\(\int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1475]

Optimal result

Integrand size = 27, antiderivative size = 402 \[ \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {2 b^3 \left (3 a^2-b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a \left (a^2-b^2\right )^{7/2} d}+\frac {b^3 \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a \left (a^2-b^2\right )^{7/2} d}+\frac {2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^4 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \]

[Out]

2*b^3*(3*a^2-b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a/(a^2-b^2)^(7/2)/d+b^3*(2*a^2+b^2)*arctan(
(b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a/(a^2-b^2)^(7/2)/d+2*b^3*(6*a^4-3*a^2*b^2+b^4)*arctan((b+a*tan(1/2*
d*x+1/2*c))/(a^2-b^2)^(1/2))/a^3/(a^2-b^2)^(7/2)/d-arctanh(cos(d*x+c))/a^3/d+1/2*cos(d*x+c)/(a+b)^3/d/(1-sin(d
*x+c))+1/2*cos(d*x+c)/(a-b)^3/d/(1+sin(d*x+c))+1/2*b^4*cos(d*x+c)/a/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^2+3/2*b^4*c
os(d*x+c)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))+b^4*(3*a^2-b^2)*cos(d*x+c)/a^2/(a^2-b^2)^3/d/(a+b*sin(d*x+c))

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2976, 3855, 2727, 2743, 2833, 12, 2739, 632, 210} \[ \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {2 b^3 \left (3 a^2-b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a d \left (a^2-b^2\right )^{7/2}}+\frac {b^3 \left (2 a^2+b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a d \left (a^2-b^2\right )^{7/2}}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {3 b^4 \cos (c+d x)}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \left (a^2-b^2\right )^{7/2}}+\frac {\cos (c+d x)}{2 d (a+b)^3 (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 d (a-b)^3 (\sin (c+d x)+1)} \]

[In]

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

[Out]

(2*b^3*(3*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(7/2)*d) + (b^3*(2*a^2 +
b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(7/2)*d) + (2*b^3*(6*a^4 - 3*a^2*b^2 +
b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d*x]]/(a^3*
d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (b^4*
Cos[c + d*x])/(2*a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^4*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Si
n[c + d*x])) + (b^4*(3*a^2 - b^2)*Cos[c + d*x])/(a^2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*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 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rule 2833

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

Rule 2976

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

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\csc (c+d x)}{a^3}-\frac {1}{2 (a+b)^3 (-1+\sin (c+d x))}-\frac {1}{2 (a-b)^3 (1+\sin (c+d x))}-\frac {b^3}{a \left (-a^2+b^2\right ) (a+b \sin (c+d x))^3}+\frac {b^3 \left (3 a^2-b^2\right )}{a^2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {6 a^4 b^3-3 a^2 b^5+b^7}{a^3 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {\int \csc (c+d x) \, dx}{a^3}-\frac {\int \frac {1}{1+\sin (c+d x)} \, dx}{2 (a-b)^3}-\frac {\int \frac {1}{-1+\sin (c+d x)} \, dx}{2 (a+b)^3}+\frac {b^3 \int \frac {1}{(a+b \sin (c+d x))^3} \, dx}{a \left (a^2-b^2\right )}+\frac {\left (b^3 \left (3 a^2-b^2\right )\right ) \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^2 \left (a^2-b^2\right )^2}+\frac {\left (b^3 \left (6 a^4-3 a^2 b^2+b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )^3} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {b^3 \int \frac {-2 a+b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )^2}+\frac {\left (b^3 \left (3 a^2-b^2\right )\right ) \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^3}+\frac {\left (2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^3 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^4 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^3 \int \frac {2 a^2+b^2}{a+b \sin (c+d x)} \, dx}{2 a \left (a^2-b^2\right )^3}+\frac {\left (b^3 \left (3 a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a \left (a^2-b^2\right )^3}-\frac {\left (4 b^3 \left (6 a^4-3 a^2 b^2+b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^3 d} \\ & = \frac {2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^4 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {\left (b^3 \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a \left (a^2-b^2\right )^3}+\frac {\left (2 b^3 \left (3 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right )^3 d} \\ & = \frac {2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^4 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\left (4 b^3 \left (3 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right )^3 d}+\frac {\left (b^3 \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right )^3 d} \\ & = \frac {2 b^3 \left (3 a^2-b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a \left (a^2-b^2\right )^{7/2} d}+\frac {2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^4 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\left (2 b^3 \left (2 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a \left (a^2-b^2\right )^3 d} \\ & = \frac {2 b^3 \left (3 a^2-b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a \left (a^2-b^2\right )^{7/2} d}+\frac {b^3 \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a \left (a^2-b^2\right )^{7/2} d}+\frac {2 b^3 \left (6 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{2 (a+b)^3 d (1-\sin (c+d x))}+\frac {\cos (c+d x)}{2 (a-b)^3 d (1+\sin (c+d x))}+\frac {b^4 \cos (c+d x)}{2 a \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}+\frac {3 b^4 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac {b^4 \left (3 a^2-b^2\right ) \cos (c+d x)}{a^2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.85 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.69 \[ \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 b^3 \left (20 a^4-7 a^2 b^2+2 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{7/2}}-\frac {2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}+\frac {2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b^4 \cos (c+d x)}{a (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^2}+\frac {b^4 \left (9 a^2-2 b^2\right ) \cos (c+d x)}{a^2 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))}}{2 d} \]

