3.3.30 \(\int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [230]

Optimal. Leaf size=267 \[ \frac {\left (a^2-12 b^2\right ) x}{2 a^5}-\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {b \left (11 a^2-12 b^2\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))} \]

[Out]

1/2*(a^2-12*b^2)*x/a^5-b*(6*a^4-19*a^2*b^2+12*b^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-
b)^(3/2)/(a+b)^(3/2)/d+1/2*b*(11*a^2-12*b^2)*sin(d*x+c)/a^4/(a^2-b^2)/d-1/2*(5*a^2-6*b^2)*cos(d*x+c)*sin(d*x+c
)/a^3/(a^2-b^2)/d+1/2*cos(d*x+c)^3*sin(d*x+c)/a/d/(b+a*cos(d*x+c))^2+1/2*(3*a^2-4*b^2)*cos(d*x+c)^2*sin(d*x+c)
/a^2/(a^2-b^2)/d/(b+a*cos(d*x+c))

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Rubi [A]
time = 0.65, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2968, 3127, 3128, 3102, 2814, 2738, 214} \begin {gather*} \frac {\left (3 a^2-4 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}+\frac {x \left (a^2-12 b^2\right )}{2 a^5}+\frac {b \left (11 a^2-12 b^2\right ) \sin (c+d x)}{2 a^4 d \left (a^2-b^2\right )}-\frac {\left (5 a^2-6 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )}-\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\sin (c+d x) \cos ^3(c+d x)}{2 a d (a \cos (c+d x)+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 - 12*b^2)*x)/(2*a^5) - (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a +
b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(11*a^2 - 12*b^2)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)*d) - ((5*a^2
- 6*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*a*d*(b + a*Cos[c
+ d*x])^2) + ((3*a^2 - 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d*(b + a*Cos[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 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3127

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2
*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^2(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=-\int \frac {\cos ^3(c+d x) \left (1-\cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) \left (3 \left (a^2-b^2\right )-4 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (2 \left (3 a^4-7 a^2 b^2+4 b^4\right )+a b \left (a^2-b^2\right ) \cos (c+d x)-2 \left (5 a^2-6 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (5 a^2-6 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))}+\frac {\int \frac {2 b \left (5 a^4-11 a^2 b^2+6 b^4\right )-2 a \left (a^4-3 a^2 b^2+2 b^4\right ) \cos (c+d x)-2 b \left (11 a^2-12 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (11 a^2-12 b^2\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))}-\frac {\int \frac {-2 a b \left (5 a^4-11 a^2 b^2+6 b^4\right )+2 \left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (a^2-12 b^2\right ) x}{2 a^5}+\frac {b \left (11 a^2-12 b^2\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))}+\frac {\left (b \left (6 a^4-19 a^2 b^2+12 b^4\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )}\\ &=\frac {\left (a^2-12 b^2\right ) x}{2 a^5}+\frac {b \left (11 a^2-12 b^2\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))}+\frac {\left (b \left (6 a^4-19 a^2 b^2+12 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right ) d}\\ &=\frac {\left (a^2-12 b^2\right ) x}{2 a^5}-\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {b \left (11 a^2-12 b^2\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{2 a d (b+a \cos (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 