\(\int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx\) [708]

Optimal result

Integrand size = 30, antiderivative size = 162 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {x}{a^2}-\frac {2 b \left (4 a^4-2 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

x/a^2-2*b*(4*a^4-2*a^2*b^2+b^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^2/(a-b)^(5/2)/(a+b)^(5/2
)/d+b^2*tan(d*x+c)/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+b^2*(4*a^2-b^2)*tan(d*x+c)/a/(a^2-b^2)^2/d/(a+b*sec(d*x+c))

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {4127, 4008, 4145, 4004, 3916, 2738, 214} \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {x}{a^2}-\frac {2 b \left (4 a^4-2 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{5/2} (a+b)^{5/2}} \]

[In]

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

[Out]

x/a^2 - (2*b*(4*a^4 - 2*a^2*b^2 + b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)
*(a + b)^(5/2)*d) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(4*a^2 - b^2)*Tan[c + d*x
])/(a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 4127

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Dist[
C/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[-a + b*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x
] && EqQ[A*b^2 + a^2*C, 0]

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]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {-a+b \sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx \\ & = \frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\int \frac {2 a \left (a^2-b^2\right )-4 a^2 b \sec (c+d x)+2 a b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )} \\ & = \frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {-2 a \left (a^2-b^2\right )^2+6 a^4 b \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {x}{a^2}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b \left (4 a^4-2 a^2 b^2+b^4\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^2} \\ & = \frac {x}{a^2}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (4 a^4-2 a^2 b^2+b^4\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^2 \left (a^2-b^2\right )^2} \\ & = \frac {x}{a^2}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (2 \left (4 a^4-2 a^2 b^2+b^4\right )\right ) \text {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^2 \left (a^2-b^2\right )^2 d} \\ & = \frac {x}{a^2}-\frac {2 b \left (4 a^4-2 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (4 a^2-b^2\right ) \tan (c+d x)}{a \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.38 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^2(c+d x) (a-b \sec (c+d x)) \left ((c+d x) (b+a \cos (c+d x))^2+\frac {2 b \left (4 a^4-2 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2}{\left (a^2-b^2\right )^{5/2}}+\frac {a b^3 \sin (c+d x)}{(-a+b) (a+b)}+\frac {a b^2 \left (5 a^2-2 b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^2 (a+b)^2}\right )}{a^2 d (-b+a \cos (c+d x)) (a+b \sec (c+d x))^3} \]

[In]

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

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(a - b*Sec[c + d*x])*((c + d*x)*(b + a*Cos[c + d*x])^2 + (2*b*(4*a^4 - 2*
a^2*b^2 + b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2)
+ (a*b^3*Sin[c + d*x])/((-a + b)*(a + b)) + (a*b^2*(5*a^2 - 2*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)
^2*(a + b)^2)))/(a^2*d*(-b + a*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {-\frac {\left (5 a^{2}+a b -b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (5 a^{2}-a b -b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (4 a^{4}-2 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\)	\(234\)
default	\(\frac {\frac {2 b \left (\frac {-\frac {\left (5 a^{2}+a b -b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (5 a^{2}-a b -b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (4 a^{4}-2 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{d}\)	\(234\)
risch	\(\frac {x}{a^{2}}-\frac {2 i b^{2} \left (-6 a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+3 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-5 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-8 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+4 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-14 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+5 b^{3} {\mathrm e}^{i \left (d x +c \right )} a -5 a^{4}+2 a^{2} b^{2}\right )}{a^{2} \left (-a^{2}+b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {4 b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {2 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{2}}-\frac {4 b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{2}}\)	\(681\)

[In]

int((a^2-b^2*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*b/a^2*((-(5*a^2+a*b-b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+(5*a^2-a*b-b^2)*a*b/(a+b)/(a^2-
2*a*b+b^2)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-(4*a^4-2*a^2*b^2+b^4)/(a^
4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))+2/a^2*arctan(tan(1
/2*d*x+1/2*c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (153) = 306\).

