3.89 \(\int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx\)

Optimal. Leaf size=215 \[ -\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}-\frac {3832 \sin (c+d x) \cos (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac {31 x}{2 a^5}-\frac {577 \sin (c+d x) \cos (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac {28 \sin (c+d x) \cos (c+d x)}{45 a^2 d (a \sec (c+d x)+a)^3}-\frac {17 \sin (c+d x) \cos (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]

[Out]

31/2*x/a^5-7664/315*sin(d*x+c)/a^5/d+31/2*cos(d*x+c)*sin(d*x+c)/a^5/d-1/9*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x
+c))^5-17/63*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^4-28/45*cos(d*x+c)*sin(d*x+c)/a^2/d/(a+a*sec(d*x+c))^3
-577/315*cos(d*x+c)*sin(d*x+c)/a^3/d/(a+a*sec(d*x+c))^2-3832/315*cos(d*x+c)*sin(d*x+c)/d/(a^5+a^5*sec(d*x+c))

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Rubi [A]  time = 0.51, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3817, 4020, 3787, 2635, 8, 2637} \[ -\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}-\frac {3832 \sin (c+d x) \cos (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {577 \sin (c+d x) \cos (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac {28 \sin (c+d x) \cos (c+d x)}{45 a^2 d (a \sec (c+d x)+a)^3}+\frac {31 x}{2 a^5}-\frac {17 \sin (c+d x) \cos (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^5,x]

[Out]

(31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]*Si
n[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (17*Cos[c + d*x]*Sin[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (2
8*Cos[c + d*x]*Sin[c + d*x])/(45*a^2*d*(a + a*Sec[c + d*x])^3) - (577*Cos[c + d*x]*Sin[c + d*x])/(315*a^3*d*(a
+ a*Sec[c + d*x])^2) - (3832*Cos[c + d*x]*Sin[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))

Rule 8

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2637

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

Rule 3787

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 3817

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*Csc
[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 4020

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 ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {\int \frac {\cos ^2(c+d x) (-11 a+6 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {\int \frac {\cos ^2(c+d x) \left (-111 a^2+85 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (-947 a^3+784 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-6303 a^4+5193 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {\int \cos ^2(c+d x) \left (-29295 a^5+22992 a^5 \sec (c+d x)\right ) \, dx}{945 a^{10}}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {7664 \int \cos (c+d x) \, dx}{315 a^5}+\frac {31 \int \cos ^2(c+d x) \, dx}{a^5}\\ &=-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}+\frac {31 \int 1 \, dx}{2 a^5}\\ &=\frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 345, normalized size = 1.60 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (7194600 \sin \left (c+\frac {d x}{2}\right )-7472241 \sin \left (c+\frac {3 d x}{2}\right )+3432975 \sin \left (2 c+\frac {3 d x}{2}\right )-3871989 \sin \left (2 c+\frac {5 d x}{2}\right )+801675 \sin \left (3 c+\frac {5 d x}{2}\right )-1186056 \sin \left (3 c+\frac {7 d x}{2}\right )-17640 \sin \left (4 c+\frac {7 d x}{2}\right )-175184 \sin \left (4 c+\frac {9 d x}{2}\right )-45360 \sin \left (5 c+\frac {9 d x}{2}\right )-3465 \sin \left (5 c+\frac {11 d x}{2}\right )-3465 \sin \left (6 c+\frac {11 d x}{2}\right )+315 \sin \left (6 c+\frac {13 d x}{2}\right )+315 \sin \left (7 c+\frac {13 d x}{2}\right )+4921560 d x \cos \left (c+\frac {d x}{2}\right )+3281040 d x \cos \left (c+\frac {3 d x}{2}\right )+3281040 d x \cos \left (2 c+\frac {3 d x}{2}\right )+1406160 d x \cos \left (2 c+\frac {5 d x}{2}\right )+1406160 d x \cos \left (3 c+\frac {5 d x}{2}\right )+351540 d x \cos \left (3 c+\frac {7 d x}{2}\right )+351540 d x \cos \left (4 c+\frac {7 d x}{2}\right )+39060 d x \cos \left (4 c+\frac {9 d x}{2}\right )+39060 d x \cos \left (5 c+\frac {9 d x}{2}\right )-9163224 \sin \left (\frac {d x}{2}\right )+4921560 d x \cos \left (\frac {d x}{2}\right )\right )}{1290240 a^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^5,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^9*(4921560*d*x*Cos[(d*x)/2] + 4921560*d*x*Cos[c + (d*x)/2] + 3281040*d*x*Cos[c + (3
*d*x)/2] + 3281040*d*x*Cos[2*c + (3*d*x)/2] + 1406160*d*x*Cos[2*c + (5*d*x)/2] + 1406160*d*x*Cos[3*c + (5*d*x)
/2] + 351540*d*x*Cos[3*c + (7*d*x)/2] + 351540*d*x*Cos[4*c + (7*d*x)/2] + 39060*d*x*Cos[4*c + (9*d*x)/2] + 390
60*d*x*Cos[5*c + (9*d*x)/2] - 9163224*Sin[(d*x)/2] + 7194600*Sin[c + (d*x)/2] - 7472241*Sin[c + (3*d*x)/2] + 3
432975*Sin[2*c + (3*d*x)/2] - 3871989*Sin[2*c + (5*d*x)/2] + 801675*Sin[3*c + (5*d*x)/2] - 1186056*Sin[3*c + (
7*d*x)/2] - 17640*Sin[4*c + (7*d*x)/2] - 175184*Sin[4*c + (9*d*x)/2] - 45360*Sin[5*c + (9*d*x)/2] - 3465*Sin[5
*c + (11*d*x)/2] - 3465*Sin[6*c + (11*d*x)/2] + 315*Sin[6*c + (13*d*x)/2] + 315*Sin[7*c + (13*d*x)/2]))/(12902
40*a^5*d)

