\(\int \frac {a+b \sec (d+e x)}{(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x))^{3/2}} \, dx\) [525]

Optimal result

Integrand size = 41, antiderivative size = 330 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\left (2 a^4-3 a^2 b^2+2 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right ) (b+a \sec (d+e x))^3}{(a-b)^{3/2} b^3 (a+b)^{3/2} e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{2 b^2 \left (a^2-b^2\right ) e \left (a^2 b+a^3 \sec (d+e x)\right ) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \]

[Out]

-(2*a^4-3*a^2*b^2+2*b^4)*arctan((a-b)^(1/2)*tan(1/2*e*x+1/2*d)/(a+b)^(1/2))*(b+a*sec(e*x+d))^3/(a-b)^(3/2)/b^3
/(a+b)^(3/2)/e/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2)+x*(a*b+a^2*sec(e*x+d))^3/a^2/b^3/(b^2+2*a*b*sec(e
*x+d)+a^2*sec(e*x+d)^2)^(3/2)-1/2*(a*b+a^2*sec(e*x+d))*tan(e*x+d)/b/e/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^
(3/2)-1/2*(2*a^2-3*b^2)*(a*b+a^2*sec(e*x+d))^3*tan(e*x+d)/b^2/(a^2-b^2)/e/(a^2*b+a^3*sec(e*x+d))/(b^2+2*a*b*se
c(e*x+d)+a^2*sec(e*x+d)^2)^(3/2)

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4259, 4008, 4145, 4004, 3916, 2738, 211} \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )}{2 b e \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}}+\frac {x \left (a^2 \sec (d+e x)+a b\right )^3}{a^2 b^3 \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}}-\frac {\left (2 a^4-3 a^2 b^2+2 b^4\right ) (a \sec (d+e x)+b)^3 \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b^3 e (a-b)^{3/2} (a+b)^{3/2} \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )^3}{2 b^2 e \left (a^2-b^2\right ) \left (a^3 \sec (d+e x)+a^2 b\right ) \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}} \]

[In]

Int[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2),x]

[Out]

-(((2*a^4 - 3*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]]*(b + a*Sec[d + e*x])^3)/((a
- b)^(3/2)*b^3*(a + b)^(3/2)*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))) + (x*(a*b + a^2*Sec[d +
e*x])^3)/(a^2*b^3*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)) - ((a*b + a^2*Sec[d + e*x])*Tan[d +
e*x])/(2*b*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)) - ((2*a^2 - 3*b^2)*(a*b + a^2*Sec[d + e*x]
)^3*Tan[d + e*x])/(2*b^2*(a^2 - b^2)*e*(a^2*b + a^3*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]
^2)^(3/2))

