3.522 \(\int \frac {\cos ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx\)

Optimal. Leaf size=387 \[ \frac {5 b^2 \left (2 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {b^2 \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {x \left (a^2+20 b^2\right )}{2 a^6}+\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}-\frac {b^3 \left (40 a^6-84 a^4 b^2+69 a^2 b^4-20 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right ) \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3} \]

[Out]

1/2*(a^2+20*b^2)*x/a^6-b^3*(40*a^6-84*a^4*b^2+69*a^2*b^4-20*b^6)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^
(1/2))/a^6/(a-b)^(7/2)/(a+b)^(7/2)/d-1/6*b*(24*a^6-146*a^4*b^2+167*a^2*b^4-60*b^6)*sin(d*x+c)/a^5/(a^2-b^2)^3/
d+1/2*(a^6-23*a^4*b^2+27*a^2*b^4-10*b^6)*cos(d*x+c)*sin(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*b^2*cos(d*x+c)*sin(d*x+c)
/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+5/6*b^2*(2*a^2-b^2)*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))
^2+1/6*b^2*(48*a^4-53*a^2*b^2+20*b^4)*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

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Rubi [A]  time = 1.46, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3847, 4100, 4104, 3919, 3831, 2659, 208} \[ -\frac {b \left (-146 a^4 b^2+167 a^2 b^4+24 a^6-60 b^6\right ) \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}+\frac {\left (-23 a^4 b^2+27 a^2 b^4+a^6-10 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}-\frac {b^3 \left (-84 a^4 b^2+69 a^2 b^4+40 a^6-20 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {b^2 \left (-53 a^2 b^2+48 a^4+20 b^4\right ) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {5 b^2 \left (2 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {b^2 \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {x \left (a^2+20 b^2\right )}{2 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 + 20*b^2)*x)/(2*a^6) - (b^3*(40*a^6 - 84*a^4*b^2 + 69*a^2*b^4 - 20*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*
x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(24*a^6 - 146*a^4*b^2 + 167*a^2*b^4 - 60*b^6)*Si
n[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6 - 23*a^4*b^2 + 27*a^2*b^4 - 10*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*
a^4*(a^2 - b^2)^3*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (5*b^2*(2*
a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(48*a^4 - 53*a^2*b
^2 + 20*b^4)*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3847

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

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4100

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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

Rule 4104

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*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx &=\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 a^2+5 b^2+3 a b \sec (c+d x)-4 b^2 \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (3 a^4-18 a^2 b^2+10 b^4\right )-2 a b \left (6 a^2-b^2\right ) \sec (c+d x)+15 b^2 \left (2 a^2-b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (-6 \left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right )+a b \left (18 a^4-8 a^2 b^2+5 b^4\right ) \sec (c+d x)-2 b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (-2 b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right )+2 a \left (3 a^6+27 a^4 b^2-25 a^2 b^4+10 b^6\right ) \sec (c+d x)+6 b \left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^3}\\ &=-\frac {b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {-6 \left (a^2-b^2\right )^3 \left (a^2+20 b^2\right )-6 a b \left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^3}\\ &=\frac {\left (a^2+20 b^2\right ) x}{2 a^6}-\frac {b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (b^3 \left (40 a^6-84 a^4 b^2+69 a^2 b^4-20 b^6\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (a^2+20 b^2\right ) x}{2 a^6}-\frac {b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (b^2 \left (40 a^6-84 a^4 b^2+69 a^2 b^4-20 b^6\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (a^2+20 b^2\right ) x}{2 a^6}-\frac {b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (b^2 \left (40 a^6-84 a^4 b^2+69 a^2 b^4-20 b^6\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right )^3 d}\\ &=\frac {\left (a^2+20 b^2\right ) x}{2 a^6}-\frac {b^3 \left (40 a^6-84 a^4 b^2+69 a^2 b^4-20 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {b \left (24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {5 b^2 \left (2 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (48 a^4-53 a^2 b^2+20 b^4\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.38, size = 326, normalized size = 0.84 \[ -\frac {b^6 \sin (c+d x)}{3 a^5 d (b-a) (a+b) (a \cos (c+d x)+b)^3}-\frac {4 b \sin (c+d x)}{a^5 d}+\frac {\sin (2 (c+d x))}{4 a^4 d}+\frac {\left (a^2+20 b^2\right ) (c+d x)}{2 a^6 d}+\frac {13 b^7 \sin (c+d x)-18 a^2 b^5 \sin (c+d x)}{6 a^5 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac {b^3 \left (-40 a^6+84 a^4 b^2-69 a^2 b^4+20 b^6\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2} \left (b^2-a^2\right )^3}+\frac {-90 a^4 b^4 \sin (c+d x)+122 a^2 b^6 \sin (c+d x)-47 b^8 \sin (c+d x)}{6 a^5 d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 + 20*b^2)*(c + d*x))/(2*a^6*d) + (b^3*(-40*a^6 + 84*a^4*b^2 - 69*a^2*b^4 + 20*b^6)*ArcTanh[((-a + b)*Tan
[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d) - (4*b*Sin[c + d*x])/(a^5*d) - (b^6*Si
n[c + d*x])/(3*a^5*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])^3) + (-18*a^2*b^5*Sin[c + d*x] + 13*b^7*Sin[c + d*x
])/(6*a^5*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) + (-90*a^4*b^4*Sin[c + d*x] + 122*a^2*b^6*Sin[c + d*x
] - 47*b^8*Sin[c + d*x])/(6*a^5*(-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(4*a^4*d)

