3.521 \(\int \frac {a+b \sec (d+e x)}{(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x))^2} \, dx\)
Optimal. Leaf size=230 \[ -\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 e \left (a^2-b^2\right ) (a \sec (d+e x)+b)^2}-\frac {\left (a^2-2 b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b^4 e (a-b)^{5/2} (a+b)^{5/2}}-\frac {a \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 e \left (a^2-b^2\right )^2 (a \sec (d+e x)+b)}-\frac {a^4 \tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}+\frac {a x}{b^4} \]
[Out]
a*x/b^4-(a^2-2*b^2)*(2*a^4-a^2*b^2+b^4)*arctan((a-b)^(1/2)*tan(1/2*e*x+1/2*d)/(a+b)^(1/2))/(a-b)^(5/2)/b^4/(a+
b)^(5/2)/e-1/6*a*(3*a^2-5*b^2)*tan(e*x+d)/b^2/(a^2-b^2)/e/(b+a*sec(e*x+d))^2-1/6*a*(6*a^4-11*a^2*b^2+11*b^4)*t
an(e*x+d)/b^3/(a^2-b^2)^2/e/(b+a*sec(e*x+d))-1/3*a^4*tan(e*x+d)/b/e/(a*b+a^2*sec(e*x+d))^3
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Rubi [A] time = 0.83, antiderivative size = 230, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 7, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, Rules used =
{4172, 3923, 4060, 3919, 3831, 2659, 205} \[ -\frac {\left (a^2-2 b^2\right ) \left (-a^2 b^2+2 a^4+b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b^4 e (a-b)^{5/2} (a+b)^{5/2}}-\frac {a \left (-11 a^2 b^2+6 a^4+11 b^4\right ) \tan (d+e x)}{6 b^3 e \left (a^2-b^2\right )^2 (a \sec (d+e x)+b)}-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 e \left (a^2-b^2\right ) (a \sec (d+e x)+b)^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a^2 \sec (d+e x)+a b\right )^3}+\frac {a x}{b^4} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^2,x]
[Out]
(a*x)/b^4 - ((a^2 - 2*b^2)*(2*a^4 - a^2*b^2 + b^4)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/((a - b
)^(5/2)*b^4*(a + b)^(5/2)*e) - (a*(3*a^2 - 5*b^2)*Tan[d + e*x])/(6*b^2*(a^2 - b^2)*e*(b + a*Sec[d + e*x])^2) -
(a*(6*a^4 - 11*a^2*b^2 + 11*b^4)*Tan[d + e*x])/(6*b^3*(a^2 - b^2)^2*e*(b + a*Sec[d + e*x])) - (a^4*Tan[d + e*
x])/(3*b*e*(a*b + a^2*Sec[d + e*x])^3)
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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 3923
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(b*(
b*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(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)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b
*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m,
-1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
Rule 4060
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(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))/(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)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Rule 4172
Int[((A_) + (B_.)*sec[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*sec[(d_.) + (e_.)*(x_)] + (c_.)*sec[(d_.) + (e_.)*(x_
)]^2)^(n_), x_Symbol] :> Dist[1/(4^n*c^n), Int[(A + B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x], x] /; Fr
eeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
Rubi steps
\begin {align*} \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2} \, dx &=\left (16 a^4\right ) \int \frac {a+b \sec (d+e x)}{\left (2 a b+2 a^2 \sec (d+e x)\right )^4} \, dx\\ &=-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}+\frac {(2 a) \int \frac {12 a^3 \left (a^2-b^2\right )+12 a^2 b \left (a^2-b^2\right ) \sec (d+e x)-8 a^3 \left (a^2-b^2\right ) \sec ^2(d+e x)}{\left (2 a b+2 a^2 \sec (d+e x)\right )^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}+\frac {\int \frac {96 a^5 \left (a^2-b^2\right )^2+32 a^4 b \left (a^2-3 b^2\right ) \left (a^2-b^2\right ) \sec (d+e x)-16 a^5 \left (3 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sec ^2(d+e x)}{\left (2 a b+2 a^2 \sec (d+e x)\right )^2} \, dx}{24 a^2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}-\frac {a^2 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 \left (a^2-b^2\right )^2 e \left (a b+a^2 \sec (d+e x)\right )}+\frac {\int \frac {384 a^7 \left (a^2-b^2\right )^3+192 a^6 b \left (a^6-2 a^4 b^2+3 a^2 b^4-2 b^6\right ) \sec (d+e x)}{2 a b+2 a^2 \sec (d+e x)} \, dx}{192 a^5 b^3 \left (a^2-b^2\right )^3}\\ &=\frac {a x}{b^4}-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}-\frac {a^2 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 \left (a^2-b^2\right )^2 e \left (a b+a^2 \sec (d+e x)\right )}-\frac {\left (a \left (a^2-2 b^2\right ) \left (2 a^4-a^2 b^2+b^4\right )\right ) \int \frac {\sec (d+e x)}{2 a b+2 a^2 \sec (d+e x)} \, dx}{b^4 \left (a^2-b^2\right )^2}\\ &=\frac {a x}{b^4}-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}-\frac {a^2 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 \left (a^2-b^2\right )^2 e \left (a b+a^2 \sec (d+e x)\right )}-\frac {\left (\left (a^2-2 b^2\right ) \left (2 a^4-a^2 b^2+b^4\right )\right ) \int \frac {1}{1+\frac {b \cos (d+e x)}{a}} \, dx}{2 a b^4 \left (a^2-b^2\right )^2}\\ &=\frac {a x}{b^4}-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}-\frac {a^2 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 \left (a^2-b^2\right )^2 e \left (a b+a^2 \sec (d+e x)\right )}-\frac {\left (\left (a^2-2 b^2\right ) \left (2 a^4-a^2 b^2+b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {b}{a}+\left (1-\frac {b}{a}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{a b^4 \left (a^2-b^2\right )^2 e}\\ &=\frac {a x}{b^4}-\frac {\left (a^2-2 b^2\right ) \left (2 a^4-a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} e}-\frac {a \left (3 a^2-5 b^2\right ) \tan (d+e x)}{6 b^2 \left (a^2-b^2\right ) e (b+a \sec (d+e x))^2}-\frac {a^4 \tan (d+e x)}{3 b e \left (a b+a^2 \sec (d+e x)\right )^3}-\frac {a^2 \left (6 a^4-11 a^2 b^2+11 b^4\right ) \tan (d+e x)}{6 b^3 \left (a^2-b^2\right )^2 e \left (a b+a^2 \sec (d+e x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.53, size = 276, normalized size = 1.20 \[ \frac {\sec ^3(d+e x) (a+b \cos (d+e x)) (a+b \sec (d+e x)) \left (-2 a^3 b \sin (d+e x)+\frac {a^2 b \left (7 a^2-9 b^2\right ) \sin (d+e x) (a+b \cos (d+e x))}{(a-b) (a+b)}-\frac {a b \left (11 a^4-23 a^2 b^2+18 b^4\right ) \sin (d+e x) (a+b \cos (d+e x))^2}{(a-b)^2 (a+b)^2}+\frac {6 \left (-2 a^6+5 a^4 b^2-3 a^2 b^4+2 b^6\right ) (a+b \cos (d+e x))^3 \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}+6 a (d+e x) (a+b \cos (d+e x))^3\right )}{6 b^4 e (a \cos (d+e x)+b) (a \sec (d+e x)+b)^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^2,x]
[Out]
((a + b*Cos[d + e*x])*Sec[d + e*x]^3*(a + b*Sec[d + e*x])*(6*a*(d + e*x)*(a + b*Cos[d + e*x])^3 + (6*(-2*a^6 +
5*a^4*b^2 - 3*a^2*b^4 + 2*b^6)*ArcTanh[((-a + b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]]*(a + b*Cos[d + e*x])^3)/
(-a^2 + b^2)^(5/2) - 2*a^3*b*Sin[d + e*x] + (a^2*b*(7*a^2 - 9*b^2)*(a + b*Cos[d + e*x])*Sin[d + e*x])/((a - b)
*(a + b)) - (a*b*(11*a^4 - 23*a^2*b^2 + 18*b^4)*(a + b*Cos[d + e*x])^2*Sin[d + e*x])/((a - b)^2*(a + b)^2)))/(
6*b^4*e*(b + a*Cos[d + e*x])*(b + a*Sec[d + e*x])^4)
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fricas [B] time = 2.11, size = 1335, normalized size = 5.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^2,x, algorithm="fricas")
[Out]