Time = 0.31 (sec) , antiderivative size = 906, normalized size of antiderivative = 5.59 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\left [\frac {2 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d x + {\left (4 \, a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7} + {\left (4 \, a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\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 (4 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + a b^{7} + {\left (5 \, a^{6} b^{2} - 7 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\right )} d\right )}}, \frac {{\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d x - {\left (4 \, a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7} + {\left (4 \, a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\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 (4 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + a b^{7} + {\left (5 \, a^{6} b^{2} - 7 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\right )} d}\right ] \]

[In]

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

[Out]

[1/2*(2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cos(d*x + c)^2 + 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)
*d*x*cos(d*x + c) + 2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d*x + (4*a^4*b^3 - 2*a^2*b^5 + b^7 + (4*a^6*b -
2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 + 2*(4*a^5*b^2 - 2*a^3*b^4 + a*b^6)*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*(4*a^5*b^3 - 5*a^3*b^5 + a*b^7 + (5*a^6*b^2 - 7*a^4
*b^4 + 2*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*d*cos(d*x + c)^2 + 2*(
a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*d*cos(d*x + c) + (a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*d), ((a^
8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cos(d*x + c)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x*cos(d*
x + c) + (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d*x - (4*a^4*b^3 - 2*a^2*b^5 + b^7 + (4*a^6*b - 2*a^4*b^3 + a
^2*b^5)*cos(d*x + c)^2 + 2*(4*a^5*b^2 - 2*a^3*b^4 + a*b^6)*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))) + (4*a^5*b^3 - 5*a^3*b^5 + a*b^7 + (5*a^6*b^2 - 7*a^4*b^
4 + 2*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*d*cos(d*x + c)^2 + 2*(a^9
*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*d*cos(d*x + c) + (a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*d)]

Sympy [F]

\[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\int \frac {a - b \sec {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

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

[Out]

Integral((a - b*sec(c + d*x))/(a + b*sec(c + d*x))**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a^2-b^2*sec(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 de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (153) = 306\).

Time = 0.36 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.96 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {\frac {2 \, {\left (4 \, a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {d x + c}{a^{2}} - \frac {2 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{d} \]

[In]

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

[Out]

(2*(4*a^4*b - 2*a^2*b^3 + b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*d*x + 1/2*
c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^6 - 2*a^4*b^2 + a^2*b^4)*sqrt(-a^2 + b^2)) + (d*x + c)/a^2
- 2*(5*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 4*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*a*b^4*tan(1/2*d*x + 1/2*c)^3 + b
^5*tan(1/2*d*x + 1/2*c)^3 - 5*a^3*b^2*tan(1/2*d*x + 1/2*c) - 4*a^2*b^3*tan(1/2*d*x + 1/2*c) + 2*a*b^4*tan(1/2*
d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c))/((a^5 - 2*a^3*b^2 + a*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x
+ 1/2*c)^2 - a - b)^2))/d

Mupad [B] (verification not implemented)