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fricas [A]  time = 3.03, size = 207, normalized size = 0.96 \[ \frac {9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x + {\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+a*sec(d*x+c))^5,x, algorithm="fricas")

[Out]

1/630*(9765*d*x*cos(d*x + c)^5 + 48825*d*x*cos(d*x + c)^4 + 97650*d*x*cos(d*x + c)^3 + 97650*d*x*cos(d*x + c)^
2 + 48825*d*x*cos(d*x + c) + 9765*d*x + (315*cos(d*x + c)^6 - 1575*cos(d*x + c)^5 - 28828*cos(d*x + c)^4 - 874
40*cos(d*x + c)^3 - 112119*cos(d*x + c)^2 - 66875*cos(d*x + c) - 15328)*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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giac [A]  time = 0.86, size = 145, normalized size = 0.67 \[ \frac {\frac {78120 \, {\left (d x + c\right )}}{a^{5}} - \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+a*sec(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(78120*(d*x + c)/a^5 - 5040*(11*tan(1/2*d*x + 1/2*c)^3 + 9*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)
^2 + 1)^2*a^5) - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1
/2*c)^5 - 15750*a^40*tan(1/2*d*x + 1/2*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d

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maple [A]  time = 0.66, size = 179, normalized size = 0.83 \[ -\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d \,a^{5}}+\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{5}}-\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d \,a^{5}}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{5}}-\frac {351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {31 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2/(a+a*sec(d*x+c))^5,x)

[Out]

-1/144/d/a^5*tan(1/2*d*x+1/2*c)^9+5/56/d/a^5*tan(1/2*d*x+1/2*c)^7-3/5/d/a^5*tan(1/2*d*x+1/2*c)^5+25/8/d/a^5*ta
n(1/2*d*x+1/2*c)^3-351/16/d/a^5*tan(1/2*d*x+1/2*c)-11/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3-9/
d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)+31/d/a^5*arctan(tan(1/2*d*x+1/2*c))

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maxima [A]  time = 1.13, size = 224, normalized size = 1.04 \[ -\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {156240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+a*sec(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 11*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^5 + 2*a^5*sin(d*
x + c)^2/(cos(d*x + c) + 1)^2 + a^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c)
+ 1) - 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 450*sin(d*x + c)
^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 156240*arctan(sin(d*x + c)/(cos(d*x +
c) + 1))/a^5)/d

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mupad [B]  time = 0.95, size = 181, normalized size = 0.84 \[ -\frac {35\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-590\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+4584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-23288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+129824\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+55440\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^2/(a + a/cos(c + d*x))^5,x)

[Out]

-(35*sin(c/2 + (d*x)/2) - 590*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 4584*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d
*x)/2) - 23288*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) + 129824*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 5544
0*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 10080*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2) - 78120*cos(c/2 +
(d*x)/2)^9*(c + d*x))/(5040*a^5*d*cos(c/2 + (d*x)/2)^9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2/(a+a*sec(d*x+c))**5,x)

[Out]

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

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