3.5.69 \(\int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx\) [469]

Optimal. Leaf size=300 \[ \frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]

[Out]

1/2*(12*a^2+b^2)*x/b^5-a^3*(12*a^4-29*a^2*b^2+20*b^4)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)
^(5/2)/b^5/(a+b)^(5/2)/d-3/2*a*(4*a^4-7*a^2*b^2+2*b^4)*sin(d*x+c)/b^4/(a^2-b^2)^2/d+1/2*(6*a^4-10*a^2*b^2+b^4)
*cos(d*x+c)*sin(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*a^2*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2-1/2*
a^2*(4*a^2-7*b^2)*cos(d*x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))

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Rubi [A]
time = 0.54, antiderivative size = 300, normalized size of antiderivative = 1.00, 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 = {2871, 3126, 3128, 3102, 2814, 2738, 211} \begin {gather*} -\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {x \left (12 a^2+b^2\right )}{2 b^5}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3,x]

[Out]

((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a
+ b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)
^2*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^3*S
in[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(4*a^2 - 7*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2
*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))

Rule 211

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

Rule 2738

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 2814

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

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(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 3102

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

Rule 3126

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 3128

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

Rubi steps

\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (3 a^2-2 a b \cos (c+d x)-2 \left (2 a^2-b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (-2 a^2 \left (4 a^2-7 b^2\right )+a b \left (a^2-4 b^2\right ) \cos (c+d x)+2 \left (6 a^4-10 a^2 b^2+b^4\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {2 a \left (6 a^4-10 a^2 b^2+b^4\right )-2 b \left (2 a^4-4 a^2 b^2-b^4\right ) \cos (c+d x)-6 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {2 a b \left (6 a^4-10 a^2 b^2+b^4\right )+2 \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \text {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 )^2 d}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 2.06, size = 199, normalized size = 0.66 \begin {gather*} \frac {2 \left (12 a^2+b^2\right ) (c+d x)+\frac {4 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}-12 a b \sin (c+d x)+\frac {2 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {2 a^4 b \left (-7 a^2+10 b^2\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+b^2 \sin (2 (c+d x))}{4 b^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3,x]

[Out]

(2*(12*a^2 + b^2)*(c + d*x) + (4*a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a
^2 + b^2]])/(-a^2 + b^2)^(5/2) - 12*a*b*Sin[c + d*x] + (2*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c +
d*x])^2) + (2*a^4*b*(-7*a^2 + 10*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])) + b^2*Sin[2*(c
+ d*x)])/(4*b^5*d)