[1/12*(12*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*e*x*cos(e*x + d)^3 + 36*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 -
a^2*b^8)*e*x*cos(e*x + d)^2 + 36*(a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*e*x*cos(e*x + d) + 12*(a^10 - 3*a^
8*b^2 + 3*a^6*b^4 - a^4*b^6)*e*x + 3*(2*a^9 - 5*a^7*b^2 + 3*a^5*b^4 - 2*a^3*b^6 + (2*a^6*b^3 - 5*a^4*b^5 + 3*a
^2*b^7 - 2*b^9)*cos(e*x + d)^3 + 3*(2*a^7*b^2 - 5*a^5*b^4 + 3*a^3*b^6 - 2*a*b^8)*cos(e*x + d)^2 + 3*(2*a^8*b -
5*a^6*b^3 + 3*a^4*b^5 - 2*a^2*b^7)*cos(e*x + d))*sqrt(-a^2 + b^2)*log((2*a*b*cos(e*x + d) + (2*a^2 - b^2)*cos
(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e*x + d) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*
cos(e*x + d) + a^2)) - 2*(6*a^9*b - 17*a^7*b^3 + 22*a^5*b^5 - 11*a^3*b^7 + (11*a^7*b^3 - 34*a^5*b^5 + 41*a^3*b
^7 - 18*a*b^9)*cos(e*x + d)^2 + 3*(5*a^8*b^2 - 15*a^6*b^4 + 19*a^4*b^6 - 9*a^2*b^8)*cos(e*x + d))*sin(e*x + d)
)/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*e*cos(e*x + d)^3 + 3*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*
e*cos(e*x + d)^2 + 3*(a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*e*cos(e*x + d) + (a^9*b^4 - 3*a^7*b^6 + 3*a^
5*b^8 - a^3*b^10)*e), 1/6*(6*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*e*x*cos(e*x + d)^3 + 18*(a^8*b^2 - 3*a^
6*b^4 + 3*a^4*b^6 - a^2*b^8)*e*x*cos(e*x + d)^2 + 18*(a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*e*x*cos(e*x + d
) + 6*(a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*e*x - 3*(2*a^9 - 5*a^7*b^2 + 3*a^5*b^4 - 2*a^3*b^6 + (2*a^6*b^3
- 5*a^4*b^5 + 3*a^2*b^7 - 2*b^9)*cos(e*x + d)^3 + 3*(2*a^7*b^2 - 5*a^5*b^4 + 3*a^3*b^6 - 2*a*b^8)*cos(e*x + d
)^2 + 3*(2*a^8*b - 5*a^6*b^3 + 3*a^4*b^5 - 2*a^2*b^7)*cos(e*x + d))*sqrt(a^2 - b^2)*arctan(-(a*cos(e*x + d) +
b)/(sqrt(a^2 - b^2)*sin(e*x + d))) - (6*a^9*b - 17*a^7*b^3 + 22*a^5*b^5 - 11*a^3*b^7 + (11*a^7*b^3 - 34*a^5*b^
5 + 41*a^3*b^7 - 18*a*b^9)*cos(e*x + d)^2 + 3*(5*a^8*b^2 - 15*a^6*b^4 + 19*a^4*b^6 - 9*a^2*b^8)*cos(e*x + d))*
sin(e*x + d))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*e*cos(e*x + d)^3 + 3*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^1
0 - a*b^12)*e*cos(e*x + d)^2 + 3*(a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*e*cos(e*x + d) + (a^9*b^4 - 3*a^
7*b^6 + 3*a^5*b^8 - a^3*b^10)*e)]
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giac [B] time = 0.49, size = 490, normalized size = 2.13 \[ \frac {1}{3} \, {\left (\frac {3 \, {\left (2 \, a^{6} - 5 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {x e + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, {\left (x e + d\right )} a}{b^{4}} - \frac {6 \, a^{7} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 15 \, a^{6} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 30 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 12 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 27 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 18 \, a b^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 12 \, a^{7} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 44 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 68 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 36 \, a b^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 6 \, a^{7} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 15 \, a^{6} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 30 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 12 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 27 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 18 \, a b^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + a + b\right )}^{3}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^2,x, algorithm="giac")
