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

Optimal. Leaf size=176 \[ \frac{2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{x (4 A-B)}{a^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]

[Out]

-(((4*A - B)*x)/a^4) + (2*(332*A - 80*B + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Sin[c + d*x])/
(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c +
d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)

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Rubi [A]  time = 0.559175, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{x (4 A-B)}{a^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]

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])^4,x]

[Out]

-(((4*A - B)*x)/a^4) + (2*(332*A - 80*B + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Sin[c + d*x])/
(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c +
d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)

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))^4} \, dx &=-\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac{\int \frac{\cos (c+d x) (a (8 A-B+C)-a (4 A-4 B-3 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos (c+d x) \left (a^2 (52 A-10 B+3 C)-3 a^2 (12 A-5 B-2 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos (c+d x) \left (a^3 (244 A-55 B+6 C)-2 a^3 (88 A-25 B-3 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{\int \cos (c+d x) \left (2 a^4 (332 A-80 B+3 C)-105 a^4 (4 A-B) \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac{(4 A-B) \int 1 \, dx}{a^4}+\frac{(2 (332 A-80 B+3 C)) \int \cos (c+d x) \, dx}{105 a^4}\\ &=-\frac{(4 A-B) x}{a^4}+\frac{2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 1.92801, size = 567, normalized size = 3.22 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (-7350 d x (4 A-B) \cos \left (c+\frac{d x}{2}\right )-7350 d x (4 A-B) \cos \left (\frac{d x}{2}\right )-46130 A \sin \left (c+\frac{d x}{2}\right )+46116 A \sin \left (c+\frac{3 d x}{2}\right )-18060 A \sin \left (2 c+\frac{3 d x}{2}\right )+19292 A \sin \left (2 c+\frac{5 d x}{2}\right )-2100 A \sin \left (3 c+\frac{5 d x}{2}\right )+3791 A \sin \left (3 c+\frac{7 d x}{2}\right )+735 A \sin \left (4 c+\frac{7 d x}{2}\right )+105 A \sin \left (4 c+\frac{9 d x}{2}\right )+105 A \sin \left (5 c+\frac{9 d x}{2}\right )-17640 A d x \cos \left (c+\frac{3 d x}{2}\right )-17640 A d x \cos \left (2 c+\frac{3 d x}{2}\right )-5880 A d x \cos \left (2 c+\frac{5 d x}{2}\right )-5880 A d x \cos \left (3 c+\frac{5 d x}{2}\right )-840 A d x \cos \left (3 c+\frac{7 d x}{2}\right )-840 A d x \cos \left (4 c+\frac{7 d x}{2}\right )+60830 A \sin \left (\frac{d x}{2}\right )+16520 B \sin \left (c+\frac{d x}{2}\right )-14280 B \sin \left (c+\frac{3 d x}{2}\right )+7560 B \sin \left (2 c+\frac{3 d x}{2}\right )-5600 B \sin \left (2 c+\frac{5 d x}{2}\right )+1680 B \sin \left (3 c+\frac{5 d x}{2}\right )-1040 B \sin \left (3 c+\frac{7 d x}{2}\right )+4410 B d x \cos \left (c+\frac{3 d x}{2}\right )+4410 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+1470 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+1470 B d x \cos \left (3 c+\frac{5 d x}{2}\right )+210 B d x \cos \left (3 c+\frac{7 d x}{2}\right )+210 B d x \cos \left (4 c+\frac{7 d x}{2}\right )-19880 B \sin \left (\frac{d x}{2}\right )-2520 C \sin \left (c+\frac{d x}{2}\right )+1764 C \sin \left (c+\frac{3 d x}{2}\right )-1260 C \sin \left (2 c+\frac{3 d x}{2}\right )+588 C \sin \left (2 c+\frac{5 d x}{2}\right )-420 C \sin \left (3 c+\frac{5 d x}{2}\right )+144 C \sin \left (3 c+\frac{7 d x}{2}\right )+2520 C \sin \left (\frac{d x}{2}\right )\right )}{26880 a^4 d} \]

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])^4,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^7*(-7350*(4*A - B)*d*x*Cos[(d*x)/2] - 7350*(4*A - B)*d*x*Cos[c + (d*x)/2] - 17640*A
*d*x*Cos[c + (3*d*x)/2] + 4410*B*d*x*Cos[c + (3*d*x)/2] - 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 4410*B*d*x*Cos[2*
c + (3*d*x)/2] - 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 1470*B*d*x*Cos[2*c + (5*d*x)/2] - 5880*A*d*x*Cos[3*c + (5*d
*x)/2] + 1470*B*d*x*Cos[3*c + (5*d*x)/2] - 840*A*d*x*Cos[3*c + (7*d*x)/2] + 210*B*d*x*Cos[3*c + (7*d*x)/2] - 8
40*A*d*x*Cos[4*c + (7*d*x)/2] + 210*B*d*x*Cos[4*c + (7*d*x)/2] + 60830*A*Sin[(d*x)/2] - 19880*B*Sin[(d*x)/2] +
2520*C*Sin[(d*x)/2] - 46130*A*Sin[c + (d*x)/2] + 16520*B*Sin[c + (d*x)/2] - 2520*C*Sin[c + (d*x)/2] + 46116*A
*Sin[c + (3*d*x)/2] - 14280*B*Sin[c + (3*d*x)/2] + 1764*C*Sin[c + (3*d*x)/2] - 18060*A*Sin[2*c + (3*d*x)/2] +
7560*B*Sin[2*c + (3*d*x)/2] - 1260*C*Sin[2*c + (3*d*x)/2] + 19292*A*Sin[2*c + (5*d*x)/2] - 5600*B*Sin[2*c + (5
*d*x)/2] + 588*C*Sin[2*c + (5*d*x)/2] - 2100*A*Sin[3*c + (5*d*x)/2] + 1680*B*Sin[3*c + (5*d*x)/2] - 420*C*Sin[
3*c + (5*d*x)/2] + 3791*A*Sin[3*c + (7*d*x)/2] - 1040*B*Sin[3*c + (7*d*x)/2] + 144*C*Sin[3*c + (7*d*x)/2] + 73
5*A*Sin[4*c + (7*d*x)/2] + 105*A*Sin[4*c + (9*d*x)/2] + 105*A*Sin[5*c + (9*d*x)/2]))/(26880*a^4*d)

