3.471 \(\int \frac{\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=141 \[ \frac{2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac{(3 A-B) \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{x (3 A-B)}{a^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]

[Out]

-(((3*A - B)*x)/a^3) + (2*(36*A - 11*B + C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*
Sec[c + d*x])^3) - ((9*A - 4*B - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(
d*(a^3 + a^3*Sec[c + d*x]))

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Rubi [A]  time = 0.403664, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {4084, 4020, 3787, 2637, 8} \[ \frac{2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac{(3 A-B) \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{x (3 A-B)}{a^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]

[Out]

-(((3*A - B)*x)/a^3) + (2*(36*A - 11*B + C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*
Sec[c + d*x])^3) - ((9*A - 4*B - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(
d*(a^3 + a^3*Sec[c + d*x]))

Rule 4084

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[((a*A - b*B + a*C)*Cot[e + f*x]*(a + b*Cs
c[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[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(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, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]

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 2637

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

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos (c+d x) (a (6 A-B+C)-a (3 A-3 B-2 C) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{\cos (c+d x) \left (a^2 (27 A-7 B+2 C)-2 a^2 (9 A-4 B-C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(3 A-B) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \cos (c+d x) \left (2 a^3 (36 A-11 B+C)-15 a^3 (3 A-B) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(3 A-B) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(3 A-B) \int 1 \, dx}{a^3}+\frac{(2 (36 A-11 B+C)) \int \cos (c+d x) \, dx}{15 a^3}\\ &=-\frac{(3 A-B) x}{a^3}+\frac{2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(3 A-B) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 1.63649, size = 419, normalized size = 2.97 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) \left (-300 d x (3 A-B) \cos \left (c+\frac{d x}{2}\right )-300 d x (3 A-B) \cos \left (\frac{d x}{2}\right )-1125 A \sin \left (c+\frac{d x}{2}\right )+1215 A \sin \left (c+\frac{3 d x}{2}\right )-225 A \sin \left (2 c+\frac{3 d x}{2}\right )+363 A \sin \left (2 c+\frac{5 d x}{2}\right )+75 A \sin \left (3 c+\frac{5 d x}{2}\right )+15 A \sin \left (3 c+\frac{7 d x}{2}\right )+15 A \sin \left (4 c+\frac{7 d x}{2}\right )-450 A d x \cos \left (c+\frac{3 d x}{2}\right )-450 A d x \cos \left (2 c+\frac{3 d x}{2}\right )-90 A d x \cos \left (2 c+\frac{5 d x}{2}\right )-90 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+1755 A \sin \left (\frac{d x}{2}\right )+540 B \sin \left (c+\frac{d x}{2}\right )-460 B \sin \left (c+\frac{3 d x}{2}\right )+180 B \sin \left (2 c+\frac{3 d x}{2}\right )-128 B \sin \left (2 c+\frac{5 d x}{2}\right )+150 B d x \cos \left (c+\frac{3 d x}{2}\right )+150 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+30 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+30 B d x \cos \left (3 c+\frac{5 d x}{2}\right )-740 B \sin \left (\frac{d x}{2}\right )-120 C \sin \left (c+\frac{d x}{2}\right )+80 C \sin \left (c+\frac{3 d x}{2}\right )-60 C \sin \left (2 c+\frac{3 d x}{2}\right )+28 C \sin \left (2 c+\frac{5 d x}{2}\right )+160 C \sin \left (\frac{d x}{2}\right )\right )}{960 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^5*(-300*(3*A - B)*d*x*Cos[(d*x)/2] - 300*(3*A - B)*d*x*Cos[c + (d*x)/2] - 450*A*d*x
*Cos[c + (3*d*x)/2] + 150*B*d*x*Cos[c + (3*d*x)/2] - 450*A*d*x*Cos[2*c + (3*d*x)/2] + 150*B*d*x*Cos[2*c + (3*d
*x)/2] - 90*A*d*x*Cos[2*c + (5*d*x)/2] + 30*B*d*x*Cos[2*c + (5*d*x)/2] - 90*A*d*x*Cos[3*c + (5*d*x)/2] + 30*B*
d*x*Cos[3*c + (5*d*x)/2] + 1755*A*Sin[(d*x)/2] - 740*B*Sin[(d*x)/2] + 160*C*Sin[(d*x)/2] - 1125*A*Sin[c + (d*x
)/2] + 540*B*Sin[c + (d*x)/2] - 120*C*Sin[c + (d*x)/2] + 1215*A*Sin[c + (3*d*x)/2] - 460*B*Sin[c + (3*d*x)/2]
+ 80*C*Sin[c + (3*d*x)/2] - 225*A*Sin[2*c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] - 60*C*Sin[2*c + (3*d*x)/2
] + 363*A*Sin[2*c + (5*d*x)/2] - 128*B*Sin[2*c + (5*d*x)/2] + 28*C*Sin[2*c + (5*d*x)/2] + 75*A*Sin[3*c + (5*d*
x)/2] + 15*A*Sin[3*c + (7*d*x)/2] + 15*A*Sin[4*c + (7*d*x)/2]))/(960*a^3*d)

