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

Optimal. Leaf size=156 \[ \frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}-\frac{(4 A+3 C) \sin (c+d x)}{a d}+\frac{(5 A+4 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (5 A+4 C) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x (5 A+4 C)}{8 a} \]

[Out]

(3*(5*A + 4*C)*x)/(8*a) - ((4*A + 3*C)*Sin[c + d*x])/(a*d) + (3*(5*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d)
+ ((5*A + 4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c +
d*x])) + ((4*A + 3*C)*Sin[c + d*x]^3)/(3*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.186714, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {4085, 3787, 2635, 8, 2633} \[ \frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}-\frac{(4 A+3 C) \sin (c+d x)}{a d}+\frac{(5 A+4 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (5 A+4 C) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x (5 A+4 C)}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(5*A + 4*C)*x)/(8*a) - ((4*A + 3*C)*Sin[c + d*x])/(a*d) + (3*(5*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d)
+ ((5*A + 4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c +
d*x])) + ((4*A + 3*C)*Sin[c + d*x]^3)/(3*a*d)

Rule 4085

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)]

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 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]

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]

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{\int \cos ^4(c+d x) (-a (5 A+4 C)+a (4 A+3 C) \sec (c+d x)) \, dx}{a^2}\\ &=-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(4 A+3 C) \int \cos ^3(c+d x) \, dx}{a}+\frac{(5 A+4 C) \int \cos ^4(c+d x) \, dx}{a}\\ &=\frac{(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac{(3 (5 A+4 C)) \int \cos ^2(c+d x) \, dx}{4 a}+\frac{(4 A+3 C) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac{(4 A+3 C) \sin (c+d x)}{a d}+\frac{3 (5 A+4 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac{(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 (5 A+4 C)) \int 1 \, dx}{8 a}\\ &=\frac{3 (5 A+4 C) x}{8 a}-\frac{(4 A+3 C) \sin (c+d x)}{a d}+\frac{3 (5 A+4 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac{(5 A+4 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{(A+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}\\ \end{align*}

Mathematica [A]  time = 0.695898, size = 283, normalized size = 1.81 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (72 d x (5 A+4 C) \cos \left (c+\frac{d x}{2}\right )-168 A \sin \left (c+\frac{d x}{2}\right )-120 A \sin \left (c+\frac{3 d x}{2}\right )-120 A \sin \left (2 c+\frac{3 d x}{2}\right )+40 A \sin \left (2 c+\frac{5 d x}{2}\right )+40 A \sin \left (3 c+\frac{5 d x}{2}\right )-5 A \sin \left (3 c+\frac{7 d x}{2}\right )-5 A \sin \left (4 c+\frac{7 d x}{2}\right )+3 A \sin \left (4 c+\frac{9 d x}{2}\right )+3 A \sin \left (5 c+\frac{9 d x}{2}\right )+72 d x (5 A+4 C) \cos \left (\frac{d x}{2}\right )-552 A \sin \left (\frac{d x}{2}\right )-96 C \sin \left (c+\frac{d x}{2}\right )-72 C \sin \left (c+\frac{3 d x}{2}\right )-72 C \sin \left (2 c+\frac{3 d x}{2}\right )+24 C \sin \left (2 c+\frac{5 d x}{2}\right )+24 C \sin \left (3 c+\frac{5 d x}{2}\right )-480 C \sin \left (\frac{d x}{2}\right )\right )}{192 a d (\cos (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(72*(5*A + 4*C)*d*x*Cos[(d*x)/2] + 72*(5*A + 4*C)*d*x*Cos[c + (d*x)/2] - 552*A*Sin[
(d*x)/2] - 480*C*Sin[(d*x)/2] - 168*A*Sin[c + (d*x)/2] - 96*C*Sin[c + (d*x)/2] - 120*A*Sin[c + (3*d*x)/2] - 72
*C*Sin[c + (3*d*x)/2] - 120*A*Sin[2*c + (3*d*x)/2] - 72*C*Sin[2*c + (3*d*x)/2] + 40*A*Sin[2*c + (5*d*x)/2] + 2
4*C*Sin[2*c + (5*d*x)/2] + 40*A*Sin[3*c + (5*d*x)/2] + 24*C*Sin[3*c + (5*d*x)/2] - 5*A*Sin[3*c + (7*d*x)/2] -
5*A*Sin[4*c + (7*d*x)/2] + 3*A*Sin[4*c + (9*d*x)/2] + 3*A*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))

