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

Optimal. Leaf size=171 \[ -\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 d \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {2 \left (a^4 C+3 a^2 A b^2-2 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

-2*A*b*x/a^3+2*(3*A*a^2*b^2-2*A*b^4+C*a^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^3/(a-b)^(3/2)
/(a+b)^(3/2)/d-(2*A*b^2-a^2*(A-C))*sin(d*x+c)/a^2/(a^2-b^2)/d+(A*b^2+C*a^2)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(
d*x+c))

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Rubi [A]  time = 0.43, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4101, 4104, 3919, 3831, 2659, 208} \[ -\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 d \left (a^2-b^2\right )}+\frac {2 \left (3 a^2 A b^2+a^4 C-2 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {2 A b x}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b*x)/a^3 + (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*
(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b^2 - a^2*(A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)
*Sin[c + d*x])/(a*(a^2 - b^2)*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 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 4101

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*b^2 + 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)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e
+ f*x])^n*Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2
+ a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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 (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (2 A b^2-a^2 (A-C)+a b (A+C) \sec (c+d x)-\left (A b^2+a^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {-2 A b \left (a^2-b^2\right )+a \left (A b^2+a^2 C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (3 a^2 A b^2-2 A b^4+a^4 C\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )}\\ &=-\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (3 a^2 A b^2-2 A b^4+a^4 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^3 b \left (a^2-b^2\right )}\\ &=-\frac {2 A b x}{a^3}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (2 \left (3 a^2 A b^2-2 A b^4+a^4 C\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^3 b \left (a^2-b^2\right ) d}\\ &=-\frac {2 A b x}{a^3}+\frac {2 \left (3 a^2 A b^2-2 A b^4+a^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 1.10, size = 137, normalized size = 0.80 \[ \frac {-\frac {a b \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-\frac {2 \left (a^4 C+3 a^2 A b^2-2 A b^4\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+a A \sin (c+d x)-2 A b (c+d x)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b*(c + d*x) - (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(
a^2 - b^2)^(3/2) + a*A*Sin[c + d*x] - (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])
))/(a^3*d)

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fricas [A]  time = 0.53, size = 642, normalized size = 3.75 \[ \left [-\frac {4 \, {\left (A a^{5} b - 2 \, A a^{3} b^{3} + A a b^{5}\right )} d x \cos \left (d x + c\right ) + 4 \, {\left (A a^{4} b^{2} - 2 \, A a^{2} b^{4} + A b^{6}\right )} d x + {\left (C a^{4} b + 3 \, A a^{2} b^{3} - 2 \, A b^{5} + {\left (C a^{5} + 3 \, A a^{3} b^{2} - 2 \, A a b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left ({\left (A - C\right )} a^{5} b - {\left (3 \, A - C\right )} a^{3} b^{3} + 2 \, A a b^{5} + {\left (A a^{6} - 2 \, A a^{4} b^{2} + A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d\right )}}, -\frac {2 \, {\left (A a^{5} b - 2 \, A a^{3} b^{3} + A a b^{5}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (A a^{4} b^{2} - 2 \, A a^{2} b^{4} + A b^{6}\right )} d x - {\left (C a^{4} b + 3 \, A a^{2} b^{3} - 2 \, A b^{5} + {\left (C a^{5} + 3 \, A a^{3} b^{2} - 2 \, A a b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left ({\left (A - C\right )} a^{5} b - {\left (3 \, A - C\right )} a^{3} b^{3} + 2 \, A a b^{5} + {\left (A a^{6} - 2 \, A a^{4} b^{2} + A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(4*(A*a^5*b - 2*A*a^3*b^3 + A*a*b^5)*d*x*cos(d*x + c) + 4*(A*a^4*b^2 - 2*A*a^2*b^4 + A*b^6)*d*x + (C*a^4
*b + 3*A*a^2*b^3 - 2*A*b^5 + (C*a^5 + 3*A*a^3*b^2 - 2*A*a*b^4)*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*((A - C)*a^5*b - (3*A - C)*a^3*b^3 + 2*A*a*b^5 + (A*a^6 - 2*
A*a^4*b^2 + A*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*d*cos(d*x + c) + (a^7*b - 2*a^
5*b^3 + a^3*b^5)*d), -(2*(A*a^5*b - 2*A*a^3*b^3 + A*a*b^5)*d*x*cos(d*x + c) + 2*(A*a^4*b^2 - 2*A*a^2*b^4 + A*b
^6)*d*x - (C*a^4*b + 3*A*a^2*b^3 - 2*A*b^5 + (C*a^5 + 3*A*a^3*b^2 - 2*A*a*b^4)*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))) - ((A - C)*a^5*b - (3*A - C)*a^3*b^3
+ 2*A*a*b^5 + (A*a^6 - 2*A*a^4*b^2 + A*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*d*co
s(d*x + c) + (a^7*b - 2*a^5*b^3 + a^3*b^5)*d)]