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Maple [A]  time = 0.11, size = 247, normalized size = 1.8 \begin{align*}{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{B}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{C}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{B}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{17\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{7\,B}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-6\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}+2\,{\frac{B\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x)

[Out]

1/20/d/a^3*tan(1/2*d*x+1/2*c)^5*A-1/20/d/a^3*tan(1/2*d*x+1/2*c)^5*B+1/20/d/a^3*C*tan(1/2*d*x+1/2*c)^5-1/2/d/a^
3*tan(1/2*d*x+1/2*c)^3*A+1/3/d/a^3*tan(1/2*d*x+1/2*c)^3*B-1/6/d/a^3*C*tan(1/2*d*x+1/2*c)^3+17/4/d/a^3*A*tan(1/
2*d*x+1/2*c)-7/4/d/a^3*B*tan(1/2*d*x+1/2*c)+1/4/d/a^3*C*tan(1/2*d*x+1/2*c)+2/d/a^3*A*tan(1/2*d*x+1/2*c)/(1+tan
(1/2*d*x+1/2*c)^2)-6/d/a^3*A*arctan(tan(1/2*d*x+1/2*c))+2/d/a^3*B*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.45872, size = 398, normalized size = 2.82 \begin{align*} \frac{3 \, A{\left (\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac{a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B{\left (\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + \frac{C{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*(3*A*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (85*sin(d*x
+ c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 -
120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) - B*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 20*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 120*arctan(sin(d*x + c)/(cos(d*x + c) + 1
))/a^3) + C*(15*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(c
os(d*x + c) + 1)^5)/a^3)/d

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Fricas [A]  time = 0.498218, size = 455, normalized size = 3.23 \begin{align*} -\frac{15 \,{\left (3 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 45 \,{\left (3 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 45 \,{\left (3 \, A - B\right )} d x \cos \left (d x + c\right ) + 15 \,{\left (3 \, A - B\right )} d x -{\left (15 \, A \cos \left (d x + c\right )^{3} +{\left (117 \, A - 32 \, B + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (57 \, A - 17 \, B + 2 \, C\right )} \cos \left (d x + c\right ) + 72 \, A - 22 \, B + 2 \, C\right )} \sin \left (d x + c\right )}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/15*(15*(3*A - B)*d*x*cos(d*x + c)^3 + 45*(3*A - B)*d*x*cos(d*x + c)^2 + 45*(3*A - B)*d*x*cos(d*x + c) + 15*
(3*A - B)*d*x - (15*A*cos(d*x + c)^3 + (117*A - 32*B + 7*C)*cos(d*x + c)^2 + 3*(57*A - 17*B + 2*C)*cos(d*x + c
) + 72*A - 22*B + 2*C)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a
^3*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)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.22984, size = 278, normalized size = 1.97 \begin{align*} -\frac{\frac{60 \,{\left (d x + c\right )}{\left (3 \, A - B\right )}}{a^{3}} - \frac{120 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 20 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 10 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/60*(60*(d*x + c)*(3*A - B)/a^3 - 120*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^3) - (3*A*a^12*
tan(1/2*d*x + 1/2*c)^5 - 3*B*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^12*tan(1/2*d*x + 1/2*c)^5 - 30*A*a^12*tan(1/2
*d*x + 1/2*c)^3 + 20*B*a^12*tan(1/2*d*x + 1/2*c)^3 - 10*C*a^12*tan(1/2*d*x + 1/2*c)^3 + 255*A*a^12*tan(1/2*d*x
+ 1/2*c) - 105*B*a^12*tan(1/2*d*x + 1/2*c) + 15*C*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d