2.68, size = 282, normalized size = 1.06 \begin {gather*} \frac {\frac {4 b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {4 a b \left (a^4-13 a^2 b^2+12 b^4\right ) (c+d x) \cos (c+d x)-2 a^4 \left (a^2-b^2\right ) \cos ^3(c+d x) \sin (c+d x)+2 a^2 \left (a^2-b^2\right ) \cos ^2(c+d x) \left (\left (a^2-12 b^2\right ) (c+d x)+4 a b \sin (c+d x)\right )+b^2 \left (2 \left (a^4-13 a^2 b^2+12 b^4\right ) (c+d x)+\left (22 a^3 b-24 a b^3\right ) \sin (c+d x)+\left (17 a^4-18 a^2 b^2\right ) \sin (2 (c+d x))\right )}{(b+a \cos (c+d x))^2}}{4 a^5 (a-b) (a+b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (4
*a*b*(a^4 - 13*a^2*b^2 + 12*b^4)*(c + d*x)*Cos[c + d*x] - 2*a^4*(a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x] + 2*a^
2*(a^2 - b^2)*Cos[c + d*x]^2*((a^2 - 12*b^2)*(c + d*x) + 4*a*b*Sin[c + d*x]) + b^2*(2*(a^4 - 13*a^2*b^2 + 12*b
^4)*(c + d*x) + (22*a^3*b - 24*a*b^3)*Sin[c + d*x] + (17*a^4 - 18*a^2*b^2)*Sin[2*(c + d*x)]))/(b + a*Cos[c + d
*x])^2)/(4*a^5*(a - b)*(a + b)*d)

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Maple [A]
time = 0.25, size = 276, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {-\frac {\left (6 a^{2}+b a -6 b^{2}\right ) b a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )}+\frac {\left (6 a^{2}-b a -6 b^{2}\right ) b a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (6 a^{4}-19 b^{2} a^{2}+12 b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{2}-b^{2}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2}+3 b a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 b a -\frac {1}{2} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (a^{2}-12 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\)	\(276\)
default	\(\frac {\frac {2 b \left (\frac {-\frac {\left (6 a^{2}+b a -6 b^{2}\right ) b a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a +b \right )}+\frac {\left (6 a^{2}-b a -6 b^{2}\right ) b a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (6 a^{4}-19 b^{2} a^{2}+12 b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{2}-b^{2}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2}+3 b a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 b a -\frac {1}{2} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (a^{2}-12 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\)	\(276\)
risch	\(\frac {x}{2 a^{3}}-\frac {6 x \,b^{2}}{a^{5}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {3 i b \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{4} d}+\frac {3 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{4} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i b^{2} \left (-7 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+8 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-5 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+14 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-17 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+20 b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}-6 a^{4}+7 b^{2} a^{2}\right )}{a^{5} \left (-a^{2}+b^{2}\right ) 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 {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {19 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {6 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {19 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {6 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}\)	\(772\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*b/a^5*((-1/2*(6*a^2+a*b-6*b^2)*b*a/(a+b)*tan(1/2*d*x+1/2*c)^3+1/2*(6*a^2-a*b-6*b^2)*b*a/(a-b)*tan(1/2*d