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Maple [A]
time = 0.62, size = 306, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-3 a b -\frac {1}{2} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (12 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {2 a^{3} \left (\frac {\frac {\left (6 a^{2}-a b -10 b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (6 a^{2}+a b -10 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (12 a^{4}-29 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(306\)
default	\(\frac {\frac {\frac {2 \left (\left (-3 a b -\frac {1}{2} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (12 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {2 a^{3} \left (\frac {\frac {\left (6 a^{2}-a b -10 b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (6 a^{2}+a b -10 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (12 a^{4}-29 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(306\)
risch	\(\frac {6 x \,a^{2}}{b^{5}}+\frac {x}{2 b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {3 i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {3 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {i a^{4} \left (8 a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}-11 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-13 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-10 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+20 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-29 a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+7 a^{2} b^{2}-10 b^{4}\right )}{b^{5} \left (a^{2}-b^{2}\right )^{2} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )^{2}}-\frac {6 a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{5}}+\frac {29 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}-\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}+\frac {6 a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{5}}-\frac {29 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}+\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}\)	\(776\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^5*(((-3*a*b-1/2*b^2)*tan(1/2*d*x+1/2*c)^3+(-3*a*b+1/2*b^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^
2)^2+1/2*(12*a^2+b^2)*arctan(tan(1/2*d*x+1/2*c)))-2*a^3/b^5*((1/2*(6*a^2-a*b-10*b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)
*tan(1/2*d*x+1/2*c)^3+1/2*(6*a^2+a*b-10*b^2)*a*b/(a+b)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*x+1/2*
c)^2-b*tan(1/2*d*x+1/2*c)^2+a+b)^2+1/2*(12*a^4-29*a^2*b^2+20*b^4)/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arct
an(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.49, size = 1161, normalized size = 3.87 \begin {gather*} \left [\frac {2 \, {\left (12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (12 \, a^{10} - 35 \, a^{8} b^{2} + 33 \, a^{6} b^{4} - 9 \, a^{4} b^{6} - a^{2} b^{8}\right )} d x - {\left (12 \, a^{9} - 29 \, a^{7} b^{2} + 20 \, a^{5} b^{4} + {\left (12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (12 \, a^{9} b - 33 \, a^{7} b^{3} + 27 \, a^{5} b^{5} - 6 \, a^{3} b^{7} - {\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} \cos \left (d x + c\right )^{2} + {\left (18 \, a^{8} b^{2} - 50 \, a^{6} b^{4} + 43 \, a^{4} b^{6} - 11 \, a^{2} b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{5} - 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} - a^{2} b^{11}\right )} d\right )}}, \frac {{\left (12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right )} d x \cos \left (d x + c\right ) + {\left (12 \, a^{10} - 35 \, a^{8} b^{2} + 33 \, a^{6} b^{4} - 9 \, a^{4} b^{6} - a^{2} b^{8}\right )} d x - {\left (12 \, a^{9} - 29 \, a^{7} b^{2} + 20 \, a^{5} b^{4} + {\left (12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (12 \, a^{9} b - 33 \, a^{7} b^{3} + 27 \, a^{5} b^{5} - 6 \, a^{3} b^{7} - {\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} \cos \left (d x + c\right )^{2} + {\left (18 \, a^{8} b^{2} - 50 \, a^{6} b^{4} + 43 \, a^{4} b^{6} - 11 \, a^{2} b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{5} - 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} - a^{2} b^{11}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/4*(2*(12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d*x*cos(d*x + c)^2 + 4*(12*a^9*b - 35*a^7*b^
3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d*x*cos(d*x + c) + 2*(12*a^10 - 35*a^8*b^2 + 33*a^6*b^4 - 9*a^4*b^6 - a^2*
b^8)*d*x - (12*a^9 - 29*a^7*b^2 + 20*a^5*b^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(d*x + c)^2 + 2*(12*a
^8*b - 29*a^6*b^3 + 20*a^4*b^5)*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*(12*a^9*b - 33*a^7*b^3 + 27*a^5*b^5 - 6*a^3*b^7 - (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10