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fricas [B]  time = 0.75, size = 1767, normalized size = 4.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(6*(a^13 + 16*a^11*b^2 - 74*a^9*b^4 + 116*a^7*b^6 - 79*a^5*b^8 + 20*a^3*b^10)*d*x*cos(d*x + c)^3 + 18*(a
^12*b + 16*a^10*b^3 - 74*a^8*b^5 + 116*a^6*b^7 - 79*a^4*b^9 + 20*a^2*b^11)*d*x*cos(d*x + c)^2 + 18*(a^11*b^2 +
16*a^9*b^4 - 74*a^7*b^6 + 116*a^5*b^8 - 79*a^3*b^10 + 20*a*b^12)*d*x*cos(d*x + c) + 6*(a^10*b^3 + 16*a^8*b^5
- 74*a^6*b^7 + 116*a^4*b^9 - 79*a^2*b^11 + 20*b^13)*d*x + 3*(40*a^6*b^6 - 84*a^4*b^8 + 69*a^2*b^10 - 20*b^12 +
(40*a^9*b^3 - 84*a^7*b^5 + 69*a^5*b^7 - 20*a^3*b^9)*cos(d*x + c)^3 + 3*(40*a^8*b^4 - 84*a^6*b^6 + 69*a^4*b^8
- 20*a^2*b^10)*cos(d*x + c)^2 + 3*(40*a^7*b^5 - 84*a^5*b^7 + 69*a^3*b^9 - 20*a*b^11)*cos(d*x + c))*sqrt(a^2 -
b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x +
c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*(24*a^9*b^4 - 170*a^7*b^6 + 313*a^5*b^
8 - 227*a^3*b^10 + 60*a*b^12 - 3*(a^13 - 4*a^11*b^2 + 6*a^9*b^4 - 4*a^7*b^6 + a^5*b^8)*cos(d*x + c)^4 + 15*(a^
12*b - 4*a^10*b^3 + 6*a^8*b^5 - 4*a^6*b^7 + a^4*b^9)*cos(d*x + c)^3 + (63*a^11*b^2 - 342*a^9*b^4 + 590*a^7*b^6
- 421*a^5*b^8 + 110*a^3*b^10)*cos(d*x + c)^2 + 3*(23*a^10*b^3 - 146*a^8*b^5 + 263*a^6*b^7 - 190*a^4*b^9 + 50*
a^2*b^11)*cos(d*x + c))*sin(d*x + c))/((a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*cos(d*x + c)^
3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4*a^13*b^4 +
6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c) + (a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6*b^11
)*d), 1/6*(3*(a^13 + 16*a^11*b^2 - 74*a^9*b^4 + 116*a^7*b^6 - 79*a^5*b^8 + 20*a^3*b^10)*d*x*cos(d*x + c)^3 + 9
*(a^12*b + 16*a^10*b^3 - 74*a^8*b^5 + 116*a^6*b^7 - 79*a^4*b^9 + 20*a^2*b^11)*d*x*cos(d*x + c)^2 + 9*(a^11*b^2
+ 16*a^9*b^4 - 74*a^7*b^6 + 116*a^5*b^8 - 79*a^3*b^10 + 20*a*b^12)*d*x*cos(d*x + c) + 3*(a^10*b^3 + 16*a^8*b^
5 - 74*a^6*b^7 + 116*a^4*b^9 - 79*a^2*b^11 + 20*b^13)*d*x - 3*(40*a^6*b^6 - 84*a^4*b^8 + 69*a^2*b^10 - 20*b^12
+ (40*a^9*b^3 - 84*a^7*b^5 + 69*a^5*b^7 - 20*a^3*b^9)*cos(d*x + c)^3 + 3*(40*a^8*b^4 - 84*a^6*b^6 + 69*a^4*b^
8 - 20*a^2*b^10)*cos(d*x + c)^2 + 3*(40*a^7*b^5 - 84*a^5*b^7 + 69*a^3*b^9 - 20*a*b^11)*cos(d*x + c))*sqrt(-a^2
+ b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (24*a^9*b^4 - 170*a^7*b^6
+ 313*a^5*b^8 - 227*a^3*b^10 + 60*a*b^12 - 3*(a^13 - 4*a^11*b^2 + 6*a^9*b^4 - 4*a^7*b^6 + a^5*b^8)*cos(d*x + c
)^4 + 15*(a^12*b - 4*a^10*b^3 + 6*a^8*b^5 - 4*a^6*b^7 + a^4*b^9)*cos(d*x + c)^3 + (63*a^11*b^2 - 342*a^9*b^4 +
590*a^7*b^6 - 421*a^5*b^8 + 110*a^3*b^10)*cos(d*x + c)^2 + 3*(23*a^10*b^3 - 146*a^8*b^5 + 263*a^6*b^7 - 190*a
^4*b^9 + 50*a^2*b^11)*cos(d*x + c))*sin(d*x + c))/((a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*c
os(d*x + c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4
*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c) + (a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^
9 + a^6*b^11)*d)]