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giac [B]  time = 0.45, size = 988, normalized size = 5.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((C*a^8 - 2*A*a^7*b + 5*A*a^6*b^2 - C*a^6*b^2 + 4*A*a^5*b^3 - 9*A*a^4*b^4 - 2*A*a^3*b^5 + 4*A*a^2*b^6 + C*a^3*
abs(-a^5 + a^3*b^2) + 2*A*a^2*b*abs(-a^5 + a^3*b^2) + A*a*b^2*abs(-a^5 + a^3*b^2) - 2*A*b^3*abs(-a^5 + a^3*b^2
))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^4*b - a^2*b^3 + sqrt((a^5 + a^4*b
- a^3*b^2 - a^2*b^3)*(a^5 - a^4*b - a^3*b^2 + a^2*b^3) + (a^4*b - a^2*b^3)^2))/(a^5 - a^4*b - a^3*b^2 + a^2*b^
3))))/(a^4*b*abs(-a^5 + a^3*b^2) - a^2*b^3*abs(-a^5 + a^3*b^2) + (a^5 - a^3*b^2)^2) - (sqrt(-a^2 + b^2)*C*a^3*
abs(-a^5 + a^3*b^2)*abs(-a + b) + (2*a^2*b + a*b^2 - 2*b^3)*sqrt(-a^2 + b^2)*A*abs(-a^5 + a^3*b^2)*abs(-a + b)
+ (2*a^7*b - 5*a^6*b^2 - 4*a^5*b^3 + 9*a^4*b^4 + 2*a^3*b^5 - 4*a^2*b^6)*sqrt(-a^2 + b^2)*A*abs(-a + b) - (a^8
- a^6*b^2)*sqrt(-a^2 + b^2)*C*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sq
rt(-(a^4*b - a^2*b^3 - sqrt((a^5 + a^4*b - a^3*b^2 - a^2*b^3)*(a^5 - a^4*b - a^3*b^2 + a^2*b^3) + (a^4*b - a^2
*b^3)^2))/(a^5 - a^4*b - a^3*b^2 + a^2*b^3))))/((a^5 - a^3*b^2)^2*(a^2 - 2*a*b + b^2) - (a^6*b - 2*a^5*b^2 + 2
*a^3*b^4 - a^2*b^5)*abs(-a^5 + a^3*b^2)) + 2*(A*a^3*tan(1/2*d*x + 1/2*c)^3 - A*a^2*b*tan(1/2*d*x + 1/2*c)^3 +
C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^3*tan(1/2*d*x + 1/2*c)^3 - A*a^3*tan(1
/2*d*x + 1/2*c) - A*a^2*b*tan(1/2*d*x + 1/2*c) + C*a^2*b*tan(1/2*d*x + 1/2*c) + A*a*b^2*tan(1/2*d*x + 1/2*c) +
2*A*b^3*tan(1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*b*tan(1/2*d*x + 1/2*c
)^2 - a - b)*(a^4 - a^2*b^2)))/d

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maple [B]  time = 0.90, size = 367, normalized size = 2.15 \[ \frac {2 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{d \,a^{2} \left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{d \left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}+\frac {6 \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) A \,b^{2}}{d a \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {4 \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) A \,b^{4}}{d \,a^{3} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 a \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) C}{d \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4 A b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^2*b^3/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)*A+2/d*b/(a^2-b^2)
*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)*C+6/d/a/(a-b)/(a+b)/((a-b)*(a+b))^(1/2
)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^2-4/d/a^3/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(
tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^4+2/d*a/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+
1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C+2/d*A/a^2*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-4/d*A/a^3*b*arctan(t
an(1/2*d*x+1/2*c))