________________________________________________________________________________________

Maple [B]  time = 0.1, size = 352, normalized size = 2.3 \begin{align*} -{\frac{A}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{25\,A}{4\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-3\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}C}{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{115\,A}{12\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-7\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}C}{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{109\,A}{12\,ad} \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 ) ^{-4}}-5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}C}{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{7\,A}{4\,ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{15\,A}{4\,ad}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{ad}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/d*A*tan(1/2*d*x+1/2*c)-1/a/d*C*tan(1/2*d*x+1/2*c)-25/4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^
7*A-3/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*C-115/12/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+
1/2*c)^5*A-7/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*C-109/12/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1
/2*d*x+1/2*c)^3*A-5/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*C-7/4/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*A
*tan(1/2*d*x+1/2*c)-1/a/d/(1+tan(1/2*d*x+1/2*c)^2)^4*C*tan(1/2*d*x+1/2*c)+15/4/a/d*A*arctan(tan(1/2*d*x+1/2*c)
)+3/a/d*arctan(tan(1/2*d*x+1/2*c))*C

________________________________________________________________________________________

Maxima [B]  time = 1.43518, size = 474, normalized size = 3.04 \begin{align*} -\frac{A{\left (\frac{\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{45 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{12 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, C{\left (\frac{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(A*((21*sin(d*x + c)/(cos(d*x + c) + 1) + 109*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 115*sin(d*x + c)^5/(
cos(d*x + c) + 1)^5 + 75*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6
*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8) - 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 12*sin(d*x + c)/(a*(cos(d*x + c) + 1))) + 12*C*((
sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a + 2*a*sin(d*x + c)^2/(cos(d*x + c)
+ 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + sin(d*x + c)/
(a*(cos(d*x + c) + 1))))/d

________________________________________________________________________________________

Fricas [A]  time = 0.497758, size = 290, normalized size = 1.86 \begin{align*} \frac{9 \,{\left (5 \, A + 4 \, C\right )} d x \cos \left (d x + c\right ) + 9 \,{\left (5 \, A + 4 \, C\right )} d x +{\left (6 \, A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} +{\left (13 \, A + 12 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (19 \, A + 12 \, C\right )} \cos \left (d x + c\right ) - 64 \, A - 48 \, C\right )} \sin \left (d x + c\right )}{24 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(9*(5*A + 4*C)*d*x*cos(d*x + c) + 9*(5*A + 4*C)*d*x + (6*A*cos(d*x + c)^4 - 2*A*cos(d*x + c)^3 + (13*A +
12*C)*cos(d*x + c)^2 - (19*A + 12*C)*cos(d*x + c) - 64*A - 48*C)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.16628, size = 243, normalized size = 1.56 \begin{align*} \frac{\frac{9 \,{\left (d x + c\right )}{\left (5 \, A + 4 \, C\right )}}{a} - \frac{24 \,{\left (A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a} - \frac{2 \,{\left (75 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 36 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 115 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 84 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 109 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 60 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, C \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 )}^{4} a}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(9*(d*x + c)*(5*A + 4*C)/a - 24*(A*tan(1/2*d*x + 1/2*c) + C*tan(1/2*d*x + 1/2*c))/a - 2*(75*A*tan(1/2*d*x
+ 1/2*c)^7 + 36*C*tan(1/2*d*x + 1/2*c)^7 + 115*A*tan(1/2*d*x + 1/2*c)^5 + 84*C*tan(1/2*d*x + 1/2*c)^5 + 109*A
*tan(1/2*d*x + 1/2*c)^3 + 60*C*tan(1/2*d*x + 1/2*c)^3 + 21*A*tan(1/2*d*x + 1/2*c) + 12*C*tan(1/2*d*x + 1/2*c))
/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d