)*cos(d*x + c)^3 + 4*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(d*x + c)^2 + (18*a^8*b^2 - 50*a^6*b^4 + 43*
a^4*b^6 - 11*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2
+ 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c) + (a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*
d), 1/2*((12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d*x*cos(d*x + c)^2 + 2*(12*a^9*b - 35*a^7*b
^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d*x*cos(d*x + c) + (12*a^10 - 35*a^8*b^2 + 33*a^6*b^4 - 9*a^4*b^6 - a^2*b
^8)*d*x - (12*a^9 - 29*a^7*b^2 + 20*a^5*b^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(d*x + c)^2 + 2*(12*a^
8*b - 29*a^6*b^3 + 20*a^4*b^5)*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))) - (12*a^9*b - 33*a^7*b^3 + 27*a^5*b^5 - 6*a^3*b^7 - (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(d
*x + c)^3 + 4*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(d*x + c)^2 + (18*a^8*b^2 - 50*a^6*b^4 + 43*a^4*b^6
- 11*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2 + 2*(a^
7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c) + (a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*d)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1735 vs. \(2 (281) = 562\).
time = 0.62, size = 1735, normalized size = 5.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(((12*a^6 - 6*a^5*b - 23*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 - a*b^5 + b^6)*sqrt(a^2 - b^2)*abs(a^4*b^5 - 2
*a^2*b^7 + b^9)*abs(-a + b) + (24*a^11*b^4 - 12*a^10*b^5 - 100*a^9*b^6 + 47*a^8*b^7 + 158*a^7*b^8 - 68*a^6*b^9
- 111*a^5*b^10 + 42*a^4*b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14 - b^15)*sqrt(a^2 - b^2)*abs(-a + b))*(pi*flo
or(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/2*c)/sqrt((4*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 + sqrt(-16*(a
^5*b^4 + a^4*b^5 - 2*a^3*b^6 - 2*a^2*b^7 + a*b^8 + b^9)*(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b
^9) + 16*(a^5*b^4 - 2*a^3*b^6 + a*b^8)^2))/(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9))))/((a^4*
b^5 - 2*a^2*b^7 + b^9)^2*(a^2 - 2*a*b + b^2) + (a^7*b^4 - 2*a^6*b^5 - a^5*b^6 + 4*a^4*b^7 - a^3*b^8 - 2*a^2*b^
9 + a*b^10)*abs(a^4*b^5 - 2*a^2*b^7 + b^9)) - (24*a^11*b^4 - 12*a^10*b^5 - 100*a^9*b^6 + 47*a^8*b^7 + 158*a^7*
b^8 - 68*a^6*b^9 - 111*a^5*b^10 + 42*a^4*b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14 - b^15 - 12*a^6*abs(a^4*b^5
- 2*a^2*b^7 + b^9) + 6*a^5*b*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + 23*a^4*b^2*abs(a^4*b^5 - 2*a^2*b^7 + b^9) - 10*a
^3*b^3*abs(a^4*b^5 - 2*a^2*b^7 + b^9) - 10*a^2*b^4*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + a*b^5*abs(a^4*b^5 - 2*a^2*
b^7 + b^9) - b^6*abs(a^4*b^5 - 2*a^2*b^7 + b^9))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/
2*c)/sqrt((4*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 - sqrt(-16*(a^5*b^4 + a^4*b^5 - 2*a^3*b^6 - 2*a^2*b^7 + a*b^8 + b^9
)*(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9) + 16*(a^5*b^4 - 2*a^3*b^6 + a*b^8)^2))/(a^5*b^4 -
a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9))))/(a^5*b^4*abs(a^4*b^5 - 2*a^2*b^7 + b^9) - 2*a^3*b^6*abs(a^4*
b^5 - 2*a^2*b^7 + b^9) + a*b^8*abs(a^4*b^5 - 2*a^2*b^7 + b^9) - (a^4*b^5 - 2*a^2*b^7 + b^9)^2) + 2*(12*a^7*tan
(1/2*d*x + 1/2*c)^7 - 18*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 17*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 33*a^4*b^3*tan(1/2
*d*x + 1/2*c)^7 - 2*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 13*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 4*a*b^6*tan(1/2*d*x +
1/2*c)^7 + b^7*tan(1/2*d*x + 1/2*c)^7 + 36*a^7*tan(1/2*d*x + 1/2*c)^5 - 18*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 67*
a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 29*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 26*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 5*a^2
*b^5*tan(1/2*d*x + 1/2*c)^5 - 4*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 3*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*a^7*tan(1/2*d
*x + 1/2*c)^3 + 18*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 67*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 29*a^4*b^3*tan(1/2*d*x +
1/2*c)^3 + 26*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 5*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*a*b^6*tan(1/2*d*x + 1/2*c
)^3 + 3*b^7*tan(1/2*d*x + 1/2*c)^3 + 12*a^7*tan(1/2*d*x + 1/2*c) + 18*a^6*b*tan(1/2*d*x + 1/2*c) - 17*a^5*b^2*
tan(1/2*d*x + 1/2*c) - 33*a^4*b^3*tan(1/2*d*x + 1/2*c) - 2*a^3*b^4*tan(1/2*d*x + 1/2*c) + 13*a^2*b^5*tan(1/2*d
*x + 1/2*c) + 4*a*b^6*tan(1/2*d*x + 1/2*c) - b^7*tan(1/2*d*x + 1/2*c))/((a^4*b^4 - 2*a^2*b^6 + b^8)*(a*tan(1/2
*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 + 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)^2))/d

