3.117 \(\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx\)

Optimal. Leaf size=256 \[ -\frac{8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d}+\frac{8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac{(44 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{(44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac{(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{x (44 A-21 B)}{2 a^4}-\frac{(16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4} \]

[Out]

-((44*A - 21*B)*x)/(2*a^4) + (8*(227*A - 108*B)*Sin[c + d*x])/(35*a^4*d) - ((44*A - 21*B)*Cos[c + d*x]*Sin[c +
d*x])/(2*a^4*d) - ((178*A - 87*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((44*A - 21
*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a
+ a*Sec[c + d*x])^4) - ((16*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*(227*A
- 108*B)*Sin[c + d*x]^3)/(105*a^4*d)

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Rubi [A]  time = 0.705455, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4020, 3787, 2633, 2635, 8} \[ -\frac{8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d}+\frac{8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac{(44 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{(44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac{(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{x (44 A-21 B)}{2 a^4}-\frac{(16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]

[Out]

-((44*A - 21*B)*x)/(2*a^4) + (8*(227*A - 108*B)*Sin[c + d*x])/(35*a^4*d) - ((44*A - 21*B)*Cos[c + d*x]*Sin[c +
d*x])/(2*a^4*d) - ((178*A - 87*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((44*A - 21
*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a
+ a*Sec[c + d*x])^4) - ((16*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*(227*A
- 108*B)*Sin[c + d*x]^3)/(105*a^4*d)

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]

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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 8

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

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx &=-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{\int \frac{\cos ^3(c+d x) (a (10 A-3 B)-6 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos ^3(c+d x) \left (14 a^2 (7 A-3 B)-5 a^2 (16 A-9 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos ^3(c+d x) \left (9 a^3 (92 A-43 B)-4 a^3 (178 A-87 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{\int \cos ^3(c+d x) \left (24 a^4 (227 A-108 B)-105 a^4 (44 A-21 B) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac{(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{(8 (227 A-108 B)) \int \cos ^3(c+d x) \, dx}{35 a^4}-\frac{(44 A-21 B) \int \cos ^2(c+d x) \, dx}{a^4}\\ &=-\frac{(44 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac{(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac{(44 A-21 B) \int 1 \, dx}{2 a^4}-\frac{(8 (227 A-108 B)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 a^4 d}\\ &=-\frac{(44 A-21 B) x}{2 a^4}+\frac{8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac{(44 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac{(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac{8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d}\\ \end{align*}