[Out]
1/3*(3*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*b^6)*(pi*floor(1/2*(x*e + d)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*
tan(1/2*x*e + 1/2*d) - b*tan(1/2*x*e + 1/2*d))/sqrt(a^2 - b^2)))/((a^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2))
+ 3*(x*e + d)*a/b^4 - (6*a^7*tan(1/2*x*e + 1/2*d)^5 - 15*a^6*b*tan(1/2*x*e + 1/2*d)^5 + 30*a^4*b^3*tan(1/2*x*
e + 1/2*d)^5 - 12*a^3*b^4*tan(1/2*x*e + 1/2*d)^5 - 27*a^2*b^5*tan(1/2*x*e + 1/2*d)^5 + 18*a*b^6*tan(1/2*x*e +
1/2*d)^5 + 12*a^7*tan(1/2*x*e + 1/2*d)^3 - 44*a^5*b^2*tan(1/2*x*e + 1/2*d)^3 + 68*a^3*b^4*tan(1/2*x*e + 1/2*d)
^3 - 36*a*b^6*tan(1/2*x*e + 1/2*d)^3 + 6*a^7*tan(1/2*x*e + 1/2*d) + 15*a^6*b*tan(1/2*x*e + 1/2*d) - 30*a^4*b^3
*tan(1/2*x*e + 1/2*d) - 12*a^3*b^4*tan(1/2*x*e + 1/2*d) + 27*a^2*b^5*tan(1/2*x*e + 1/2*d) + 18*a*b^6*tan(1/2*x
*e + 1/2*d))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*x*e + 1/2*d)^2 - b*tan(1/2*x*e + 1/2*d)^2 + a + b)^3))*e^
(-1)
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maple [B] time = 0.35, size = 1118, normalized size = 4.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^2,x)
[Out]
-2/e/b^3/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^5/(a^2+2*a*b+b^2)*tan(1/2*d+1/2*e*x)^5+1/e/b^
2/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^4/(a^2+2*a*b+b^2)*tan(1/2*d+1/2*e*x)^5+4/e/b/(a*tan(
1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^3/(a^2+2*a*b+b^2)*tan(1/2*d+1/2*e*x)^5-3/e/(a*tan(1/2*d+1/2*e
*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^2/(a^2+2*a*b+b^2)*tan(1/2*d+1/2*e*x)^5-6/e*b/(a*tan(1/2*d+1/2*e*x)^2-b*t
an(1/2*d+1/2*e*x)^2+a+b)^3*a/(a^2+2*a*b+b^2)*tan(1/2*d+1/2*e*x)^5-4/e/b^3/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+
1/2*e*x)^2+a+b)^3*a^5/(a+b)/(a-b)*tan(1/2*d+1/2*e*x)^3+32/3/e/b/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2
+a+b)^3*a^3/(a+b)/(a-b)*tan(1/2*d+1/2*e*x)^3-12/e*b/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a/(a
+b)/(a-b)*tan(1/2*d+1/2*e*x)^3-2/e/b^3/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^5/(a^2-2*a*b+b^
2)*tan(1/2*d+1/2*e*x)-1/e/b^2/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^4/(a^2-2*a*b+b^2)*tan(1/
2*d+1/2*e*x)+4/e/b/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^3/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x
)+3/e/(a*tan(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a^2/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)-6/e*b/(a*ta
n(1/2*d+1/2*e*x)^2-b*tan(1/2*d+1/2*e*x)^2+a+b)^3*a/(a^2-2*a*b+b^2)*tan(1/2*d+1/2*e*x)-2/e/b^4/(a^4-2*a^2*b^2+b
^4)/((a+b)*(a-b))^(1/2)*arctan(tan(1/2*d+1/2*e*x)*(a-b)/((a+b)*(a-b))^(1/2))*a^6+5/e/b^2/(a^4-2*a^2*b^2+b^4)/(
(a+b)*(a-b))^(1/2)*arctan(tan(1/2*d+1/2*e*x)*(a-b)/((a+b)*(a-b))^(1/2))*a^4-3/e/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-
b))^(1/2)*arctan(tan(1/2*d+1/2*e*x)*(a-b)/((a+b)*(a-b))^(1/2))*a^2+2/e*b^2/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(
1/2)*arctan(tan(1/2*d+1/2*e*x)*(a-b)/((a+b)*(a-b))^(1/2))+2/e*a/b^4*arctan(tan(1/2*d+1/2*e*x))
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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((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^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*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?
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mupad [B] time = 11.47, size = 5469, normalized size = 23.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b/cos(d + e*x))/(b^2 + a^2/cos(d + e*x)^2 + (2*a*b)/cos(d + e*x))^2,x)
[Out]
