\(\int \frac {\cos ^2(c+d x) (A+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [153]

Optimal result

Integrand size = 33, antiderivative size = 215 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(21 A+2 C) x}{2 a^4}-\frac {32 (54 A+5 C) \sin (c+d x)}{105 a^4 d}+\frac {(21 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {16 (54 A+5 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]

[Out]

1/2*(21*A+2*C)*x/a^4-32/105*(54*A+5*C)*sin(d*x+c)/a^4/d+1/2*(21*A+2*C)*cos(d*x+c)*sin(d*x+c)/a^4/d-1/105*(129*
A+10*C)*cos(d*x+c)*sin(d*x+c)/a^4/d/(1+sec(d*x+c))^2-16/105*(54*A+5*C)*cos(d*x+c)*sin(d*x+c)/a^4/d/(1+sec(d*x+
c))-1/7*(A+C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-2/5*A*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^3

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4170, 4105, 3872, 2715, 8, 2717} \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {32 (54 A+5 C) \sin (c+d x)}{105 a^4 d}+\frac {(21 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac {16 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(129 A+10 C) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {x (21 A+2 C)}{2 a^4}-\frac {(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac {2 A \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]

[In]

Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]

[Out]

((21*A + 2*C)*x)/(2*a^4) - (32*(54*A + 5*C)*Sin[c + d*x])/(105*a^4*d) + ((21*A + 2*C)*Cos[c + d*x]*Sin[c + d*x
])/(2*a^4*d) - ((129*A + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (16*(54*A + 5*C)*
Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Se
c[c + d*x])^4) - (2*A*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 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]

Rule 4170

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \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) (-a (9 A+2 C)+a (5 A-2 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \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 (-a^2 (73 A+10 C)+56 a^2 A \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \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 (-a^3 (477 A+50 C)+3 a^3 (129 A+10 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (54 A+5 C) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {\int \cos ^2(c+d x) \left (-105 a^4 (21 A+2 C)+32 a^4 (54 A+5 C) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (54 A+5 C) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(21 A+2 C) \int \cos ^2(c+d x) \, dx}{a^4}-\frac {(32 (54 A+5 C)) \int \cos (c+d x) \, dx}{105 a^4} \\ & = -\frac {32 (54 A+5 C) \sin (c+d x)}{105 a^4 d}+\frac {(21 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (54 A+5 C) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(21 A+2 C) \int 1 \, dx}{2 a^4} \\ & = \frac {(21 A+2 C) x}{2 a^4}-\frac {32 (54 A+5 C) \sin (c+d x)}{105 a^4 d}+\frac {(21 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 A \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {16 (54 A+5 C) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(505\) vs. \(2(215)=430\).

Time = 8.70 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.35 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (14700 (21 A+2 C) d x \cos \left (\frac {d x}{2}\right )+14700 (21 A+2 C) d x \cos \left (c+\frac {d x}{2}\right )+185220 A d x \cos \left (c+\frac {3 d x}{2}\right )+17640 C d x \cos \left (c+\frac {3 d x}{2}\right )+185220 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+17640 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+61740 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+5880 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+5880 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+8820 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+840 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+840 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-539490 A \sin \left (\frac {d x}{2}\right )-79520 C \sin \left (\frac {d x}{2}\right )+386190 A \sin \left (c+\frac {d x}{2}\right )+66080 C \sin \left (c+\frac {d x}{2}\right )-422478 A \sin \left (c+\frac {3 d x}{2}\right )-57120 C \sin \left (c+\frac {3 d x}{2}\right )+132930 A \sin \left (2 c+\frac {3 d x}{2}\right )+30240 C \sin \left (2 c+\frac {3 d x}{2}\right )-181461 A \sin \left (2 c+\frac {5 d x}{2}\right )-22400 C \sin \left (2 c+\frac {5 d x}{2}\right )+3675 A \sin \left (3 c+\frac {5 d x}{2}\right )+6720 C \sin \left (3 c+\frac {5 d x}{2}\right )-36003 A \sin \left (3 c+\frac {7 d x}{2}\right )-4160 C \sin \left (3 c+\frac {7 d x}{2}\right )-9555 A \sin \left (4 c+\frac {7 d x}{2}\right )-945 A \sin \left (4 c+\frac {9 d x}{2}\right )-945 A \sin \left (5 c+\frac {9 d x}{2}\right )+105 A \sin \left (5 c+\frac {11 d x}{2}\right )+105 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{107520 a^4 d} \]