Mathematica [B]  time = 1.62759, size = 611, normalized size = 2.39 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-14700 d x (44 A-21 B) \cos \left (c+\frac{d x}{2}\right )-14700 d x (44 A-21 B) \cos \left (\frac{d x}{2}\right )-687260 A \sin \left (c+\frac{d x}{2}\right )+814107 A \sin \left (c+\frac{3 d x}{2}\right )-204645 A \sin \left (2 c+\frac{3 d x}{2}\right )+357609 A \sin \left (2 c+\frac{5 d x}{2}\right )+18025 A \sin \left (3 c+\frac{5 d x}{2}\right )+72522 A \sin \left (3 c+\frac{7 d x}{2}\right )+24010 A \sin \left (4 c+\frac{7 d x}{2}\right )+2310 A \sin \left (4 c+\frac{9 d x}{2}\right )+2310 A \sin \left (5 c+\frac{9 d x}{2}\right )-175 A \sin \left (5 c+\frac{11 d x}{2}\right )-175 A \sin \left (6 c+\frac{11 d x}{2}\right )+35 A \sin \left (6 c+\frac{13 d x}{2}\right )+35 A \sin \left (7 c+\frac{13 d x}{2}\right )-388080 A d x \cos \left (c+\frac{3 d x}{2}\right )-388080 A d x \cos \left (2 c+\frac{3 d x}{2}\right )-129360 A d x \cos \left (2 c+\frac{5 d x}{2}\right )-129360 A d x \cos \left (3 c+\frac{5 d x}{2}\right )-18480 A d x \cos \left (3 c+\frac{7 d x}{2}\right )-18480 A d x \cos \left (4 c+\frac{7 d x}{2}\right )+1010660 A \sin \left (\frac{d x}{2}\right )+386190 B \sin \left (c+\frac{d x}{2}\right )-422478 B \sin \left (c+\frac{3 d x}{2}\right )+132930 B \sin \left (2 c+\frac{3 d x}{2}\right )-181461 B \sin \left (2 c+\frac{5 d x}{2}\right )+3675 B \sin \left (3 c+\frac{5 d x}{2}\right )-36003 B \sin \left (3 c+\frac{7 d x}{2}\right )-9555 B \sin \left (4 c+\frac{7 d x}{2}\right )-945 B \sin \left (4 c+\frac{9 d x}{2}\right )-945 B \sin \left (5 c+\frac{9 d x}{2}\right )+105 B \sin \left (5 c+\frac{11 d x}{2}\right )+105 B \sin \left (6 c+\frac{11 d x}{2}\right )+185220 B d x \cos \left (c+\frac{3 d x}{2}\right )+185220 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+61740 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+61740 B d x \cos \left (3 c+\frac{5 d x}{2}\right )+8820 B d x \cos \left (3 c+\frac{7 d x}{2}\right )+8820 B d x \cos \left (4 c+\frac{7 d x}{2}\right )-539490 B \sin \left (\frac{d x}{2}\right )\right )}{6720 a^4 d (\cos (c+d x)+1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(-14700*(44*A - 21*B)*d*x*Cos[(d*x)/2] - 14700*(44*A - 21*B)*d*x*Cos[c + (d*x)/2] -
388080*A*d*x*Cos[c + (3*d*x)/2] + 185220*B*d*x*Cos[c + (3*d*x)/2] - 388080*A*d*x*Cos[2*c + (3*d*x)/2] + 18522
0*B*d*x*Cos[2*c + (3*d*x)/2] - 129360*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*B*d*x*Cos[2*c + (5*d*x)/2] - 129360*A
*d*x*Cos[3*c + (5*d*x)/2] + 61740*B*d*x*Cos[3*c + (5*d*x)/2] - 18480*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*B*d*x*C
os[3*c + (7*d*x)/2] - 18480*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*B*d*x*Cos[4*c + (7*d*x)/2] + 1010660*A*Sin[(d*x)
/2] - 539490*B*Sin[(d*x)/2] - 687260*A*Sin[c + (d*x)/2] + 386190*B*Sin[c + (d*x)/2] + 814107*A*Sin[c + (3*d*x)
/2] - 422478*B*Sin[c + (3*d*x)/2] - 204645*A*Sin[2*c + (3*d*x)/2] + 132930*B*Sin[2*c + (3*d*x)/2] + 357609*A*S
in[2*c + (5*d*x)/2] - 181461*B*Sin[2*c + (5*d*x)/2] + 18025*A*Sin[3*c + (5*d*x)/2] + 3675*B*Sin[3*c + (5*d*x)/
2] + 72522*A*Sin[3*c + (7*d*x)/2] - 36003*B*Sin[3*c + (7*d*x)/2] + 24010*A*Sin[4*c + (7*d*x)/2] - 9555*B*Sin[4
*c + (7*d*x)/2] + 2310*A*Sin[4*c + (9*d*x)/2] - 945*B*Sin[4*c + (9*d*x)/2] + 2310*A*Sin[5*c + (9*d*x)/2] - 945
*B*Sin[5*c + (9*d*x)/2] - 175*A*Sin[5*c + (11*d*x)/2] + 105*B*Sin[5*c + (11*d*x)/2] - 175*A*Sin[6*c + (11*d*x)
/2] + 105*B*Sin[6*c + (11*d*x)/2] + 35*A*Sin[6*c + (13*d*x)/2] + 35*A*Sin[7*c + (13*d*x)/2]))/(6720*a^4*d*(1 +
Cos[c + d*x])^4)

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Maple [A]  time = 0.111, size = 402, normalized size = 1.6 \begin{align*} -{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{B}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{11\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{9\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{59\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{13\,B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{209\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{111\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+26\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}A}{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-9\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}B}{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{124\,A}{3\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}-16\,{\frac{B \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+18\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-7\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-44\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{4}}}+21\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x)

[Out]

