3.1.88 \(\int \frac {\cos (c+d x)}{(a+a \sec (c+d x))^5} \, dx\) [88]

Optimal. Leaf size=159 \[ -\frac {5 x}{a^5}+\frac {496 \sin (c+d x)}{63 a^5 d}-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )} \]

[Out]

-5*x/a^5+496/63*sin(d*x+c)/a^5/d-1/9*sin(d*x+c)/d/(a+a*sec(d*x+c))^5-5/21*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^4-29
/63*sin(d*x+c)/a^2/d/(a+a*sec(d*x+c))^3-67/63*sin(d*x+c)/a^3/d/(a+a*sec(d*x+c))^2-5*sin(d*x+c)/d/(a^5+a^5*sec(
d*x+c))

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Rubi [A]
time = 0.27, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3902, 4105, 3872, 2717, 8} \begin {gather*} \frac {496 \sin (c+d x)}{63 a^5 d}-\frac {5 \sin (c+d x)}{d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {5 x}{a^5}-\frac {67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac {29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac {5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac {\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*x)/a^5 + (496*Sin[c + d*x])/(63*a^5*d) - Sin[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sin[c + d*x])/(21*
a*d*(a + a*Sec[c + d*x])^4) - (29*Sin[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sin[c + d*x])/(63*a^3*
d*(a + a*Sec[c + d*x])^2) - (5*Sin[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3902

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-Cot[
e + f*x])*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*(2*m + 1))), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*C
sc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b,
d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])

Rule 4105

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

Rubi steps

\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {\int \frac {\cos (c+d x) (-10 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {\int \frac {\cos (c+d x) \left (-85 a^2+60 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (-570 a^3+435 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (-2715 a^4+2010 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {\int \cos (c+d x) \left (-7440 a^5+4725 a^5 \sec (c+d x)\right ) \, dx}{945 a^{10}}\\ &=-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {5 \int 1 \, dx}{a^5}+\frac {496 \int \cos (c+d x) \, dx}{63 a^5}\\ &=-\frac {5 x}{a^5}+\frac {496 \sin (c+d x)}{63 a^5 d}-\frac {\sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {5 \sin (c+d x)}{21 a d (a+a \sec (c+d x))^4}-\frac {29 \sin (c+d x)}{63 a^2 d (a+a \sec (c+d x))^3}-\frac {67 \sin (c+d x)}{63 a^3 d (a+a \sec (c+d x))^2}-\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(319\) vs. \(2(159)=318\).
time = 0.68, size = 319, normalized size = 2.01 \begin {gather*} -\frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (79380 d x \cos \left (\frac {d x}{2}\right )+79380 d x \cos \left (c+\frac {d x}{2}\right )+52920 d x \cos \left (c+\frac {3 d x}{2}\right )+52920 d x \cos \left (2 c+\frac {3 d x}{2}\right )+22680 d x \cos \left (2 c+\frac {5 d x}{2}\right )+22680 d x \cos \left (3 c+\frac {5 d x}{2}\right )+5670 d x \cos \left (3 c+\frac {7 d x}{2}\right )+5670 d x \cos \left (4 c+\frac {7 d x}{2}\right )+630 d x \cos \left (4 c+\frac {9 d x}{2}\right )+630 d x \cos \left (5 c+\frac {9 d x}{2}\right )-175014 \sin \left (\frac {d x}{2}\right )+143010 \sin \left (c+\frac {d x}{2}\right )-138726 \sin \left (c+\frac {3 d x}{2}\right )+73290 \sin \left (2 c+\frac {3 d x}{2}\right )-70389 \sin \left (2 c+\frac {5 d x}{2}\right )+20475 \sin \left (3 c+\frac {5 d x}{2}\right )-21141 \sin \left (3 c+\frac {7 d x}{2}\right )+1575 \sin \left (4 c+\frac {7 d x}{2}\right )-3091 \sin \left (4 c+\frac {9 d x}{2}\right )-567 \sin \left (5 c+\frac {9 d x}{2}\right )-63 \sin \left (5 c+\frac {11 d x}{2}\right )-63 \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{64512 a^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/64512*(Sec[c/2]*Sec[(c + d*x)/2]^9*(79380*d*x*Cos[(d*x)/2] + 79380*d*x*Cos[c + (d*x)/2] + 52920*d*x*Cos[c +
(3*d*x)/2] + 52920*d*x*Cos[2*c + (3*d*x)/2] + 22680*d*x*Cos[2*c + (5*d*x)/2] + 22680*d*x*Cos[3*c + (5*d*x)/2]
+ 5670*d*x*Cos[3*c + (7*d*x)/2] + 5670*d*x*Cos[4*c + (7*d*x)/2] + 630*d*x*Cos[4*c + (9*d*x)/2] + 630*d*x*Cos[
5*c + (9*d*x)/2] - 175014*Sin[(d*x)/2] + 143010*Sin[c + (d*x)/2] - 138726*Sin[c + (3*d*x)/2] + 73290*Sin[2*c +
(3*d*x)/2] - 70389*Sin[2*c + (5*d*x)/2] + 20475*Sin[3*c + (5*d*x)/2] - 21141*Sin[3*c + (7*d*x)/2] + 1575*Sin[
4*c + (7*d*x)/2] - 3091*Sin[4*c + (9*d*x)/2] - 567*Sin[5*c + (9*d*x)/2] - 63*Sin[5*c + (11*d*x)/2] - 63*Sin[6*
c + (11*d*x)/2]))/(a^5*d)