*x+1/2*c))/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2-a-b)^2-1/2*(6*a^4-19*a^2*b^2+12*b^4)/(a^2-b^2)/((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^5*(((1/2*a^2+3*b*a)*tan(1/2*d*x+1/2*
c)^3+(3*b*a-1/2*a^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(a^2-12*b^2)*arctan(tan(1/2*d*x+1/2*c)
)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2/(a+b*sec(d*x+c))^3,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

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Fricas [A]
time = 6.01, size = 984, normalized size = 3.69 \begin {gather*} \left [\frac {2 \, {\left (a^{8} - 14 \, a^{6} b^{2} + 25 \, a^{4} b^{4} - 12 \, a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{7} b - 14 \, a^{5} b^{3} + 25 \, a^{3} b^{5} - 12 \, a b^{7}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (a^{6} b^{2} - 14 \, a^{4} b^{4} + 25 \, a^{2} b^{6} - 12 \, b^{8}\right )} d x - {\left (6 \, a^{4} b^{3} - 19 \, a^{2} b^{5} + 12 \, b^{7} + {\left (6 \, a^{6} b - 19 \, a^{4} b^{3} + 12 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 12 \, 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 (11 \, a^{5} b^{3} - 23 \, a^{3} b^{5} + 12 \, a b^{7} - {\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (17 \, a^{6} b^{2} - 35 \, a^{4} b^{4} + 18 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{11} - 2 \, a^{9} b^{2} + a^{7} b^{4}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b - 2 \, a^{8} b^{3} + a^{6} b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b^{2} - 2 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}}, \frac {{\left (a^{8} - 14 \, a^{6} b^{2} + 25 \, a^{4} b^{4} - 12 \, a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 14 \, a^{5} b^{3} + 25 \, a^{3} b^{5} - 12 \, a b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (a^{6} b^{2} - 14 \, a^{4} b^{4} + 25 \, a^{2} b^{6} - 12 \, b^{8}\right )} d x - {\left (6 \, a^{4} b^{3} - 19 \, a^{2} b^{5} + 12 \, b^{7} + {\left (6 \, a^{6} b - 19 \, a^{4} b^{3} + 12 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 12 \, 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 (11 \, a^{5} b^{3} - 23 \, a^{3} b^{5} + 12 \, a b^{7} - {\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (17 \, a^{6} b^{2} - 35 \, a^{4} b^{4} + 18 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{11} - 2 \, a^{9} b^{2} + a^{7} b^{4}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b - 2 \, a^{8} b^{3} + a^{6} b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b^{2} - 2 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(a^8 - 14*a^6*b^2 + 25*a^4*b^4 - 12*a^2*b^6)*d*x*cos(d*x + c)^2 + 4*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 -
12*a*b^7)*d*x*cos(d*x + c) + 2*(a^6*b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*d*x - (6*a^4*b^3 - 19*a^2*b^5 + 1
2*b^7 + (6*a^6*b - 19*a^4*b^3 + 12*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*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*(11*a^5*b^3 - 23*a^3*b^5
+ 12*a*b^7 - (a^8 - 2*a^6*b^2 + a^4*b^4)*cos(d*x + c)^3 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*cos(d*x + c)^2 + (1
7*a^6*b^2 - 35*a^4*b^4 + 18*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 2*a^9*b^2 + a^7*b^4)*d*cos(d*x + c)^
2 + 2*(a^10*b - 2*a^8*b^3 + a^6*b^5)*d*cos(d*x + c) + (a^9*b^2 - 2*a^7*b^4 + a^5*b^6)*d), 1/2*((a^8 - 14*a^6*b