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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)*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,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 = 11.67, size = 4113, normalized size = 24.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*tan(c/2 + (d*x)/2)^3*(A*a^3 + 2*A*b^3 - A*a*b^2 - A*a^2*b + C*a^2*b))/(a^2*(a + b)*(a - b)) + (2*tan(c/2
+ (d*x)/2)*(2*A*b^3 - A*a^3 + A*a*b^2 - A*a^2*b + C*a^2*b))/(a^2*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x
)/2)^4*(a - b) + 2*b*tan(c/2 + (d*x)/2)^2)) - (4*A*b*atan(((2*A*b*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8
- 8*A^2*a*b^7 - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b
^4 + 6*A*C*a^6*b^2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (A*b*((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8
*b^4 + 3*A*a^9*b^3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 -
a^7*b^2) - (A*b*tan(c/2 + (d*x)/2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2)*64i
)/(a^3*(a^6*b + a^7 - a^4*b^3 - a^5*b^2)))*2i)/a^3))/a^3 + (2*A*b*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8
- 8*A^2*a*b^7 - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b
^4 + 6*A*C*a^6*b^2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - (A*b*((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8
*b^4 + 3*A*a^9*b^3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 -
a^7*b^2) + (A*b*tan(c/2 + (d*x)/2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2)*64i
)/(a^3*(a^6*b + a^7 - a^4*b^3 - a^5*b^2)))*2i)/a^3))/a^3)/((64*(8*A^3*b^8 - 4*A^3*a*b^7 - 20*A^3*a^2*b^6 + 6*A
^3*a^3*b^5 + 12*A^3*a^4*b^4 + 2*A*C^2*a^7*b - 4*A^2*C*a^3*b^5 - 4*A^2*C*a^4*b^4 + 8*A^2*C*a^5*b^3 + 4*A^2*C*a^
6*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (A*b*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8 - 8*A^2*a*b^7 -
16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^
2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (A*b*((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8*b^4 + 3*A*a^9*b^
3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (A*b*t
an(c/2 + (d*x)/2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2)*64i)/(a^3*(a^6*b + a
^7 - a^4*b^3 - a^5*b^2)))*2i)/a^3)*2i)/a^3 - (A*b*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8 - 8*A^2*a*b^7 -
16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b
^2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - (A*b*((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8*b^4 + 3*A*a^9*b
^3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (A*b*
tan(c/2 + (d*x)/2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2)*64i)/(a^3*(a^6*b +
a^7 - a^4*b^3 - a^5*b^2)))*2i)/a^3)*2i)/a^3)))/(a^3*d) - (atan(((((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8
- 8*A^2*a*b^7 - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b^
4 + 6*A*C*a^6*b^2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8*b^4
+ 3*A*a^9*b^3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b
^2) - (32*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2)*(2*a^11*b - 2*a^6*b^6
+ 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*
b^4 - 3*a^7*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3
*a^7*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2)*1i)/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*
b^2) + (((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8 - 8*A^2*a*b^7 - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a
^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - (
((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8*b^4 + 3*A*a^9*b^3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2
*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (32*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*
(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6
*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2
*A*b^4 + 3*A*a^2*b^2))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 +
3*A*a^2*b^2)*1i)/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))/((64*(8*A^3*b^8 - 4*A^3*a*b^7 - 20*A^3*a^2*b^6 + 6*
A^3*a^3*b^5 + 12*A^3*a^4*b^4 + 2*A*C^2*a^7*b - 4*A^2*C*a^3*b^5 - 4*A^2*C*a^4*b^4 + 8*A^2*C*a^5*b^3 + 4*A^2*C*a
^6*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + C^2*a^8 - 8*A^2*a*b^7 - 16*
A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2))
/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (((32*(C*a^12 + 2*A*a^6*b^6 - A*a^7*b^5 - 5*A*a^8*b^4 + 3*A*a^9*b^3 + 3*A
*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (32*tan(c/2 +
(d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^
8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*
((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*((a + b)
^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2) - (((32*tan(c/2 +
(d*x)/2)*(8*A^2*b^8 + C^2*a^8 - 8*A^2*a*b^7 - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3
+ 4*A^2*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - (((32*(C*a^12 + 2*A*a^6
*b^6 - A*a^7*b^5 - 5*A*a^8*b^4 + 3*A*a^9*b^3 + 3*A*a^10*b^2 + C*a^9*b^3 - C*a^10*b^2 - 2*A*a^11*b - C*a^11*b))
/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (32*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A
*a^2*b^2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a
^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2))
/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2))/(a^9 -
a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2)*2i)/(d*(a^9 - a
^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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