Rule 211

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

Rule 2738

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 3916

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

Rule 4004

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 4008

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 4145

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 4259

Int[((A_) + (B_.)*sec[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*sec[(d_.) + (e_.)*(x_)] + (c_.)*sec[(d_.) + (e_.)*(x_
)]^2)^(n_), x_Symbol] :> Dist[(a + b*Sec[d + e*x] + c*Sec[d + e*x]^2)^n/(b + 2*c*Sec[d + e*x])^(2*n), Int[(A +
B*Sec[d + e*x])*(b + 2*c*Sec[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0
] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a b+2 a^2 \sec (d+e x)\right )^3 \int \frac {a+b \sec (d+e x)}{\left (2 a b+2 a^2 \sec (d+e x)\right )^3} \, dx}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = -\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sec (d+e x)\right )^3 \int \frac {8 a^3 \left (a^2-b^2\right )+8 a^2 b \left (a^2-b^2\right ) \sec (d+e x)-4 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 )^2} \, dx}{16 a^3 b \left (a^2-b^2\right ) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = -\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sec (d+e x)\right )^3 \int \frac {32 a^5 \left (a^2-b^2\right )^2+16 a^4 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \sec (d+e x)}{2 a b+2 a^2 \sec (d+e x)} \, dx}{128 a^6 b^2 \left (a^2-b^2\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = \frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (\left (-32 a^5 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )+64 a^7 \left (a^2-b^2\right )^2\right ) \left (2 a b+2 a^2 \sec (d+e x)\right )^3\right ) \int \frac {\sec (d+e x)}{2 a b+2 a^2 \sec (d+e x)} \, dx}{256 a^7 b^3 \left (a^2-b^2\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = \frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (\left (-32 a^5 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )+64 a^7 \left (a^2-b^2\right )^2\right ) \left (2 a b+2 a^2 \sec (d+e x)\right )^3\right ) \int \frac {1}{1+\frac {b \cos (d+e x)}{a}} \, dx}{512 a^9 b^3 \left (a^2-b^2\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = \frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (\left (-32 a^5 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )+64 a^7 \left (a^2-b^2\right )^2\right ) \left (2 a b+2 a^2 \sec (d+e x)\right )^3\right ) \text {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 )}{256 a^9 b^3 \left (a^2-b^2\right )^2 e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = -\frac {\left (2 a^4-3 a^2 b^2+2 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right ) (b+a \sec (d+e x))^3}{(a-b)^{3/2} b^3 (a+b)^{3/2} e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.50 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\frac {(a+b \cos (d+e x)) \sec ^2(d+e x) (a+b \sec (d+e x)) \left (2 a (d+e x) (a+b \cos (d+e x))^2+\frac {2 \left (2 a^4-3 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right ) (a+b \cos (d+e x))^2}{\left (-a^2+b^2\right )^{3/2}}+a^2 b \sin (d+e x)+\frac {a b \left (3 a^2-4 b^2\right ) (a+b \cos (d+e x)) \sin (d+e x)}{(-a+b) (a+b)}\right )}{2 b^3 e (b+a \cos (d+e x)) \left ((b+a \sec (d+e x))^2\right )^{3/2}} \]

[In]

Integrate[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.60 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.05

method	result	size
risch	\(\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) a x}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, b^{3}}-\frac {i a \left (4 a^{3} b \,{\mathrm e}^{3 i \left (e x +d \right )}-5 a \,b^{3} {\mathrm e}^{3 i \left (e x +d \right )}+6 a^{4} {\mathrm e}^{2 i \left (e x +d \right )}-5 a^{2} b^{2} {\mathrm e}^{2 i \left (e x +d \right )}-4 b^{4} {\mathrm e}^{2 i \left (e x +d \right )}+8 a^{3} b \,{\mathrm e}^{i \left (e x +d \right )}-11 a \,b^{3} {\mathrm e}^{i \left (e x +d \right )}+3 a^{2} b^{2}-4 b^{4}\right )}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, b^{3} \left (a^{2}-b^{2}\right ) e}-\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) \left (2 a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) e \,b^{3}}+\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) \left (2 a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) e \,b^{3}}\)	\(678\)
default	\(-\frac {2 \sin \left (e x +d \right ) \tan \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{3} b^{2} \left (e x +d \right )+2 \sqrt {\left (a +b \right ) \left (a -b \right )}\, \cos \left (e x +d \right ) a \,b^{4} \left (e x +d \right )+4 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) \cos \left (e x +d \right ) a^{4} b^{2}-6 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) \cos \left (e x +d \right ) a^{2} b^{4}+4 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) \cos \left (e x +d \right ) b^{6}+3 \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{3} b^{2} \sin \left (e x +d \right )-4 \sqrt {\left (a +b \right ) \left (a -b \right )}\, a \,b^{4} \sin \left (e x +d \right )-4 \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (e x +d \right ) a^{4} b +4 \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (e x +d \right ) a^{2} b^{3}+8 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{5} b -12 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{3} b^{3}+8 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a \,b^{5}+2 \tan \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{4} b -3 \tan \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{2} b^{3}-2 \sec \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (e x +d \right ) a^{5}+4 \sec \left (e x +d \right ) \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{6}-6 \sec \left (e x +d \right ) \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{4} b^{2}+4 \sec \left (e x +d \right ) \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{2} b^{4}}{2 e \left (b \cos \left (e x +d \right )+a \right ) \sqrt {\sec \left (e x +d \right )^{2} \left (b \cos \left (e x +d \right )+a \right )^{2}}\, \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (a^{2}-b^{2}\right ) b^{3}}\)	\(695\)
parts	\(\text {Expression too large to display}\)	\(1210\)