^2 + 25*a^4*b^4 - 12*a^2*b^6)*d*x*cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*d*x*cos(d*x
+ c) + (a^6*b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*d*x - (6*a^4*b^3 - 19*a^2*b^5 + 12*b^7 + (6*a^6*b - 19*a^4
*b^3 + 12*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6)*cos(d*x + c))*sqrt(-a^2 + b^2)*arcta
n(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (11*a^5*b^3 - 23*a^3*b^5 + 12*a*b^7 - (
a^8 - 2*a^6*b^2 + a^4*b^4)*cos(d*x + c)^3 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*cos(d*x + c)^2 + (17*a^6*b^2 - 35*
a^4*b^4 + 18*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 2*a^9*b^2 + a^7*b^4)*d*cos(d*x + c)^2 + 2*(a^10*b -
2*a^8*b^3 + a^6*b^5)*d*cos(d*x + c) + (a^9*b^2 - 2*a^7*b^4 + a^5*b^6)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (248) = 496\).
time = 0.64, size = 1193, normalized size = 4.47 \begin {gather*} \frac {\frac {{\left (a^{11} - 7 \, a^{10} b - 14 \, a^{9} b^{2} + 39 \, a^{8} b^{3} + 25 \, a^{7} b^{4} - 56 \, a^{6} b^{5} - 12 \, a^{5} b^{6} + 24 \, a^{4} b^{7} - a^{4} {\left | -a^{7} + a^{5} b^{2} \right |} - 5 \, a^{3} b {\left | -a^{7} + a^{5} b^{2} \right |} + 13 \, a^{2} b^{2} {\left | -a^{7} + a^{5} b^{2} \right |} + 6 \, a b^{3} {\left | -a^{7} + a^{5} b^{2} \right |} - 12 \, b^{4} {\left | -a^{7} + a^{5} b^{2} \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\frac {a^{6} b - a^{4} b^{3} + \sqrt {{\left (a^{7} + a^{6} b - a^{5} b^{2} - a^{4} b^{3}\right )} {\left (a^{7} - a^{6} b - a^{5} b^{2} + a^{4} b^{3}\right )} + {\left (a^{6} b - a^{4} b^{3}\right )}^{2}}}{a^{7} - a^{6} b - a^{5} b^{2} + a^{4} b^{3}}}}\right )\right )}}{a^{6} b {\left | -a^{7} + a^{5} b^{2} \right |} - a^{4} b^{3} {\left | -a^{7} + a^{5} b^{2} \right |} + {\left (a^{7} - a^{5} b^{2}\right )}^{2}} + \frac {{\left ({\left (a^{4} + 5 \, a^{3} b - 13 \, a^{2} b^{2} - 6 \, a b^{3} + 12 \, b^{4}\right )} \sqrt {-a^{2} + b^{2}} {\left | -a^{7} + a^{5} b^{2} \right |} {\left | -a + b \right |} + {\left (a^{11} - 7 \, a^{10} b - 14 \, a^{9} b^{2} + 39 \, a^{8} b^{3} + 25 \, a^{7} b^{4} - 56 \, a^{6} b^{5} - 12 \, a^{5} b^{6} + 24 \, a^{4} b^{7}\right )} \sqrt {-a^{2} + b^{2}} {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-\frac {a^{6} b - a^{4} b^{3} - \sqrt {{\left (a^{7} + a^{6} b - a^{5} b^{2} - a^{4} b^{3}\right )} {\left (a^{7} - a^{6} b - a^{5} b^{2} + a^{4} b^{3}\right )} + {\left (a^{6} b - a^{4} b^{3}\right )}^{2}}}{a^{7} - a^{6} b - a^{5} b^{2} + a^{4} b^{3}}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )}^{2} {\left (a^{2} - 2 \, a b + b^{2}\right )} - {\left (a^{8} b - 2 \, a^{7} b^{2} + 2 \, a^{5} b^{4} - a^{4} b^{5}\right )} {\left | -a^{7} + a^{5} b^{2} \right |}} + \frac {2 \, {\left (a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 18 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 18 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 37 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 37 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^11 - 7*a^10*b - 14*a^9*b^2 + 39*a^8*b^3 + 25*a^7*b^4 - 56*a^6*b^5 - 12*a^5*b^6 + 24*a^4*b^7 - a^4*abs(
-a^7 + a^5*b^2) - 5*a^3*b*abs(-a^7 + a^5*b^2) + 13*a^2*b^2*abs(-a^7 + a^5*b^2) + 6*a*b^3*abs(-a^7 + a^5*b^2) -
12*b^4*abs(-a^7 + a^5*b^2))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^6*b - a^
4*b^3 + sqrt((a^7 + a^6*b - a^5*b^2 - a^4*b^3)*(a^7 - a^6*b - a^5*b^2 + a^4*b^3) + (a^6*b - a^4*b^3)^2))/(a^7
