3.72 \(\int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=184 \[ -\frac {576 \sin (c+d x)}{35 a^4 d}-\frac {43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac {288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac {21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac {21 x}{2 a^4}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]

[Out]

21/2*x/a^4-576/35*sin(d*x+c)/a^4/d+21/2*cos(d*x+c)*sin(d*x+c)/a^4/d-43/35*cos(d*x+c)^3*sin(d*x+c)/a^4/d/(1+cos
(d*x+c))^2-288/35*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^
4-2/5*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3

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Rubi [A]  time = 0.38, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2765, 2977, 2734} \[ -\frac {576 \sin (c+d x)}{35 a^4 d}-\frac {43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac {288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac {21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac {21 x}{2 a^4}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4,x]

[Out]

(21*x)/(2*a^4) - (576*Sin[c + d*x])/(35*a^4*d) + (21*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - (43*Cos[c + d*x]^3
*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2) - (288*Cos[c + d*x]^2*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x]
)) - (Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a +
a*Cos[c + d*x])^3)

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2765

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

Rule 2977

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

Rubi steps

\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos ^4(c+d x) (5 a-9 a \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (56 a^2-73 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (387 a^3-477 a^3 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {288 \cos ^2(c+d x) \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {\int \cos (c+d x) \left (1728 a^4-2205 a^4 \cos (c+d x)\right ) \, dx}{105 a^8}\\ &=\frac {21 x}{2 a^4}-\frac {576 \sin (c+d x)}{35 a^4 d}+\frac {21 \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac {\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {288 \cos ^2(c+d x) \sin (c+d x)}{35 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 289, normalized size = 1.57 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (128730 \sin \left (c+\frac {d x}{2}\right )-140826 \sin \left (c+\frac {3 d x}{2}\right )+44310 \sin \left (2 c+\frac {3 d x}{2}\right )-60487 \sin \left (2 c+\frac {5 d x}{2}\right )+1225 \sin \left (3 c+\frac {5 d x}{2}\right )-12001 \sin \left (3 c+\frac {7 d x}{2}\right )-3185 \sin \left (4 c+\frac {7 d x}{2}\right )-315 \sin \left (4 c+\frac {9 d x}{2}\right )-315 \sin \left (5 c+\frac {9 d x}{2}\right )+35 \sin \left (5 c+\frac {11 d x}{2}\right )+35 \sin \left (6 c+\frac {11 d x}{2}\right )+102900 d x \cos \left (c+\frac {d x}{2}\right )+61740 d x \cos \left (c+\frac {3 d x}{2}\right )+61740 d x \cos \left (2 c+\frac {3 d x}{2}\right )+20580 d x \cos \left (2 c+\frac {5 d x}{2}\right )+20580 d x \cos \left (3 c+\frac {5 d x}{2}\right )+2940 d x \cos \left (3 c+\frac {7 d x}{2}\right )+2940 d x \cos \left (4 c+\frac {7 d x}{2}\right )-179830 \sin \left (\frac {d x}{2}\right )+102900 d x \cos \left (\frac {d x}{2}\right )\right )}{35840 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^7*(102900*d*x*Cos[(d*x)/2] + 102900*d*x*Cos[c + (d*x)/2] + 61740*d*x*Cos[c + (3*d*x
)/2] + 61740*d*x*Cos[2*c + (3*d*x)/2] + 20580*d*x*Cos[2*c + (5*d*x)/2] + 20580*d*x*Cos[3*c + (5*d*x)/2] + 2940
*d*x*Cos[3*c + (7*d*x)/2] + 2940*d*x*Cos[4*c + (7*d*x)/2] - 179830*Sin[(d*x)/2] + 128730*Sin[c + (d*x)/2] - 14
0826*Sin[c + (3*d*x)/2] + 44310*Sin[2*c + (3*d*x)/2] - 60487*Sin[2*c + (5*d*x)/2] + 1225*Sin[3*c + (5*d*x)/2]
- 12001*Sin[3*c + (7*d*x)/2] - 3185*Sin[4*c + (7*d*x)/2] - 315*Sin[4*c + (9*d*x)/2] - 315*Sin[5*c + (9*d*x)/2]
+ 35*Sin[5*c + (11*d*x)/2] + 35*Sin[6*c + (11*d*x)/2]))/(35840*a^4*d)