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giac [A]  time = 0.35, size = 615, normalized size = 1.59 \[ \frac {\frac {6 \, {\left (40 \, a^{6} b^{3} - 84 \, a^{4} b^{5} + 69 \, a^{2} b^{7} - 20 \, b^{9}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {2 \, {\left (90 \, a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 162 \, a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 213 \, a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 81 \, a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 180 \, a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 392 \, a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 284 \, a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 90 \, a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 162 \, a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 213 \, a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 81 \, a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}} + \frac {3 \, {\left (a^{2} + 20 \, b^{2}\right )} {\left (d x + c\right )}}{a^{6}} - \frac {6 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, 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 )}^{2} a^{5}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(40*a^6*b^3 - 84*a^4*b^5 + 69*a^2*b^7 - 20*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(2*a - 2*b) + arct
an((a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*
b^6)*sqrt(-a^2 + b^2)) - 2*(90*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 162*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 48*a^4*b^
6*tan(1/2*d*x + 1/2*c)^5 + 213*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 48*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 81*a*b^9*t
an(1/2*d*x + 1/2*c)^5 + 36*b^10*tan(1/2*d*x + 1/2*c)^5 - 180*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 392*a^4*b^6*tan(
1/2*d*x + 1/2*c)^3 - 284*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 72*b^10*tan(1/2*d*x + 1/2*c)^3 + 90*a^6*b^4*tan(1/2*
d*x + 1/2*c) + 162*a^5*b^5*tan(1/2*d*x + 1/2*c) - 48*a^4*b^6*tan(1/2*d*x + 1/2*c) - 213*a^3*b^7*tan(1/2*d*x +
1/2*c) - 48*a^2*b^8*tan(1/2*d*x + 1/2*c) + 81*a*b^9*tan(1/2*d*x + 1/2*c) + 36*b^10*tan(1/2*d*x + 1/2*c))/((a^1
1 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) + 3*(a^2
+ 20*b^2)*(d*x + c)/a^6 - 6*(a*tan(1/2*d*x + 1/2*c)^3 + 8*b*tan(1/2*d*x + 1/2*c)^3 - a*tan(1/2*d*x + 1/2*c) +
8*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^5))/d