- ((tan(d/2 + (e*x)/2)*(6*a*b^4 + a^4*b + 2*a^5 - 3*a^2*b^3 - 4*a^3*b^2))/(b^5 - 2*a*b^4 + a^2*b^3) + (tan(d/2
+ (e*x)/2)^5*(6*a*b^4 - a^4*b + 2*a^5 + 3*a^2*b^3 - 4*a^3*b^2))/(b^3*(a + b)^2) + (4*tan(d/2 + (e*x)/2)^3*(9*
a*b^4 + 3*a^5 - 8*a^3*b^2))/(3*(a*b^3 - b^4)*(a + b)))/(e*(3*a*b^2 - tan(d/2 + (e*x)/2)^4*(3*a*b^2 + 3*a^2*b -
3*a^3 - 3*b^3) - tan(d/2 + (e*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) + 3*a^2*b + a^3 + b^3 + tan(d/2 + (
e*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) - (2*a*atan(((a*((a*((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^
4*b^14 + 14*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a
^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (a*tan(d/2 + (e*x)/2)*(8*a*b^17 - 8*a^2
*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10
*b^8)*8i)/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*1i)/b^4
- (8*tan(d/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a
^6*b^6 + 48*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4
*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))/b^4 - (a*((a*((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14
*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^14 - 3
*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (a*tan(d/2 + (e*x)/2)*(8*a*b^17 - 8*a^2*b^16 - 32*
a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8)*8i)/(
b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*1i)/b^4 + (8*tan(d
/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6 + 48
*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^
5*b^8 - a^6*b^7 - a^7*b^6)))/b^4)/((a*((a*((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14*a^5*b^13 -
30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 +
3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (a*tan(d/2 + (e*x)/2)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 +
32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8)*8i)/(b^4*(a*b^12
+ b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*1i)/b^4 - (8*tan(d/2 + (e*x)/
2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6 + 48*a^7*b^5 -
45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6
*b^7 - a^7*b^6))*1i)/b^4 - (16*(4*a*b^11 - 2*a^11*b + 4*a^12 - 4*a^2*b^10 - 8*a^3*b^9 + 14*a^4*b^8 + 15*a^5*b^
7 - 26*a^6*b^6 - 12*a^7*b^5 + 30*a^8*b^4 + 7*a^9*b^3 - 18*a^10*b^2))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13
+ 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (a*((a*((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14
+ 14*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^14
- 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (a*tan(d/2 + (e*x)/2)*(8*a*b^17 - 8*a^2*b^16 -
32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8)*8
i)/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*1i)/b^4 + (8*t
an(d/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6
+ 48*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 +
3*a^5*b^8 - a^6*b^7 - a^7*b^6))*1i)/b^4)))/(b^4*e) - (atan((((a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*((8*ta
n(d/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6 +
48*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3
*a^5*b^8 - a^6*b^7 - a^7*b^6) - (((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14*a^5*b^13 - 30*a^6*b
^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^
12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (4*tan(d/2 + (e*x)/2)*(a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^
4 + b^4 - a^2*b^2)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^1
1 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b