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Maple [A]
time = 0.10, size = 111, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-160 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\)	\(111\)
default	\(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-160 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\)	\(111\)
norman	\(\frac {-\frac {5 x}{a}+\frac {161 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {105 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {17 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {65 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1008 a d}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 a d}-\frac {5 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}\)	\(156\)
risch	\(-\frac {5 x}{a^{5}}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{5} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{5} d}+\frac {2 i \left (945 \,{\mathrm e}^{8 i \left (d x +c \right )}+6300 \,{\mathrm e}^{7 i \left (d x +c \right )}+19740 \,{\mathrm e}^{6 i \left (d x +c \right )}+36414 \,{\mathrm e}^{5 i \left (d x +c \right )}+43092 \,{\mathrm e}^{4 i \left (d x +c \right )}+33264 \,{\mathrm e}^{3 i \left (d x +c \right )}+16416 \,{\mathrm e}^{2 i \left (d x +c \right )}+4734 \,{\mathrm e}^{i \left (d x +c \right )}+631\right )}{63 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/d/a^5*(1/9*tan(1/2*d*x+1/2*c)^9-8/7*tan(1/2*d*x+1/2*c)^7+6*tan(1/2*d*x+1/2*c)^5-24*tan(1/2*d*x+1/2*c)^3+1
29*tan(1/2*d*x+1/2*c)+32*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-160*arctan(tan(1/2*d*x+1/2*c)))

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Maxima [A]
time = 0.52, size = 178, normalized size = 1.12 \begin {gather*} \frac {\frac {2016 \, \sin \left (d x + c\right )}{{\left (a^{5} + \frac {a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{1008 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1008*(2016*sin(d*x + c)/((a^5 + a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (8127*sin(d*x
+ c)/(cos(d*x + c) + 1) - 1512*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5
- 72*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 7*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 10080*arctan(sin(d*x +
c)/(cos(d*x + c) + 1))/a^5)/d

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Fricas [A]
time = 2.54, size = 198, normalized size = 1.25 \begin {gather*} -\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (63 \, \cos \left (d x + c\right )^{5} + 946 \, \cos \left (d x + c\right )^{4} + 2840 \, \cos \left (d x + c\right )^{3} + 3633 \, \cos \left (d x + c\right )^{2} + 2165 \, \cos \left (d x + c\right ) + 496\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/63*(315*d*x*cos(d*x + c)^5 + 1575*d*x*cos(d*x + c)^4 + 3150*d*x*cos(d*x + c)^3 + 3150*d*x*cos(d*x + c)^2 +
1575*d*x*cos(d*x + c) + 315*d*x - (63*cos(d*x + c)^5 + 946*cos(d*x + c)^4 + 2840*cos(d*x + c)^3 + 3633*cos(d*x
+ c)^2 + 2165*cos(d*x + c) + 496)*sin(d*x + c))/(a^5*d*cos(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos
(d*x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cos {\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(c + d*x)/(sec(c + d*x)**5 + 5*sec(c + d*x)**4 + 10*sec(c + d*x)**3 + 10*sec(c + d*x)**2 + 5*sec(c
+ d*x) + 1), x)/a**5

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Giac [A]
time = 0.48, size = 129, normalized size = 0.81 \begin {gather*} -\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{5}} - \frac {7 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 72 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1512 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{1008 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1008*(5040*(d*x + c)/a^5 - 2016*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^5) - (7*a^40*tan(1/2*d
*x + 1/2*c)^9 - 72*a^40*tan(1/2*d*x + 1/2*c)^7 + 378*a^40*tan(1/2*d*x + 1/2*c)^5 - 1512*a^40*tan(1/2*d*x + 1/2
*c)^3 + 8127*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d

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Mupad [B]
time = 0.83, size = 159, normalized size = 1.00 \begin {gather*} \frac {7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-100\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+636\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+10096\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2016\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{1008\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*sin(c/2 + (d*x)/2) - 100*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 636*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)
/2) - 2512*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) + 10096*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 2016*cos(
c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 5040*cos(c/2 + (d*x)/2)^9*(c + d*x))/(1008*a^5*d*cos(c/2 + (d*x)/2)^9)

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