[In]

int((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/(1+exp(2*I*(e*x+d)))*(b*exp(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)/((b*exp(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)^2/
(1+exp(2*I*(e*x+d)))^2)^(1/2)*a*x/b^3-I/(1+exp(2*I*(e*x+d)))/(b*exp(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)/((b*exp
(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)^2/(1+exp(2*I*(e*x+d)))^2)^(1/2)*a/b^3*(4*a^3*b*exp(3*I*(e*x+d))-5*a*b^3*ex
p(3*I*(e*x+d))+6*a^4*exp(2*I*(e*x+d))-5*a^2*b^2*exp(2*I*(e*x+d))-4*b^4*exp(2*I*(e*x+d))+8*a^3*b*exp(I*(e*x+d))
-11*a*b^3*exp(I*(e*x+d))+3*a^2*b^2-4*b^4)/(a^2-b^2)/e-1/2/(1+exp(2*I*(e*x+d)))*(b*exp(2*I*(e*x+d))+2*a*exp(I*(
e*x+d))+b)/((b*exp(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)^2/(1+exp(2*I*(e*x+d)))^2)^(1/2)/(-a^2+b^2)^(1/2)*(2*a^4-
3*a^2*b^2+2*b^4)/(a+b)/(a-b)/e/b^3*ln(exp(I*(e*x+d))+(-I*a^2+I*b^2+(-a^2+b^2)^(1/2)*a)/(-a^2+b^2)^(1/2)/b)+1/2
/(1+exp(2*I*(e*x+d)))*(b*exp(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)/((b*exp(2*I*(e*x+d))+2*a*exp(I*(e*x+d))+b)^2/(
1+exp(2*I*(e*x+d)))^2)^(1/2)/(-a^2+b^2)^(1/2)*(2*a^4-3*a^2*b^2+2*b^4)/(a+b)/(a-b)/e/b^3*ln(exp(I*(e*x+d))+(I*a
^2-I*b^2+(-a^2+b^2)^(1/2)*a)/(-a^2+b^2)^(1/2)/b)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.42 \[ \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 )^{3/2}} \, dx=\left [\frac {4 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} e x \cos \left (e x + d\right )^{2} + 8 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} e x \cos \left (e x + d\right ) + 4 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} e x + {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (e x + d\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (e x + d\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (e x + d\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (e x + d\right ) + b\right )} \sin \left (e x + d\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}}\right ) - 2 \, {\left (2 \, a^{6} b - 5 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + {\left (3 \, a^{5} b^{2} - 7 \, a^{3} b^{4} + 4 \, a b^{6}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{4 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} e \cos \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} e \cos \left (e x + d\right ) + {\left (a^{6} b^{3} - 2 \, a^{4} b^{5} + a^{2} b^{7}\right )} e\right )}}, \frac {2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} e x \cos \left (e x + d\right )^{2} + 4 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} e x \cos \left (e x + d\right ) + 2 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} e x - {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (e x + d\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (e x + d\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (e x + d\right )}\right ) - {\left (2 \, a^{6} b - 5 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + {\left (3 \, a^{5} b^{2} - 7 \, a^{3} b^{4} + 4 \, a b^{6}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} e \cos \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} e \cos \left (e x + d\right ) + {\left (a^{6} b^{3} - 2 \, a^{4} b^{5} + a^{2} b^{7}\right )} e\right )}}\right ] \]

[In]