-1/56/d/a^4*tan(1/2*d*x+1/2*c)^7*A+1/56/d/a^4*tan(1/2*d*x+1/2*c)^7*B+11/40/d/a^4*tan(1/2*d*x+1/2*c)^5*A-9/40/d
/a^4*tan(1/2*d*x+1/2*c)^5*B-59/24/d/a^4*A*tan(1/2*d*x+1/2*c)^3+13/8/d/a^4*B*tan(1/2*d*x+1/2*c)^3+209/8/d/a^4*A
*tan(1/2*d*x+1/2*c)-111/8/d/a^4*B*tan(1/2*d*x+1/2*c)+26/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*
A-9/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*B+124/3/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d
*x+1/2*c)^3-16/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*B*tan(1/2*d*x+1/2*c)^3+18/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*A*t
an(1/2*d*x+1/2*c)-7/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*B*tan(1/2*d*x+1/2*c)-44/d/a^4*A*arctan(tan(1/2*d*x+1/2*c)
)+21/d/a^4*arctan(tan(1/2*d*x+1/2*c))*B

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Maxima [A]  time = 1.64083, size = 610, normalized size = 2.38 \begin{align*} \frac{A{\left (\frac{560 \,{\left (\frac{27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} + \frac{3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\frac{21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{231 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{36960 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 3 \, B{\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 )}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

1/840*(A*(560*(27*sin(d*x + c)/(cos(d*x + c) + 1) + 62*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 39*sin(d*x + c)^5
/(cos(d*x + c) + 1)^5)/(a^4 + 3*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) +
1)^4 + a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (21945*sin(d*x + c)/(cos(d*x + c) + 1) - 2065*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 + 231*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4
- 36960*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) - 3*B*(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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Fricas [A]  time = 0.50868, size = 701, normalized size = 2.74 \begin{align*} -\frac{105 \,{\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \,{\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \,{\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \,{\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right ) + 105 \,{\left (44 \, A - 21 \, B\right )} d x -{\left (70 \, A \cos \left (d x + c\right )^{6} - 35 \,{\left (4 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{5} + 140 \,{\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (3196 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (7184 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (24436 \, A - 11619 \, B\right )} \cos \left (d x + c\right ) + 7264 \, A - 3456 \, B\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/210*(105*(44*A - 21*B)*d*x*cos(d*x + c)^4 + 420*(44*A - 21*B)*d*x*cos(d*x + c)^3 + 630*(44*A - 21*B)*d*x*co
s(d*x + c)^2 + 420*(44*A - 21*B)*d*x*cos(d*x + c) + 105*(44*A - 21*B)*d*x - (70*A*cos(d*x + c)^6 - 35*(4*A - 3
*B)*cos(d*x + c)^5 + 140*(7*A - 3*B)*cos(d*x + c)^4 + 4*(3196*A - 1509*B)*cos(d*x + c)^3 + 4*(7184*A - 3411*B)
*cos(d*x + c)^2 + (24436*A - 11619*B)*cos(d*x + c) + 7264*A - 3456*B)*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.25121, size = 352, normalized size = 1.38 \begin{align*} -\frac{\frac{420 \,{\left (d x + c\right )}{\left (44 \, A - 21 \, B\right )}}{a^{4}} - \frac{280 \,{\left (78 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 27 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 124 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 48 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 54 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 21 \, B \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 )}^{3} a^{4}} + \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 231 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 189 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 21945 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 11655 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

-1/840*(420*(d*x + c)*(44*A - 21*B)/a^4 - 280*(78*A*tan(1/2*d*x + 1/2*c)^5 - 27*B*tan(1/2*d*x + 1/2*c)^5 + 124
*A*tan(1/2*d*x + 1/2*c)^3 - 48*B*tan(1/2*d*x + 1/2*c)^3 + 54*A*tan(1/2*d*x + 1/2*c) - 21*B*tan(1/2*d*x + 1/2*c
))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^4) + (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 - 15*B*a^24*tan(1/2*d*x + 1/2*c)^7
- 231*A*a^24*tan(1/2*d*x + 1/2*c)^5 + 189*B*a^24*tan(1/2*d*x + 1/2*c)^5 + 2065*A*a^24*tan(1/2*d*x + 1/2*c)^3
- 1365*B*a^24*tan(1/2*d*x + 1/2*c)^3 - 21945*A*a^24*tan(1/2*d*x + 1/2*c) + 11655*B*a^24*tan(1/2*d*x + 1/2*c))/
a^28)/d