[In]

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

[Out]

((2*b^3*(20*a^4 - 7*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2))
- (2*Log[Cos[(c + d*x)/2]])/a^3 + (2*Log[Sin[(c + d*x)/2]])/a^3 + (2*Sin[(c + d*x)/2])/((a + b)^3*(Cos[(c + d
*x)/2] - Sin[(c + d*x)/2])) - (2*Sin[(c + d*x)/2])/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^4*Co
s[c + d*x])/(a*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^2) + (b^4*(9*a^2 - 2*b^2)*Cos[c + d*x])/(a^2*(a - b)^3
*(a + b)^3*(a + b*Sin[c + d*x])))/(2*d)

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {2 b^{3} \left (\frac {\left (\frac {11}{2} a^{3} b^{2}-2 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (10 a^{4}+17 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (29 a^{2}-8 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (10 a^{2}-3 b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-7 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a^{3}}+\frac {1}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(278\)
default	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {2 b^{3} \left (\frac {\left (\frac {11}{2} a^{3} b^{2}-2 a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (10 a^{4}+17 a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (29 a^{2}-8 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (10 a^{2}-3 b^{2}\right )}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (20 a^{4}-7 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a^{3}}+\frac {1}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(278\)
risch	\(\frac {6 i a^{4} b^{3}-16 i a^{6} b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 i b^{7} {\mathrm e}^{4 i \left (d x +c \right )}+11 i a^{2} b^{5}-2 a^{5} {\mathrm e}^{5 i \left (d x +c \right )} b^{2}-14 a^{3} b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+a \,b^{6} {\mathrm e}^{5 i \left (d x +c \right )}-36 i a^{4} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-8 i a^{6} {\mathrm e}^{4 i \left (d x +c \right )} b +8 a^{7} {\mathrm e}^{3 i \left (d x +c \right )}+4 a^{5} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+24 a^{3} b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-6 a \,b^{6} {\mathrm e}^{3 i \left (d x +c \right )}-2 i b^{7}-14 i a^{4} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{2} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+22 a^{5} b^{2} {\mathrm e}^{i \left (d x +c \right )}+30 a^{3} b^{4} {\mathrm e}^{i \left (d x +c \right )}-7 a \,b^{6} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} a^{2} d \left (a^{2}-b^{2}\right )^{3}}+\frac {10 i b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {7 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d a}+\frac {i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d \,a^{3}}-\frac {10 i b^{3} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {7 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d a}-\frac {i b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\)	\(896\)

[In]

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

[Out]

1/d*(1/a^3*ln(tan(1/2*d*x+1/2*c))+2*b^3/(a-b)^3/(a+b)^3/a^3*(((11/2*a^3*b^2-2*a*b^4)*tan(1/2*d*x+1/2*c)^3+1/2*
b*(10*a^4+17*a^2*b^2-6*b^4)*tan(1/2*d*x+1/2*c)^2+1/2*a*b^2*(29*a^2-8*b^2)*tan(1/2*d*x+1/2*c)+1/2*a^2*b*(10*a^2
-3*b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(20*a^4-7*a^2*b^2+2*b^4)/(a^2-b^2)^(1/2)*arct
an(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))+1/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)-1/(a+b)^3/(tan(1/2*d*x+
1/2*c)-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (377) = 754\).