integrate((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(4*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*e*x*cos(e*x + d)^2 + 8*(a^6*b - 2*a^4*b^3 + a^2*b^5)*e*x*cos(e*x + d) +
4*(a^7 - 2*a^5*b^2 + a^3*b^4)*e*x + (2*a^6 - 3*a^4*b^2 + 2*a^2*b^4 + (2*a^4*b^2 - 3*a^2*b^4 + 2*b^6)*cos(e*x +
d)^2 + 2*(2*a^5*b - 3*a^3*b^3 + 2*a*b^5)*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*(2*a^6*b - 5*a^4*b^3 + 3*a^2*b^5 + (3*a^5*b^2 - 7*a^3*b^4 + 4*a*b^6)*cos(e*x +
d))*sin(e*x + d))/((a^4*b^5 - 2*a^2*b^7 + b^9)*e*cos(e*x + d)^2 + 2*(a^5*b^4 - 2*a^3*b^6 + a*b^8)*e*cos(e*x +
d) + (a^6*b^3 - 2*a^4*b^5 + a^2*b^7)*e), 1/2*(2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*e*x*cos(e*x + d)^2 + 4*(a^6*b -
2*a^4*b^3 + a^2*b^5)*e*x*cos(e*x + d) + 2*(a^7 - 2*a^5*b^2 + a^3*b^4)*e*x - (2*a^6 - 3*a^4*b^2 + 2*a^2*b^4 + (
2*a^4*b^2 - 3*a^2*b^4 + 2*b^6)*cos(e*x + d)^2 + 2*(2*a^5*b - 3*a^3*b^3 + 2*a*b^5)*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))) - (2*a^6*b - 5*a^4*b^3 + 3*a^2*b^5 + (3*a^5*b^2
- 7*a^3*b^4 + 4*a*b^6)*cos(e*x + d))*sin(e*x + d))/((a^4*b^5 - 2*a^2*b^7 + b^9)*e*cos(e*x + d)^2 + 2*(a^5*b^4
- 2*a^3*b^6 + a*b^8)*e*cos(e*x + d) + (a^6*b^3 - 2*a^4*b^5 + a^2*b^7)*e)]

Sympy [F]

\[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {a + b \sec {\left (d + e x \right )}}{\left (\left (a \sec {\left (d + e x \right )} + b\right )^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*sec(e*x+d))/(b**2+2*a*b*sec(e*x+d)+a**2*sec(e*x+d)**2)**(3/2),x)

[Out]

Integral((a + b*sec(d + e*x))/((a*sec(d + e*x) + b)**2)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),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 de

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (2 \, a^{4} - 3 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {e x + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right ) - b^{5} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right )\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 3 \, a^{3} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 4 \, a b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 2 \, a^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 3 \, a^{3} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 4 \, a b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{{\left (a^{2} b^{2} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right ) - b^{4} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right )\right )} {\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + a + b\right )}^{2}} - \frac {{\left (e x + d\right )} a}{b^{3} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right )}}{e} \]

[In]

integrate((a+b*sec(e*x+d))/(b^2+2*a*b*sec(e*x+d)+a^2*sec(e*x+d)^2)^(3/2),x, algorithm="giac")

[Out]

-((2*a^4 - 3*a^2*b^2 + 2*b^4)*(pi*floor(1/2*(e*x + d)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*e*x + 1/2*d
) - b*tan(1/2*e*x + 1/2*d))/sqrt(a^2 - b^2)))/((a^2*b^3*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d)) - b^5*sgn(b*cos
(e*x + d)^2 + a*cos(e*x + d)))*sqrt(a^2 - b^2)) + (2*a^4*tan(1/2*e*x + 1/2*d)^3 - 3*a^3*b*tan(1/2*e*x + 1/2*d)
^3 - 3*a^2*b^2*tan(1/2*e*x + 1/2*d)^3 + 4*a*b^3*tan(1/2*e*x + 1/2*d)^3 + 2*a^4*tan(1/2*e*x + 1/2*d) + 3*a^3*b*
tan(1/2*e*x + 1/2*d) - 3*a^2*b^2*tan(1/2*e*x + 1/2*d) - 4*a*b^3*tan(1/2*e*x + 1/2*d))/((a^2*b^2*sgn(b*cos(e*x
+ d)^2 + a*cos(e*x + d)) - b^4*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d)))*(a*tan(1/2*e*x + 1/2*d)^2 - b*tan(1/2*e
*x + 1/2*d)^2 + a + b)^2) - (e*x + d)*a/(b^3*sgn(b*cos(e*x + d)^2 + a*cos(e*x + d))))/e

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {a+\frac {b}{\cos \left (d+e\,x\right )}}{{\left (b^2+\frac {a^2}{{\cos \left (d+e\,x\right )}^2}+\frac {2\,a\,b}{\cos \left (d+e\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((a + b/cos(d + e*x))/(b^2 + a^2/cos(d + e*x)^2 + (2*a*b)/cos(d + e*x))^(3/2),x)

[Out]

int((a + b/cos(d + e*x))/(b^2 + a^2/cos(d + e*x)^2 + (2*a*b)/cos(d + e*x))^(3/2), x)