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Maple [A]  time = 0.122, size = 307, normalized size = 1.7 \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{C}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{7\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3\,C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{23\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{11\,B}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{49\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{15\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\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 ) }}-8\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{4}}}+2\,{\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)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(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-1/56/d/a^4*C*tan(1/2*d*x+1/2*c)^7+7/40/d/
a^4*tan(1/2*d*x+1/2*c)^5*A-1/8/d/a^4*tan(1/2*d*x+1/2*c)^5*B+3/40/d/a^4*C*tan(1/2*d*x+1/2*c)^5-23/24/d/a^4*A*ta
n(1/2*d*x+1/2*c)^3+11/24/d/a^4*B*tan(1/2*d*x+1/2*c)^3-1/8/d/a^4*C*tan(1/2*d*x+1/2*c)^3+49/8/d/a^4*A*tan(1/2*d*
x+1/2*c)-15/8/d/a^4*B*tan(1/2*d*x+1/2*c)+1/8/d/a^4*C*tan(1/2*d*x+1/2*c)+2/d/a^4*A*tan(1/2*d*x+1/2*c)/(1+tan(1/
2*d*x+1/2*c)^2)-8/d/a^4*A*arctan(tan(1/2*d*x+1/2*c))+2/d/a^4*arctan(tan(1/2*d*x+1/2*c))*B

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Maxima [B]  time = 1.46788, size = 481, normalized size = 2.73 \begin{align*} \frac{A{\left (\frac{1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac{a^{4} \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{5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{147 \, \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{6720 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 5 \, B{\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 )} + \frac{3 \, C{\left (\frac{35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{35 \, \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{5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, 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))^4,x, algorithm="maxima")

[Out]

1/840*(A*(1680*sin(d*x + c)/((a^4 + a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*sin(d
*x + c)/(cos(d*x + c) + 1) - 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*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 - 6720*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) - 5*B*((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) + 3*C*(35
*sin(d*x + c)/(cos(d*x + c) + 1) - 35*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*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)/d

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Fricas [A]  time = 0.512233, size = 608, normalized size = 3.45 \begin{align*} -\frac{105 \,{\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{4} + 420 \,{\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 630 \,{\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 420 \,{\left (4 \, A - B\right )} d x \cos \left (d x + c\right ) + 105 \,{\left (4 \, A - B\right )} d x -{\left (105 \, A \cos \left (d x + c\right )^{4} + 4 \,{\left (296 \, A - 65 \, B + 9 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (2636 \, A - 620 \, B + 39 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (2236 \, A - 535 \, B + 24 \, C\right )} \cos \left (d x + c\right ) + 664 \, A - 160 \, B + 6 \, C\right )} \sin \left (d x + c\right )}{105 \,{\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)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/105*(105*(4*A - B)*d*x*cos(d*x + c)^4 + 420*(4*A - B)*d*x*cos(d*x + c)^3 + 630*(4*A - B)*d*x*cos(d*x + c)^2
+ 420*(4*A - B)*d*x*cos(d*x + c) + 105*(4*A - B)*d*x - (105*A*cos(d*x + c)^4 + 4*(296*A - 65*B + 9*C)*cos(d*x
+ c)^3 + (2636*A - 620*B + 39*C)*cos(d*x + c)^2 + (2236*A - 535*B + 24*C)*cos(d*x + c) + 664*A - 160*B + 6*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)

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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))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.16047, size = 346, normalized size = 1.97 \begin{align*} -\frac{\frac{840 \,{\left (d x + c\right )}{\left (4 \, A - B\right )}}{a^{4}} - \frac{1680 \, 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^{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} + 15 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 147 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 105 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 63 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5145 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1575 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105 \, C 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)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

-1/840*(840*(d*x + c)*(4*A - B)/a^4 - 1680*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*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 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 147*A*a^24*
tan(1/2*d*x + 1/2*c)^5 + 105*B*a^24*tan(1/2*d*x + 1/2*c)^5 - 63*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 805*A*a^24*tan
(1/2*d*x + 1/2*c)^3 - 385*B*a^24*tan(1/2*d*x + 1/2*c)^3 + 105*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 5145*A*a^24*tan(
1/2*d*x + 1/2*c) + 1575*B*a^24*tan(1/2*d*x + 1/2*c) - 105*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d