^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(a^2 - 2*b^2)*(-(a
+ b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6
- a^10*b^4)))*(2*a^4 + b^4 - a^2*b^2)*1i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10
*b^4)) + ((a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(d/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^
2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6 + 48*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2
))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (((8*(4*b^18 - 14*a
^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^1
0*b^8))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (4*tan(d/2
+ (e*x)/2)*(a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*
b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 -
5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^
4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2))/(
2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(2*a^4 + b^4 - a^2*b^2)*1i)/(2*(b^14
- 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))/((16*(4*a*b^11 - 2*a^11*b + 4*a^12 - 4*a^2*
b^10 - 8*a^3*b^9 + 14*a^4*b^8 + 15*a^5*b^7 - 26*a^6*b^6 - 12*a^7*b^5 + 30*a^8*b^4 + 7*a^9*b^3 - 18*a^10*b^2))/
(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + ((a^2 - 2*b^2)*(-(a
+ b)^5*(a - b)^5)^(1/2)*((8*tan(d/2 + (e*x)/2)*(8*a^11*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*
b^8 - 32*a^5*b^7 + 30*a^6*b^6 + 48*a^7*b^5 - 45*a^8*b^4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^
11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^
4*b^14 + 14*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a
^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (4*tan(d/2 + (e*x)/2)*(a^2 - 2*b^2)*(-(
a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*
b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 -
10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7
- a^7*b^6)))*(a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^
4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(2*a^4 + b^4 - a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 1
0*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)) - ((a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(d/2 + (e*x)/2)*(8*a^1
1*b - 8*a^12 - 4*b^12 + 8*a^2*b^10 + 8*a^3*b^9 - 17*a^4*b^8 - 32*a^5*b^7 + 30*a^6*b^6 + 48*a^7*b^5 - 45*a^8*b^
4 - 32*a^9*b^3 + 32*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^
7*b^6) + (((8*(4*b^18 - 14*a^2*b^16 - 6*a^3*b^15 + 26*a^4*b^14 + 14*a^5*b^13 - 30*a^6*b^12 - 10*a^7*b^11 + 18*
a^8*b^10 + 2*a^9*b^9 - 4*a^10*b^8))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b
^10 - a^7*b^9) + (4*tan(d/2 + (e*x)/2)*(a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2)*(8*a
*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9
*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13 - 3*
a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(a^2 - 2*b^2)*(-(a + b)^5*(a - b)^5)^(1/2
)*(2*a^4 + b^4 - a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(2*a^4 +
b^4 - a^2*b^2))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))))*(a^2 - 2*b^2)*(-(
a + b)^5*(a - b)^5)^(1/2)*(2*a^4 + b^4 - a^2*b^2)*1i)/(e*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8
*b^6 - a^10*b^4))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \sec {\left (d + e x \right )}}{\left (a \sec {\left (d + e x \right )} + b\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(e*x+d))/(b**2+2*a*b*sec(e*x+d)+a**2*sec(e*x+d)**2)**2,x)
[Out]
Integral((a + b*sec(d + e*x))/(a*sec(d + e*x) + b)**4, x)
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