- a^6*b - a^5*b^2 + a^4*b^3))))/(a^6*b*abs(-a^7 + a^5*b^2) - a^4*b^3*abs(-a^7 + a^5*b^2) + (a^7 - a^5*b^2)^2)
+ ((a^4 + 5*a^3*b - 13*a^2*b^2 - 6*a*b^3 + 12*b^4)*sqrt(-a^2 + b^2)*abs(-a^7 + a^5*b^2)*abs(-a + b) + (a^11 -
7*a^10*b - 14*a^9*b^2 + 39*a^8*b^3 + 25*a^7*b^4 - 56*a^6*b^5 - 12*a^5*b^6 + 24*a^4*b^7)*sqrt(-a^2 + b^2)*abs(-
a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^6*b - a^4*b^3 - sqrt((a^7 + a
^6*b - a^5*b^2 - a^4*b^3)*(a^7 - a^6*b - a^5*b^2 + a^4*b^3) + (a^6*b - a^4*b^3)^2))/(a^7 - a^6*b - a^5*b^2 + a
^4*b^3))))/((a^7 - a^5*b^2)^2*(a^2 - 2*a*b + b^2) - (a^8*b - 2*a^7*b^2 + 2*a^5*b^4 - a^4*b^5)*abs(-a^7 + a^5*b
^2)) + 2*(a^5*tan(1/2*d*x + 1/2*c)^7 + 4*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 18*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 + 7*
a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 18*a*b^4*tan(1/2*d*x + 1/2*c)^7 - 12*b^5*tan(1/2*d*x + 1/2*c)^7 - 3*a^5*tan(1
/2*d*x + 1/2*c)^5 - 4*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 14*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 + 37*a^2*b^3*tan(1/2*d*
x + 1/2*c)^5 + 18*a*b^4*tan(1/2*d*x + 1/2*c)^5 - 36*b^5*tan(1/2*d*x + 1/2*c)^5 + 3*a^5*tan(1/2*d*x + 1/2*c)^3
- 4*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 14*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + 37*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 18*
a*b^4*tan(1/2*d*x + 1/2*c)^3 - 36*b^5*tan(1/2*d*x + 1/2*c)^3 - a^5*tan(1/2*d*x + 1/2*c) + 4*a^4*b*tan(1/2*d*x
+ 1/2*c) + 18*a^3*b^2*tan(1/2*d*x + 1/2*c) + 7*a^2*b^3*tan(1/2*d*x + 1/2*c) - 18*a*b^4*tan(1/2*d*x + 1/2*c) -
12*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - a^4*b^2)*(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)^2))/d

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Mupad [B]
time = 9.10, size = 2500, normalized size = 9.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(6*a*b^3 - 5*a^3*b + a^4 + 12*b^4 - 13*a^2*b^2))/(a^4*b - a^5) + (tan(c/2 + (d*x)/2)^3*(1
8*a*b^4 + 4*a^4*b - 3*a^5 + 36*b^5 - 37*a^2*b^3 - 14*a^3*b^2))/((a^4*b - a^5)*(a + b)) + (tan(c/2 + (d*x)/2)^5
*(4*a^4*b - 18*a*b^4 + 3*a^5 + 36*b^5 - 37*a^2*b^3 + 14*a^3*b^2))/((a^4*b - a^5)*(a + b)) + (tan(c/2 + (d*x)/2
)^7*(5*a^3*b - 6*a*b^3 + a^4 + 12*b^4 - 13*a^2*b^2))/(a^4*(a + b)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^4*(2*a^2 -
6*b^2) + tan(c/2 + (d*x)/2)^2*(4*a*b + 4*b^2) - tan(c/2 + (d*x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2)^8*(a
^2 - 2*a*b + b^2) + a^2 + b^2)) + (atan((((a^2*1i - b^2*12i)*((((4*(24*a^16*b - 4*a^17 - 48*a^10*b^7 + 24*a^11
*b^6 + 124*a^12*b^5 - 56*a^13*b^4 - 100*a^14*b^3 + 36*a^15*b^2))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (4*ta
n(c/2 + (d*x)/2)*(a^2*1i - b^2*12i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b
^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(a^2*1i - b^2*12i))/(2*a^5) + (8*tan(c/2 + (d*x)/2)*(a^10 - 2*
a^9*b - 288*a*b^9 + 288*b^10 - 624*a^2*b^8 + 624*a^3*b^7 + 386*a^4*b^6 - 386*a^5*b^5 - 61*a^6*b^4 + 52*a^7*b^3
+ 11*a^8*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*1i)/(2*a^5) - ((a^2*1i - b^2*12i)*((((4*(24*a^16*b - 4*a^
17 - 48*a^10*b^7 + 24*a^11*b^6 + 124*a^12*b^5 - 56*a^13*b^4 - 100*a^14*b^3 + 36*a^15*b^2))/(a^14*b + a^15 - a^
12*b^3 - a^13*b^2) + (4*tan(c/2 + (d*x)/2)*(a^2*1i - b^2*12i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^
4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(a^2*1i - b^2*12i))/(2*a^5) - (8*tan