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maple [B]  time = 0.80, size = 1576, normalized size = 4.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+
1/2*c)^5-6/d*b^5/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan
(1/2*d*x+1/2*c)^5+34/d*b^6/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^
2+b^3)*tan(1/2*d*x+1/2*c)^5+3/d*b^7/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2
*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-12/d*b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/
(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5+60/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^
3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-212/3/d*b^6/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2
*c)^2*b-a-b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+24/d*b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-tan(1
/2*d*x+1/2*c)^2*b-a-b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-30/d*b^4/a/(a*tan(1/2*d*x+1/2*c)
^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)+6/d*b^5/a^2/(a*tan(1/2*d*x
+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)+34/d*b^6/a^3/(a*tan
(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)-3/d*b^7/a^4
/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)-12/d
*b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2
*c)-40/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^
(1/2))+84/d*b^5/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*
(a+b))^(1/2))-69/d*b^7/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/
((a-b)*(a+b))^(1/2))+20/d*b^9/a^6/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)
*(a-b)/((a-b)*(a+b))^(1/2))-1/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3-8/d/a^5/(1+tan(1/2*d*x+1/2
*c)^2)^2*tan(1/2*d*x+1/2*c)^3*b+1/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)-8/d/a^5/(1+tan(1/2*d*x+1
/2*c)^2)^2*tan(1/2*d*x+1/2*c)*b+1/d/a^4*arctan(tan(1/2*d*x+1/2*c))+20/d/a^6*arctan(tan(1/2*d*x+1/2*c))*b^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 10.80, size = 8133, normalized size = 21.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^9*(7*a^7*b - 10*a*b^7 + a^8 + 20*b^8 - 59*a^2*b^6 + 27*a^3*b^5 + 57*a^4*b^4 - 21*a^5*b^3
- 11*a^6*b^2))/(a^5*(a + b)^3*(a - b)) + (2*tan(c/2 + (d*x)/2)^3*(30*a*b^8 + 21*a^8*b - 6*a^9 + 120*b^9 - 364*
a^2*b^7 - 71*a^3*b^6 + 369*a^4*b^5 + 45*a^5*b^4 - 111*a^6*b^3 - 3*a^7*b^2))/(3*a^5*(a + b)^2*(a - b)^3) - (2*t
an(c/2 + (d*x)/2)^7*(21*a^8*b - 30*a*b^8 + 6*a^9 + 120*b^9 - 364*a^2*b^7 + 71*a^3*b^6 + 369*a^4*b^5 - 45*a^5*b
^4 - 111*a^6*b^3 + 3*a^7*b^2))/(3*a^5*(a + b)^3*(a - b)^2) + (2*tan(c/2 + (d*x)/2)^5*(9*a^10 + 180*b^10 - 611*
a^2*b^8 + 740*a^4*b^6 - 324*a^6*b^4 + 36*a^8*b^2))/(3*a^5*(a + b)^3*(a - b)^3) + (tan(c/2 + (d*x)/2)*(10*a*b^7
- 7*a^7*b + a^8 + 20*b^8 - 59*a^2*b^6 - 27*a^3*b^5 + 57*a^4*b^4 + 21*a^5*b^3 - 11*a^6*b^2))/(a^5*(a + b)*(a -
b)^3))/(d*(tan(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a^3 + 5*b^3) + tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^2*b -
2*a^3 + 10*b^3) - tan(c/2 + (d*x)/2)^6*(6*a*b^2 + 6*a^2*b - 2*a^3 - 10*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 -
tan(c/2 + (d*x)/2)^10*(3*a*b^2 - 3*a^2*b + a^3 - b^3) + tan(c/2 + (d*x)/2)^8*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3
))) - (atan(((((((4*(4*a^27 - 80*a^12*b^15 + 40*a^13*b^14 + 516*a^14*b^13 - 248*a^15*b^12 - 1404*a^16*b^11 + 6
40*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764*a^20*b^7 + 724*a^21*b^6 + 816*a^22*b^5 - 316*a^23*b^4 - 160
*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*
a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (4*tan(c/2 + (d*x)/2)*(a^2*1i + b^2*20i)*(8*
a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8
- 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 -
a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a
^18*b^3 - 5*a^19*b^2)))*(a^2*1i + b^2*20i))/(2*a^6) - (8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 80
0*b^18 + 4720*a^2*b^16 - 4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 - 1
0385*a^8*b^10 + 10430*a^9*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 68
*a^15*b^3 - 35*a^16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*
a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2))*(a^2*1i + b^2*20i)*1i)/(2*a^6) - (((((4*(4*a^
27 - 80*a^12*b^15 + 40*a^13*b^14 + 516*a^14*b^13 - 248*a^15*b^12 - 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*
b^9 - 896*a^19*b^8 - 1764*a^20*b^7 + 724*a^21*b^6 + 816*a^22*b^5 - 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))
