2 b^2+12 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{-3 a^2 b^2 \left (4 a^4-11 a^2 b^2+12 b^4\right )-24 a b \left (a^2-b^2\right )^3 \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac{4 a x}{b^5}+\frac{\left (12 a^4-23 a^2 b^2+6 b^4\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a^2 \left (4 a^2-9 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a^3 \left (4 a^4-11 a^2 b^2+12 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (a^2 \left (8 a^6-28 a^4 b^2+35 a^2 b^4-20 b^6\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac{4 a x}{b^5}+\frac{\left (12 a^4-23 a^2 b^2+6 b^4\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a^2 \left (4 a^2-9 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a^3 \left (4 a^4-11 a^2 b^2+12 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (a^2 \left (8 a^6-28 a^4 b^2+35 a^2 b^4-20 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right )^3 d}\\ &=-\frac{4 a x}{b^5}+\frac{a^2 \left (8 a^6-28 a^4 b^2+35 a^2 b^4-20 b^6\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}+\frac{\left (12 a^4-23 a^2 b^2+6 b^4\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{a^2 \left (4 a^2-9 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a^3 \left (4 a^4-11 a^2 b^2+12 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 5.65826, size = 240, normalized size = 0.78 \[ \frac{\frac{5 a^4 b \left (3 b^2-2 a^2\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}+\frac{a^3 b \left (-71 a^2 b^2+26 a^4+60 b^4\right ) \sin (c+d x)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))}+\frac{6 a^2 \left (-28 a^4 b^2+35 a^2 b^4+8 a^6-20 b^6\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+\frac{2 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^3}-24 a (c+d x)+6 b \sin (c+d x)}{6 b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^4,x]

[Out]

(-24*a*(c + d*x) + (6*a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-
a^2 + b^2]])/(-a^2 + b^2)^(7/2) + 6*b*Sin[c + d*x] + (2*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*
x])^3) + (5*a^4*b*(-2*a^2 + 3*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2) + (a^3*b*(26*a^4
- 71*a^2*b^2 + 60*b^4)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])))/(6*b^5*d)

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Maple [B]  time = 0.101, size = 1396, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+b*cos(d*x+c))^4,x)

[Out]

2/d/b^4*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-8/d/b^5*a*arctan(tan(1/2*d*x+1/2*c))+6/d*a^7/b^4/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/d*a^6/b^3/
(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-18/
d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/
2*c)^5+5/d*a^4/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2
*d*x+1/2*c)^5+20/d*a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*t
an(1/2*d*x+1/2*c)^5+12/d*a^7/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2-2*a*b+b^2)/(a^2+2*
a*b+b^2)*tan(1/2*d*x+1/2*c)^3-116/3/d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2-2*a*b
+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+40/d*a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2
-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+6/d*a^7/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+
b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)+2/d*a^6/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)
^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)-18/d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*
x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)-5/d*a^4/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1
/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)+20/d*a^3/(tan(1/2*d*x+1/2*c)^2*a-t
an(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)+8/d*a^8/b^5/(a^6-3*a^4*b^2+3*a
^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-28/d*a^6/b^3/(a^6-3*a^4*b
^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+35/d*a^4/b/(a^6-3*a
^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-20/d*a^2*b/(a^6
-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.85581, size = 3557, normalized size = 11.59 \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)^5/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