(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 288*a*b^9 + 288*b^10 - 624*a^2*b^8 + 624*a^3*b^7 + 386*a^4*b^6 - 386*a^5*b^5
- 61*a^6*b^4 + 52*a^7*b^3 + 11*a^8*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*1i)/(2*a^5))/((8*(864*a*b^10 +
6*a^10*b - 1728*b^11 + 4752*a^2*b^9 - 2160*a^3*b^8 - 4356*a^4*b^7 + 1746*a^5*b^6 + 1495*a^6*b^5 - 491*a^7*b^4
- 169*a^8*b^3 + 30*a^9*b^2))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + ((a^2*1i - b^2*12i)*((((4*(24*a^16*b - 4*
a^17 - 48*a^10*b^7 + 24*a^11*b^6 + 124*a^12*b^5 - 56*a^13*b^4 - 100*a^14*b^3 + 36*a^15*b^2))/(a^14*b + a^15 -
a^12*b^3 - a^13*b^2) - (4*tan(c/2 + (d*x)/2)*(a^2*1i - b^2*12i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*
b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(a^2*1i - b^2*12i))/(2*a^5) + (8*t
an(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 288*a*b^9 + 288*b^10 - 624*a^2*b^8 + 624*a^3*b^7 + 386*a^4*b^6 - 386*a^5*b
^5 - 61*a^6*b^4 + 52*a^7*b^3 + 11*a^8*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))/(2*a^5) + ((a^2*1i - b^2*12i
)*((((4*(24*a^16*b - 4*a^17 - 48*a^10*b^7 + 24*a^11*b^6 + 124*a^12*b^5 - 56*a^13*b^4 - 100*a^14*b^3 + 36*a^15*
b^2))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (4*tan(c/2 + (d*x)/2)*(a^2*1i - b^2*12i)*(8*a^15*b - 8*a^10*b^6
+ 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(a^2*1i - b
^2*12i))/(2*a^5) - (8*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 288*a*b^9 + 288*b^10 - 624*a^2*b^8 + 624*a^3*b^7 +
386*a^4*b^6 - 386*a^5*b^5 - 61*a^6*b^4 + 52*a^7*b^3 + 11*a^8*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))/(2*a^
5)))*(a^2*1i - b^2*12i)*1i)/(a^5*d) + (b*atan(((b*((8*tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 288*a*b^9 + 288*b^1
0 - 624*a^2*b^8 + 624*a^3*b^7 + 386*a^4*b^6 - 386*a^5*b^5 - 61*a^6*b^4 + 52*a^7*b^3 + 11*a^8*b^2))/(a^10*b + a
^11 - a^8*b^3 - a^9*b^2) + (b*((4*(24*a^16*b - 4*a^17 - 48*a^10*b^7 + 24*a^11*b^6 + 124*a^12*b^5 - 56*a^13*b^4
- 100*a^14*b^3 + 36*a^15*b^2))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (4*b*tan(c/2 + (d*x)/2)*((a + b)^3*(a
- b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 -
8*a^14*b^2))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b
)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2))/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^
(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*1i)/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)) + (b*((8*tan(c/2 + (d*x)/
2)*(a^10 - 2*a^9*b - 288*a*b^9 + 288*b^10 - 624*a^2*b^8 + 624*a^3*b^7 + 386*a^4*b^6 - 386*a^5*b^5 - 61*a^6*b^4
+ 52*a^7*b^3 + 11*a^8*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) - (b*((4*(24*a^16*b - 4*a^17 - 48*a^10*b^7 +
24*a^11*b^6 + 124*a^12*b^5 - 56*a^13*b^4 - 100*a^14*b^3 + 36*a^15*b^2))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2)
+ (4*b*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*(8*a^15*b - 8*a^10*b^6 + 8
*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*
a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2))/(2*(a^11 - a^5*b^6 + 3*a^7*b
^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*1i)/(2*(a^11 - a^5*b^6 + 3*a^7*b^4
- 3*a^9*b^2)))/((8*(864*a*b^10 + 6*a^10*b - 1728*b^11 + 4752*a^2*b^9 - 2160*a^3*b^8 - 4356*a^4*b^7 + 1746*a^5
*b^6 + 1495*a^6*b^5 - 491*a^7*b^4 - 169*a^8*b^3 + 30*a^9*b^2))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (b*((8*
tan(c/2 + (d*x)/2)*(a^10 - 2*a^9*b - 288*a*b^9 ...

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