Time = 25.37 (sec) , antiderivative size = 4924, normalized size of antiderivative = 30.40 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*atan((((((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10*b^3 + a^11*b^
2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) - (tan(c/2 + (d*x)/2)*(
2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b^4 - 8*a^11*b^3
- 2*a^12*b^2)*32i)/(a^2*(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2)))*1i
)/a^2 + (32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a^4*b^6 - 12*a^
5*b^5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6
*b^3 - 3*a^7*b^2))/a^2 - ((((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^
10*b^3 + a^11*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) + (tan(
c/2 + (d*x)/2)*(2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b
^4 - 8*a^11*b^3 - 2*a^12*b^2)*32i)/(a^2*(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 -
3*a^7*b^2)))*1i)/a^2 - (32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14
*a^4*b^6 - 12*a^5*b^5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3
*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2))/a^2)/((64*(4*a^8*b + b^9 - 4*a^2*b^7 + 3*a^3*b^6 + 9*a^4*b^5 - 6*a^5*b^4 -
10*a^6*b^3 + 12*a^7*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) +
(((((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10*b^3 + a^11*b^2))/(a^
9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) - (tan(c/2 + (d*x)/2)*(2*a^13*
b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b^4 - 8*a^11*b^3 - 2*a^1
2*b^2)*32i)/(a^2*(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2)))*1i)/a^2 +
(32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a^4*b^6 - 12*a^5*b^5 -
14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 -
3*a^7*b^2))*1i)/a^2 + (((((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10
*b^3 + a^11*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) + (tan(c/
2 + (d*x)/2)*(2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b^4
- 8*a^11*b^3 - 2*a^12*b^2)*32i)/(a^2*(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3
*a^7*b^2)))*1i)/a^2 - (32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a
^4*b^6 - 12*a^5*b^5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a
^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2))*1i)/a^2)))/(a^2*d) - ((2*tan(c/2 + (d*x)/2)*(a*b^3 + b^4 - 5*a^2*b^2))/((a +
b)*(a*b^2 - 2*a^2*b + a^3)) - (2*tan(c/2 + (d*x)/2)^3*(a*b^3 - b^4 + 5*a^2*b^2))/((a + b)^2*(a*b - a^2)))/(d*(
2*a*b - tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (b*ata
n(((b*((32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a^4*b^6 - 12*a^5
*b^5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*
b^3 - 3*a^7*b^2) + (b*((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10*b^
3 + a^11*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) - (32*b*tan(
c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2)*(2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^
6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b^4 - 8*a^11*b^3 - 2*a^12*b^2))/((a^12 - a^2*b^10 + 5*a^4
*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)*(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*
b^3 - 3*a^7*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2))/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^
6*b^6 + 10*a^8*b^4 - 5*a^10*b^2))*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2)*1i)/(a^12 - a^2*b^10 +
5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2) + (b*((32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*
b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a^4*b^6 - 12*a^5*b^5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 -
a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2) - (b*((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*
b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10*b^3 + a^11*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5
+ 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) + (32*b*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*
a^2*b^2)*(2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b^4 - 8
*a^11*b^3 - 2*a^12*b^2))/((a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)*(a^8*b + a^9 -
a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4
- 2*a^2*b^2))/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2))*((a + b)^5*(a - b)^5)^(1/2
)*(4*a^4 + b^4 - 2*a^2*b^2)*1i)/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2))/((64*(4*
a^8*b + b^9 - 4*a^2*b^7 + 3*a^3*b^6 + 9*a^4*b^5 - 6*a^5*b^4 - 10*a^6*b^3 + 12*a^7*b^2))/(a^9*b + a^10 - a^3*b^
7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) + (b*((32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*
a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a^4*b^6 - 12*a^5*b^5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*
b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2) + (b*((32*(4*a^12*b - a^13 + a^4*
b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10*b^3 + a^11*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 +
3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) - (32*b*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4
+ b^4 - 2*a^2*b^2)*(2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^
10*b^4 - 8*a^11*b^3 - 2*a^12*b^2))/((a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)*(a^8*
b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3 - 3*a^7*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(4*
a^4 + b^4 - 2*a^2*b^2))/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2))*((a + b)^5*(a -
b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2))/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2) -
(b*((32*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 2*a*b^9 + 2*b^10 - 7*a^2*b^8 + 8*a^3*b^7 + 14*a^4*b^6 - 12*a^5*b^
5 - 14*a^6*b^4 + 8*a^7*b^3 + 13*a^8*b^2))/(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3
- 3*a^7*b^2) - (b*((32*(4*a^12*b - a^13 + a^4*b^9 - a^5*b^8 - 4*a^6*b^7 + a^7*b^6 + 9*a^8*b^5 - 10*a^10*b^3 +
a^11*b^2))/(a^9*b + a^10 - a^3*b^7 - a^4*b^6 + 3*a^5*b^5 + 3*a^6*b^4 - 3*a^7*b^3 - 3*a^8*b^2) + (32*b*tan(c/2
+ (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2)*(2*a^13*b - 2*a^4*b^10 + 2*a^5*b^9 + 8*a^6*b
^8 - 8*a^7*b^7 - 12*a^8*b^6 + 12*a^9*b^5 + 8*a^10*b^4 - 8*a^11*b^3 - 2*a^12*b^2))/((a^12 - a^2*b^10 + 5*a^4*b^
8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)*(a^8*b + a^9 - a^2*b^7 - a^3*b^6 + 3*a^4*b^5 + 3*a^5*b^4 - 3*a^6*b^3
- 3*a^7*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2))/(a^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b
^6 + 10*a^8*b^4 - 5*a^10*b^2))*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2))/(a^12 - a^2*b^10 + 5*a^4
*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(4*a^4 + b^4 - 2*a^2*b^2)*2i)/(d*(a
^12 - a^2*b^10 + 5*a^4*b^8 - 10*a^6*b^6 + 10*a^8*b^4 - 5*a^10*b^2))