[-1/12*(48*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*d*x*cos(d*x + c)^3 + 144*(a^10*b^2 - 4*a^8*b
^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*d*x*cos(d*x + c)^2 + 144*(a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 +
a^3*b^9)*d*x*cos(d*x + c) + 48*(a^12 - 4*a^10*b^2 + 6*a^8*b^4 - 4*a^6*b^6 + a^4*b^8)*d*x + 3*(8*a^11 - 28*a^9*
b^2 + 35*a^7*b^4 - 20*a^5*b^6 + (8*a^8*b^3 - 28*a^6*b^5 + 35*a^4*b^7 - 20*a^2*b^9)*cos(d*x + c)^3 + 3*(8*a^9*b
^2 - 28*a^7*b^4 + 35*a^5*b^6 - 20*a^3*b^8)*cos(d*x + c)^2 + 3*(8*a^10*b - 28*a^8*b^3 + 35*a^6*b^5 - 20*a^4*b^7
)*cos(d*x + c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 + 2*sqrt(-a^2 + b^2)*(
a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - 2*(24*a^11*
b - 92*a^9*b^3 + 133*a^7*b^5 - 71*a^5*b^7 + 6*a^3*b^9 + 6*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12
)*cos(d*x + c)^3 + (44*a^9*b^3 - 169*a^7*b^5 + 239*a^5*b^7 - 132*a^3*b^9 + 18*a*b^11)*cos(d*x + c)^2 + 3*(20*a
^10*b^2 - 77*a^8*b^4 + 110*a^6*b^6 - 59*a^4*b^8 + 6*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^
10 + 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b
^15)*d*cos(d*x + c)^2 + 3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4*a^4*b^12 + a^2*b^14)*d*cos(d*x + c) + (a^11*b
^5 - 4*a^9*b^7 + 6*a^7*b^9 - 4*a^5*b^11 + a^3*b^13)*d), -1/6*(24*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9
+ a*b^11)*d*x*cos(d*x + c)^3 + 72*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*d*x*cos(d*x + c)^2
+ 72*(a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*d*x*cos(d*x + c) + 24*(a^12 - 4*a^10*b^2 + 6*a^8*
b^4 - 4*a^6*b^6 + a^4*b^8)*d*x - 3*(8*a^11 - 28*a^9*b^2 + 35*a^7*b^4 - 20*a^5*b^6 + (8*a^8*b^3 - 28*a^6*b^5 +
35*a^4*b^7 - 20*a^2*b^9)*cos(d*x + c)^3 + 3*(8*a^9*b^2 - 28*a^7*b^4 + 35*a^5*b^6 - 20*a^3*b^8)*cos(d*x + c)^2
+ 3*(8*a^10*b - 28*a^8*b^3 + 35*a^6*b^5 - 20*a^4*b^7)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) +
b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (24*a^11*b - 92*a^9*b^3 + 133*a^7*b^5 - 71*a^5*b^7 + 6*a^3*b^9 + 6*(a^8*b
^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cos(d*x + c)^3 + (44*a^9*b^3 - 169*a^7*b^5 + 239*a^5*b^7 - 132
*a^3*b^9 + 18*a*b^11)*cos(d*x + c)^2 + 3*(20*a^10*b^2 - 77*a^8*b^4 + 110*a^6*b^6 - 59*a^4*b^8 + 6*a^2*b^10)*co
s(d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^10 + 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b
^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^15)*d*cos(d*x + c)^2 + 3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4
*a^4*b^12 + a^2*b^14)*d*cos(d*x + c) + (a^11*b^5 - 4*a^9*b^7 + 6*a^7*b^9 - 4*a^5*b^11 + a^3*b^13)*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)**5/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.2901, size = 760, normalized size = 2.48 \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)^5/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*(8*a^8 - 28*a^6*b^2 + 35*a^4*b^4 - 20*a^2*b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arc
tan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b
^11)*sqrt(a^2 - b^2)) - (18*a^9*tan(1/2*d*x + 1/2*c)^5 - 42*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 24*a^7*b^2*tan(1/2*
d*x + 1/2*c)^5 + 117*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 - 24*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 105*a^4*b^5*tan(1/2*
d*x + 1/2*c)^5 + 60*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 36*a^9*tan(1/2*d*x + 1/2*c)^3 - 152*a^7*b^2*tan(1/2*d*x +
1/2*c)^3 + 236*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 120*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 18*a^9*tan(1/2*d*x + 1/2
*c) + 42*a^8*b*tan(1/2*d*x + 1/2*c) - 24*a^7*b^2*tan(1/2*d*x + 1/2*c) - 117*a^6*b^3*tan(1/2*d*x + 1/2*c) - 24*
a^5*b^4*tan(1/2*d*x + 1/2*c) + 105*a^4*b^5*tan(1/2*d*x + 1/2*c) + 60*a^3*b^6*tan(1/2*d*x + 1/2*c))/((a^6*b^4 -
3*a^4*b^6 + 3*a^2*b^8 - b^10)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^3) + 12*(d*x + c)
*a/b^5 - 6*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*b^4))/d