/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 +
10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) + (4*tan(c/2 + (d*x)/2)*(a^2*1i + b^2*20i)*(8*a^25*b - 8*a^12*b^14 + 8*
a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120*a^2
0*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 +
5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*(
a^2*1i + b^2*20i))/(2*a^6) + (8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 800*b^18 + 4720*a^2*b^16 -
4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 - 10385*a^8*b^10 + 10430*a^9
*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 68*a^15*b^3 - 35*a^16*b^2))
/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 +
10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2))*(a^2*1i + b^2*20i)*1i)/(2*a^6))/((((((4*(4*a^27 - 80*a^12*b^15 + 40*a^
13*b^14 + 516*a^14*b^13 - 248*a^15*b^12 - 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764
*a^20*b^7 + 724*a^21*b^6 + 816*a^22*b^5 - 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^
11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3
- 5*a^24*b^2) - (4*tan(c/2 + (d*x)/2)*(a^2*1i + b^2*20i)*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12
- 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48
*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 -
10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*(a^2*1i + b^2*20i))/(2*a^6
) - (8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 800*b^18 + 4720*a^2*b^16 - 4720*a^3*b^15 - 11522*a^4
*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 - 10385*a^8*b^10 + 10430*a^9*b^9 + 3325*a^10*b^8 - 36
40*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 68*a^15*b^3 - 35*a^16*b^2))/(a^20*b + a^21 - a^10*b^
11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3
- 5*a^19*b^2))*(a^2*1i + b^2*20i))/(2*a^6) - (8*(8000*b^19 - 4000*a*b^18 - 50800*a^2*b^17 + 24400*a^3*b^16 + 1
35260*a^4*b^15 - 62030*a^5*b^14 - 193689*a^6*b^13 + 82337*a^7*b^12 + 155991*a^8*b^11 - 57345*a^9*b^10 - 64479*
a^10*b^9 + 16999*a^11*b^8 + 8281*a^12*b^7 + 204*a^13*b^6 + 1396*a^14*b^5 - 40*a^15*b^4 + 40*a^16*b^3))/(a^25*b
+ a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*
b^4 - 5*a^23*b^3 - 5*a^24*b^2) + (((((4*(4*a^27 - 80*a^12*b^15 + 40*a^13*b^14 + 516*a^14*b^13 - 248*a^15*b^12
- 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764*a^20*b^7 + 724*a^21*b^6 + 816*a^22*b^5
- 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8
- 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) + (4*tan(c/2 + (d*x)/2)*(a
^2*1i + b^2*20i)*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^1
7*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(
a^6*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5
+ 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*(a^2*1i + b^2*20i))/(2*a^6) + (8*tan(c/2 + (d*x)/2)*(800*a*b^17 +
2*a^17*b - a^18 - 800*b^18 + 4720*a^2*b^16 - 4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12
- 14812*a^7*b^11 - 10385*a^8*b^10 + 10430*a^9*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5
- 209*a^14*b^4 + 68*a^15*b^3 - 35*a^16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8
- 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2))*(a^2*1i + b^2*20i))/(2*a^
6)))*(a^2*1i + b^2*20i)*1i)/(a^6*d) - (b^3*atan(((b^3*((8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 8
00*b^18 + 4720*a^2*b^16 - 4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 -
10385*a^8*b^10 + 10430*a^9*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 6
8*a^15*b^3 - 35*a^16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10
*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) - (b^3*((4*(4*a^27 - 80*a^12*b^15 + 40*a^13*b
^14 + 516*a^14*b^13 - 248*a^15*b^12 - 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764*a^2
0*b^7 + 724*a^21*b^6 + 816*a^22*b^5 - 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^11 -
a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*
a^24*b^2) - (4*b^3*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*
(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*
b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/((a^20 - a^6*b^14
+ 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)*(a^20*b + a^21 - a^10*b^11
- a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 -
5*a^19*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2))/(2*(a^20 - a^6*b^14 + 7
*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b)^7)^(1/2)
*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*1i)/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8
