3.1.79 \(\int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx\) [79]

Optimal. Leaf size=176 \[ \frac {21 x}{2 a^4}-\frac {576 \sin (c+d x)}{35 a^4 d}+\frac {21 \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {43 \cos (c+d x) \sin (c+d x)}{35 a^4 d (1+\sec (c+d x))^2}-\frac {288 \cos (c+d x) \sin (c+d x)}{35 a^4 d (1+\sec (c+d x))}-\frac {\cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]

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

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3902, 4105, 3872, 2715, 8, 2717} \begin {gather*} -\frac {576 \sin (c+d x)}{35 a^4 d}+\frac {21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac {288 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)}-\frac {43 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}+\frac {21 x}{2 a^4}-\frac {2 \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(a + a*Sec[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]*S
in[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])^2) - (288*Cos[c + d*x]*Sin[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])) -
(Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c
+ d*x])^3)

Rule 8

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

Rule 2715

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] && Integ
erQ[2*n]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.56, size = 289, normalized size = 1.64 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (102900 d x \cos \left (\frac {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 )+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 )\right )}{35840 a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2/(a + a*Sec[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)

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 114, normalized size = 0.65

method	result	size
derivativedivides	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-72 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+168 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(114\)
default	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-72 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+168 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(114\)
risch	\(\frac {21 x}{2 a^{4}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (700 \,{\mathrm e}^{6 i \left (d x +c \right )}+3675 \,{\mathrm e}^{5 i \left (d x +c \right )}+8505 \,{\mathrm e}^{4 i \left (d x +c \right )}+10780 \,{\mathrm e}^{3 i \left (d x +c \right )}+7896 \,{\mathrm e}^{2 i \left (d x +c \right )}+3157 \,{\mathrm e}^{i \left (d x +c \right )}+551\right )}{35 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(170\)
norman	\(\frac {\frac {21 x}{2 a}-\frac {167 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {281 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}-\frac {217 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {167 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}-\frac {53 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a d}+\frac {21 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {21 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{3}}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2/(a+a*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/8/d/a^4*(1/7*tan(1/2*d*x+1/2*c)^7-9/5*tan(1/2*d*x+1/2*c)^5+13*tan(1/2*d*x+1/2*c)^3-111*tan(1/2*d*x+1/2*c)+16
*(-9/2*tan(1/2*d*x+1/2*c)^3-7/2*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+168*arctan(tan(1/2*d*x+1/2*c)))

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 204, normalized size = 1.16 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [A]
time = 8.73, size = 171, normalized size = 0.97 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2/(a+a*sec(d*x+c))**4,x)

[Out]

Integral(cos(c + d*x)**2/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x)/a*
*4

________________________________________________________________________________________

Giac [A]
time = 0.48, size = 128, normalized size = 0.73 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Mupad [B]
time = 0.81, size = 159, normalized size = 0.90 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^2/(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)

________________________________________________________________________________________