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fricas [A]  time = 1.73, size = 171, normalized size = 0.93 \[ \frac {735 \, d x \cos \left (d x + c\right )^{4} + 2940 \, d x \cos \left (d x + c\right )^{3} + 4410 \, d x \cos \left (d x + c\right )^{2} + 2940 \, d x \cos \left (d x + c\right ) + 735 \, d x + {\left (35 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 2012 \, \cos \left (d x + c\right )^{3} - 4548 \, \cos \left (d x + c\right )^{2} - 3873 \, \cos \left (d x + c\right ) - 1152\right )} \sin \left (d x + c\right )}{70 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/70*(735*d*x*cos(d*x + c)^4 + 2940*d*x*cos(d*x + c)^3 + 4410*d*x*cos(d*x + c)^2 + 2940*d*x*cos(d*x + c) + 735
*d*x + (35*cos(d*x + c)^5 - 140*cos(d*x + c)^4 - 2012*cos(d*x + c)^3 - 4548*cos(d*x + c)^2 - 3873*cos(d*x + c)
- 1152)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d
*x + c) + a^4*d)

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giac [A]  time = 0.60, size = 128, normalized size = 0.70 \[ \frac {\frac {2940 \, {\left (d x + c\right )}}{a^{4}} - \frac {280 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {5 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 455 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3885 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{280 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/280*(2940*(d*x + c)/a^4 - 280*(9*tan(1/2*d*x + 1/2*c)^3 + 7*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 +
1)^2*a^4) + (5*a^24*tan(1/2*d*x + 1/2*c)^7 - 63*a^24*tan(1/2*d*x + 1/2*c)^5 + 455*a^24*tan(1/2*d*x + 1/2*c)^3
- 3885*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d

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maple [A]  time = 0.06, size = 160, normalized size = 0.87 \[ \frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 d \,a^{4}}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {9 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {21 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x)

[Out]

1/56/d/a^4*tan(1/2*d*x+1/2*c)^7-9/40/d/a^4*tan(1/2*d*x+1/2*c)^5+13/8/d/a^4*tan(1/2*d*x+1/2*c)^3-111/8/d/a^4*ta
n(1/2*d*x+1/2*c)-9/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3-7/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*ta
n(1/2*d*x+1/2*c)+21/d/a^4*arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 1.12, size = 204, normalized size = 1.11 \[ -\frac {\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{280 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/280*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^4 + 2*a^4*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 + a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1)
- 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7/(cos(d*x
+ c) + 1)^7)/a^4 - 5880*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4)/d

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mupad [B]  time = 0.52, size = 159, normalized size = 0.86 \[ \frac {5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+596\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-4408\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2940\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )}{280\,a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6/(a + a*cos(c + d*x))^4,x)

[Out]

(5*sin(c/2 + (d*x)/2) - 78*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 596*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/
2) - 4408*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) - 2520*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 560*cos(c/2
+ (d*x)/2)^10*sin(c/2 + (d*x)/2) + 2940*cos(c/2 + (d*x)/2)^7*(c + d*x))/(280*a^4*d*cos(c/2 + (d*x)/2)^7)

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sympy [A]  time = 29.67, size = 530, normalized size = 2.88 \[ \begin {cases} \frac {2940 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {5880 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {2940 d x}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {5 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {53 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {334 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {3038 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {9835 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {5845 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{6}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6/(a+a*cos(d*x+c))**4,x)

[Out]

Piecewise((2940*d*x*tan(c/2 + d*x/2)**4/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280
*a**4*d) + 5880*d*x*tan(c/2 + d*x/2)**2/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280
*a**4*d) + 2940*d*x/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) + 5*tan(c/2
+ d*x/2)**11/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 53*tan(c/2 + d*
x/2)**9/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) + 334*tan(c/2 + d*x/2)*
*7/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 3038*tan(c/2 + d*x/2)**5/(
280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 9835*tan(c/2 + d*x/2)**3/(280*
a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 5845*tan(c/2 + d*x/2)/(280*a**4*d*
tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d), Ne(d, 0)), (x*cos(c)**6/(a*cos(c) + a)**4,
True))

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