[In]

Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^7*(14700*(21*A + 2*C)*d*x*Cos[(d*x)/2] + 14700*(21*A + 2*C)*d*x*Cos[c + (d*x)/2] +
185220*A*d*x*Cos[c + (3*d*x)/2] + 17640*C*d*x*Cos[c + (3*d*x)/2] + 185220*A*d*x*Cos[2*c + (3*d*x)/2] + 17640*C
*d*x*Cos[2*c + (3*d*x)/2] + 61740*A*d*x*Cos[2*c + (5*d*x)/2] + 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 61740*A*d*x*C
os[3*c + (5*d*x)/2] + 5880*C*d*x*Cos[3*c + (5*d*x)/2] + 8820*A*d*x*Cos[3*c + (7*d*x)/2] + 840*C*d*x*Cos[3*c +
(7*d*x)/2] + 8820*A*d*x*Cos[4*c + (7*d*x)/2] + 840*C*d*x*Cos[4*c + (7*d*x)/2] - 539490*A*Sin[(d*x)/2] - 79520*
C*Sin[(d*x)/2] + 386190*A*Sin[c + (d*x)/2] + 66080*C*Sin[c + (d*x)/2] - 422478*A*Sin[c + (3*d*x)/2] - 57120*C*
Sin[c + (3*d*x)/2] + 132930*A*Sin[2*c + (3*d*x)/2] + 30240*C*Sin[2*c + (3*d*x)/2] - 181461*A*Sin[2*c + (5*d*x)
/2] - 22400*C*Sin[2*c + (5*d*x)/2] + 3675*A*Sin[3*c + (5*d*x)/2] + 6720*C*Sin[3*c + (5*d*x)/2] - 36003*A*Sin[3
*c + (7*d*x)/2] - 4160*C*Sin[3*c + (7*d*x)/2] - 9555*A*Sin[4*c + (7*d*x)/2] - 945*A*Sin[4*c + (9*d*x)/2] - 945
*A*Sin[5*c + (9*d*x)/2] + 105*A*Sin[5*c + (11*d*x)/2] + 105*A*Sin[6*c + (11*d*x)/2]))/(107520*a^4*d)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55

method	result	size
parallelrisch	\(\frac {-840 \left (\left (\frac {4688 A}{35}+\frac {248 C}{21}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {7873 A}{280}+\frac {52 C}{21}\right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )-\frac {A \cos \left (5 d x +5 c \right )}{8}+\left (\frac {42881 A}{140}+\frac {584 C}{21}\right ) \cos \left (d x +c \right )+\frac {6957 A}{35}+\frac {376 C}{21}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+282240 x \left (A +\frac {2 C}{21}\right ) d}{26880 a^{4} d}\)	\(119\)
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C +13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(181\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C +13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(181\)
risch	\(\frac {21 A x}{2 a^{4}}+\frac {x C}{a^{4}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {2 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (2100 A \,{\mathrm e}^{6 i \left (d x +c \right )}+420 C \,{\mathrm e}^{6 i \left (d x +c \right )}+11025 A \,{\mathrm e}^{5 i \left (d x +c \right )}+1890 C \,{\mathrm e}^{5 i \left (d x +c \right )}+25515 A \,{\mathrm e}^{4 i \left (d x +c \right )}+4130 C \,{\mathrm e}^{4 i \left (d x +c \right )}+32340 A \,{\mathrm e}^{3 i \left (d x +c \right )}+4970 C \,{\mathrm e}^{3 i \left (d x +c \right )}+23688 A \,{\mathrm e}^{2 i \left (d x +c \right )}+3570 C \,{\mathrm e}^{2 i \left (d x +c \right )}+9471 A \,{\mathrm e}^{i \left (d x +c \right )}+1400 C \,{\mathrm e}^{i \left (d x +c \right )}+1653 A +260 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(264\)
norman	\(\frac {-\frac {\left (21 A +2 C \right ) x}{2 a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{56 a d}-\frac {\left (21 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}+\frac {\left (21 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}+\frac {\left (21 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}-\frac {\left (29 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{140 a d}+\frac {\left (167 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (171 A +17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}+\frac {\left (1161 A +265 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{840 a d}-\frac {\left (2529 A +275 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{210 a d}-\frac {\left (2913 A +265 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\)	\(297\)

[In]