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Mupad [B]
time = 8.67, size = 2500, normalized size = 8.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(a + b*cos(c + d*x))^3,x)

[Out]

(atan(((((8*tan(c/2 + (d*x)/2)*(288*a^14 - 288*a^13*b - 2*a*b^13 + b^14 + 21*a^2*b^12 - 40*a^3*b^11 + 74*a^4*b
^10 - 108*a^5*b^9 + 18*a^6*b^8 + 872*a^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 1538*a^10*b^4 + 1104*a^11*b^3 - 11
04*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) + (((4*(
4*b^21 + 28*a^2*b^19 - 80*a^3*b^18 - 120*a^4*b^17 + 276*a^5*b^16 + 164*a^6*b^15 - 360*a^7*b^14 - 100*a^8*b^13
+ 212*a^9*b^12 + 24*a^10*b^11 - 48*a^11*b^10))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b
^14 - a^6*b^13 - a^7*b^12) - (4*tan(c/2 + (d*x)/2)*(a^2*12i + b^2*1i)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 3
2*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/(b^5*(a*b^14 +
b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(a^2*12i + b^2*1i))/(2*b^5))*
(a^2*12i + b^2*1i)*1i)/(2*b^5) + (((8*tan(c/2 + (d*x)/2)*(288*a^14 - 288*a^13*b - 2*a*b^13 + b^14 + 21*a^2*b^1
2 - 40*a^3*b^11 + 74*a^4*b^10 - 108*a^5*b^9 + 18*a^6*b^8 + 872*a^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 1538*a^1
0*b^4 + 1104*a^11*b^3 - 1104*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a
^6*b^9 - a^7*b^8) - (((4*(4*b^21 + 28*a^2*b^19 - 80*a^3*b^18 - 120*a^4*b^17 + 276*a^5*b^16 + 164*a^6*b^15 - 36
0*a^7*b^14 - 100*a^8*b^13 + 212*a^9*b^12 + 24*a^10*b^11 - 48*a^11*b^10))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b
^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) + (4*tan(c/2 + (d*x)/2)*(a^2*12i + b^2*1i)*(8*a*b^19 - 8*
a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*
a^10*b^10))/(b^5*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(a^
2*12i + b^2*1i))/(2*b^5))*(a^2*12i + b^2*1i)*1i)/(2*b^5))/((8*(1728*a^15 - 864*a^14*b + 20*a^3*b^12 - 20*a^4*b
^11 + 411*a^5*b^10 - 11*a^6*b^9 + 1314*a^7*b^8 + 2326*a^8*b^7 - 7829*a^9*b^6 - 4770*a^10*b^5 + 11700*a^11*b^4
+ 3456*a^12*b^3 - 7344*a^13*b^2))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^1
3 - a^7*b^12) - (((8*tan(c/2 + (d*x)/2)*(288*a^14 - 288*a^13*b - 2*a*b^13 + b^14 + 21*a^2*b^12 - 40*a^3*b^11 +
74*a^4*b^10 - 108*a^5*b^9 + 18*a^6*b^8 + 872*a^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 1538*a^10*b^4 + 1104*a^11
*b^3 - 1104*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)
+ (((4*(4*b^21 + 28*a^2*b^19 - 80*a^3*b^18 - 120*a^4*b^17 + 276*a^5*b^16 + 164*a^6*b^15 - 360*a^7*b^14 - 100*
a^8*b^13 + 212*a^9*b^12 + 24*a^10*b^11 - 48*a^11*b^10))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15
+ 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - (4*tan(c/2 + (d*x)/2)*(a^2*12i + b^2*1i)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3
*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/(b^5*
(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(a^2*12i + b^2*1i))/
(2*b^5))*(a^2*12i + b^2*1i))/(2*b^5) + (((8*tan(c/2 + (d*x)/2)*(288*a^14 - 288*a^13*b - 2*a*b^13 + b^14 + 21*a
^2*b^12 - 40*a^3*b^11 + 74*a^4*b^10 - 108*a^5*b^9 + 18*a^6*b^8 + 872*a^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 15
38*a^10*b^4 + 1104*a^11*b^3 - 1104*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^
10 - a^6*b^9 - a^7*b^8) - (((4*(4*b^21 + 28*a^2*b^19 - 80*a^3*b^18 - 120*a^4*b^17 + 276*a^5*b^16 + 164*a^6*b^1
5 - 360*a^7*b^14 - 100*a^8*b^13 + 212*a^9*b^12 + 24*a^10*b^11 - 48*a^11*b^10))/(a*b^18 + b^19 - 3*a^2*b^17 - 3
*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) + (4*tan(c/2 + (d*x)/2)*(a^2*12i + b^2*1i)*(8*a*b^1
9 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^1
1 - 8*a^10*b^10))/(b^5*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)
))*(a^2*12i + b^2*1i))/(2*b^5))*(a^2*12i + b^2*1i))/(2*b^5)))*(a^2*12i + b^2*1i)*1i)/(b^5*d) - ((tan(c/2 + (d*
x)/2)^3*(18*a^6*b - 4*a*b^6 + 36*a^7 + 3*b^7 + 5*a^2*b^5 + 26*a^3*b^4 - 29*a^4*b^3 - 67*a^5*b^2))/((a + b)^2*(
b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^5*(4*a*b^6 + 18*a^6*b - 36*a^7 + 3*b^7 + 5*a^2*b^5 - 26*a^3*b^
4 - 29*a^4*b^3 + 67*a^5*b^2))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^7*(5*a*b^5 + 6*a^5*b
- 12*a^6 + b^6 - 8*a^2*b^4 - 10*a^3*b^3 + 23*a^4*b^2))/((a*b^4 - b^5)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(5*a*b
^5 + 6*a^5*b + 12*a^6 - b^6 + 8*a^2*b^4 - 10*a^3*b^3 - 23*a^4*b^2))/((a + b)*(b^6 - 2*a*b^5 + a^2*b^4)))/(d*(2
*a*b + tan(c/2 + (d*x)/2)^4*(6*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^2*(4*a*b + 4*a^2) - tan(c/2 + (d*x)/2)^6*(4*a
*b - 4*a^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (a^3*atan(((a^3*((8*tan(c/2 + (d*x)/2)*
(288*a^14 - 288*a^13*b - 2*a*b^13 + b^14 + 21*a^2*b^12 - 40*a^3*b^11 + 74*a^4*b^10 - 108*a^5*b^9 + 18*a^6*b^8
+ 872*a^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 1538*a^10*b^4 + 1104*a^11*b^3 - 1104*a^12*b^2))/(a*b^14 + b^15 -
3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^...

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