- 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)) + (b^3*((8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 800*
b^18 + 4720*a^2*b^16 - 4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 - 103
85*a^8*b^10 + 10430*a^9*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 68*a
^15*b^3 - 35*a^16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^
15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) + (b^3*((4*(4*a^27 - 80*a^12*b^15 + 40*a^13*b^14
+ 516*a^14*b^13 - 248*a^15*b^12 - 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764*a^20*b
^7 + 724*a^21*b^6 + 816*a^22*b^5 - 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^11 - a^
16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^2
4*b^2) + (4*b^3*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*(8*
a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8
- 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/((a^20 - a^6*b^14 + 7
*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)*(a^20*b + a^21 - a^10*b^11 -
a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a
^19*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2))/(2*(a^20 - a^6*b^14 + 7*a^
8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(4
0*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*1i)/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 -
35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))/((8*(8000*b^19 - 4000*a*b^18 - 50800*a^2*b^17 + 24400*a^3*b^16 + 135
260*a^4*b^15 - 62030*a^5*b^14 - 193689*a^6*b^13 + 82337*a^7*b^12 + 155991*a^8*b^11 - 57345*a^9*b^10 - 64479*a^
10*b^9 + 16999*a^11*b^8 + 8281*a^12*b^7 + 204*a^13*b^6 + 1396*a^14*b^5 - 40*a^15*b^4 + 40*a^16*b^3))/(a^25*b +
a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^
4 - 5*a^23*b^3 - 5*a^24*b^2) + (b^3*((8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 800*b^18 + 4720*a^2
*b^16 - 4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 - 10385*a^8*b^10 + 1
0430*a^9*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 68*a^15*b^3 - 35*a^
16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^1
6*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) - (b^3*((4*(4*a^27 - 80*a^12*b^15 + 40*a^13*b^14 + 516*a^14*b^1
3 - 248*a^15*b^12 - 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764*a^20*b^7 + 724*a^21*b
^6 + 816*a^22*b^5 - 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^1
7*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (4*b^3
*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*(8*a^25*b - 8*a^12
*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7
- 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/((a^20 - a^6*b^14 + 7*a^8*b^12 - 21*
a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a
^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*((a
+ b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^1
0*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6
+ 69*a^2*b^4 - 84*a^4*b^2))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a
^16*b^4 - 7*a^18*b^2)) - (b^3*((8*tan(c/2 + (d*x)/2)*(800*a*b^17 + 2*a^17*b - a^18 - 800*b^18 + 4720*a^2*b^16
- 4720*a^3*b^15 - 11522*a^4*b^14 + 11522*a^5*b^13 + 14837*a^6*b^12 - 14812*a^7*b^11 - 10385*a^8*b^10 + 10430*a
^9*b^9 + 3325*a^10*b^8 - 3640*a^11*b^7 + 45*a^12*b^6 + 350*a^13*b^5 - 209*a^14*b^4 + 68*a^15*b^3 - 35*a^16*b^2
))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5
+ 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) + (b^3*((4*(4*a^27 - 80*a^12*b^15 + 40*a^13*b^14 + 516*a^14*b^13 - 24
8*a^15*b^12 - 1404*a^16*b^11 + 640*a^17*b^10 + 2076*a^18*b^9 - 896*a^19*b^8 - 1764*a^20*b^7 + 724*a^21*b^6 + 8
16*a^22*b^5 - 316*a^23*b^4 - 160*a^24*b^3 + 52*a^25*b^2))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9
+ 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) + (4*b^3*tan(c
/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*(8*a^25*b - 8*a^12*b^14
+ 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120
*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/((a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b
^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^
9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*((a + b)^7
*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10
+ 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a
^2*b^4 - 84*a^4*b^2))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^
4 - 7*a^18*b^2))))*((a + b)^7*(a - b)^7)^(1/2)*(40*a^6 - 20*b^6 + 69*a^2*b^4 - 84*a^4*b^2)*1i)/(d*(a^20 - a^6*
b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(c + d*x)**2/(a + b*sec(c + d*x))**4, x)

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