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

[Out]

1/26880*(-840*((4688/35*A+248/21*C)*cos(2*d*x+2*c)+(7873/280*A+52/21*C)*cos(3*d*x+3*c)+A*cos(4*d*x+4*c)-1/8*A*
cos(5*d*x+5*c)+(42881/140*A+584/21*C)*cos(d*x+c)+6957/35*A+376/21*C)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6+2
82240*x*(A+2/21*C)*d)/a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (21 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (21 \, A + 2 \, C\right )} d x + {\left (105 \, A \cos \left (d x + c\right )^{5} - 420 \, A \cos \left (d x + c\right )^{4} - 4 \, {\left (1509 \, A + 130 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (3411 \, A + 310 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (11619 \, A + 1070 \, C\right )} \cos \left (d x + c\right ) - 3456 \, A - 320 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

integrate(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

1/210*(105*(21*A + 2*C)*d*x*cos(d*x + c)^4 + 420*(21*A + 2*C)*d*x*cos(d*x + c)^3 + 630*(21*A + 2*C)*d*x*cos(d*
x + c)^2 + 420*(21*A + 2*C)*d*x*cos(d*x + c) + 105*(21*A + 2*C)*d*x + (105*A*cos(d*x + c)^5 - 420*A*cos(d*x +
c)^4 - 4*(1509*A + 130*C)*cos(d*x + c)^3 - 4*(3411*A + 310*C)*cos(d*x + c)^2 - (11619*A + 1070*C)*cos(d*x + c)
- 3456*A - 320*C)*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]

\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]

[In]

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

[Out]

(Integral(A*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)
+ Integral(C*cos(c + d*x)**2*sec(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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + 5 \, C {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/840*(3*A*(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/(co
s(d*x + c) + 1)^7)/a^4 - 5880*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) + 5*C*((315*sin(d*x + c)/(cos(d*x +
c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/
(cos(d*x + c) + 1)^7)/a^4 - 336*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (d x + c\right )} {\left (21 \, A + 2 \, C\right )}}{a^{4}} - \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/840*(420*(d*x + c)*(21*A + 2*C)/a^4 - 840*(9*A*tan(1/2*d*x + 1/2*c)^3 + 7*A*tan(1/2*d*x + 1/2*c))/((tan(1/2*
d*x + 1/2*c)^2 + 1)^2*a^4) + (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 189*A*a^24
*tan(1/2*d*x + 1/2*c)^5 - 105*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 1365*A*a^24*tan(1/2*d*x + 1/2*c)^3 + 385*C*a^24*
tan(1/2*d*x + 1/2*c)^3 - 11655*A*a^24*tan(1/2*d*x + 1/2*c) - 1575*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d

Mupad [B] (verification not implemented)

Time = 16.03 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {x\,\left (21\,A+2\,C\right )}{2\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{4\,a^4}+\frac {6\,A+2\,C}{8\,a^4}+\frac {15\,A-C}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A+C\right )}{40\,a^4}+\frac {6\,A+2\,C}{40\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{4\,a^4}+\frac {3\,\left (6\,A+2\,C\right )}{4\,a^4}+\frac {3\,\left (15\,A-C\right )}{8\,a^4}+\frac {20\,A-4\,C}{8\,a^4}\right )}{d}-\frac {9\,A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+7\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]

[In]

int((cos(c + d*x)^2*(A + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^4,x)

[Out]

(x*(21*A + 2*C))/(2*a^4) + (tan(c/2 + (d*x)/2)^3*((A + C)/(4*a^4) + (6*A + 2*C)/(8*a^4) + (15*A - C)/(24*a^4))
)/d - (tan(c/2 + (d*x)/2)^5*((3*(A + C))/(40*a^4) + (6*A + 2*C)/(40*a^4)))/d - (tan(c/2 + (d*x)/2)*((5*(A + C)
)/(4*a^4) + (3*(6*A + 2*C))/(4*a^4) + (3*(15*A - C))/(8*a^4) + (20*A - 4*C)/(8*a^4)))/d - (7*A*tan(c/2 + (d*x)
/2) + 9*A*tan(c/2 + (d*x)/2)^3)/(d*(2*a^4*tan(c/2 + (d*x)/2)^2 + a^4*tan(c/2 + (d*x)/2)^4 + a^4)) + (tan(c/2 +
(d*x)/2)^7*(A + C))/(56*a^4*d)