Time = 1.80 (sec) , antiderivative size = 1623, normalized size of antiderivative = 4.04 \[ \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/4*(4*a^10 - 12*a^8*b^2 + 12*a^6*b^4 - 4*a^4*b^6 + 2*(10*a^8*b^2 - 2*a^6*b^4 - 11*a^4*b^6 + 3*a^2*b^8)*cos(
d*x + c)^2 - ((20*a^4*b^5 - 7*a^2*b^7 + 2*b^9)*cos(d*x + c)^3 - 2*(20*a^5*b^4 - 7*a^3*b^6 + 2*a*b^8)*cos(d*x +
c)*sin(d*x + c) - (20*a^6*b^3 + 13*a^4*b^5 - 5*a^2*b^7 + 2*b^9)*cos(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 -
b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(
-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 2*((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*
a^2*b^8 + b^10)*cos(d*x + c)^3 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)*sin(d*x +
c) - (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) -
2*((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*cos(d*x + c)^3 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4
*a^3*b^7 + a*b^9)*cos(d*x + c)*sin(d*x + c) - (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*co
s(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*a^9*b - 6*a^7*b^3 + 6*a^5*b^5 - 2*a^3*b^7 - (6*a^7*b^3 + 5*a^5
*b^5 - 13*a^3*b^7 + 2*a*b^9)*cos(d*x + c)^2)*sin(d*x + c))/((a^11*b^2 - 4*a^9*b^4 + 6*a^7*b^6 - 4*a^5*b^8 + a^
3*b^10)*d*cos(d*x + c)^3 - 2*(a^12*b - 4*a^10*b^3 + 6*a^8*b^5 - 4*a^6*b^7 + a^4*b^9)*d*cos(d*x + c)*sin(d*x +
c) - (a^13 - 3*a^11*b^2 + 2*a^9*b^4 + 2*a^7*b^6 - 3*a^5*b^8 + a^3*b^10)*d*cos(d*x + c)), -1/2*(2*a^10 - 6*a^8*
b^2 + 6*a^6*b^4 - 2*a^4*b^6 + (10*a^8*b^2 - 2*a^6*b^4 - 11*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^2 + ((20*a^4*b^5
- 7*a^2*b^7 + 2*b^9)*cos(d*x + c)^3 - 2*(20*a^5*b^4 - 7*a^3*b^6 + 2*a*b^8)*cos(d*x + c)*sin(d*x + c) - (20*a^6
*b^3 + 13*a^4*b^5 - 5*a^2*b^7 + 2*b^9)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 -
b^2)*cos(d*x + c))) + ((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*cos(d*x + c)^3 - 2*(a^9*b - 4*a^7*
b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)*sin(d*x + c) - (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3
*a^2*b^8 + b^10)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - ((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b
^10)*cos(d*x + c)^3 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)*sin(d*x + c) - (a^10
- 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - (2*a^9*b
- 6*a^7*b^3 + 6*a^5*b^5 - 2*a^3*b^7 - (6*a^7*b^3 + 5*a^5*b^5 - 13*a^3*b^7 + 2*a*b^9)*cos(d*x + c)^2)*sin(d*x +
c))/((a^11*b^2 - 4*a^9*b^4 + 6*a^7*b^6 - 4*a^5*b^8 + a^3*b^10)*d*cos(d*x + c)^3 - 2*(a^12*b - 4*a^10*b^3 + 6*
a^8*b^5 - 4*a^6*b^7 + a^4*b^9)*d*cos(d*x + c)*sin(d*x + c) - (a^13 - 3*a^11*b^2 + 2*a^9*b^4 + 2*a^7*b^6 - 3*a^
5*b^8 + a^3*b^10)*d*cos(d*x + c))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(csc(d*x+c)*sec(d*x+c)^2/(a+b*sin(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*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.02 \[ \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {{\left (20 \, a^{4} b^{3} - 7 \, a^{2} b^{5} + 2 \, b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3} - 3 \, a b^{2}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}} + \frac {11 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 17 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 29 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, a^{4} b^{4} - 3 \, a^{2} b^{6}}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}}}{d} \]

[In]

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

[Out]

((20*a^4*b^3 - 7*a^2*b^5 + 2*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) +
b)/sqrt(a^2 - b^2)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*sqrt(a^2 - b^2)) + 2*(3*a^2*b*tan(1/2*d*x + 1/2*
c) + b^3*tan(1/2*d*x + 1/2*c) - a^3 - 3*a*b^2)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(tan(1/2*d*x + 1/2*c)^2 -
1)) + (11*a^3*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*a*b^7*tan(1/2*d*x + 1/2*c)^3 + 10*a^4*b^4*tan(1/2*d*x + 1/2*c)^2
+ 17*a^2*b^6*tan(1/2*d*x + 1/2*c)^2 - 6*b^8*tan(1/2*d*x + 1/2*c)^2 + 29*a^3*b^5*tan(1/2*d*x + 1/2*c) - 8*a*b^7
*tan(1/2*d*x + 1/2*c) + 10*a^4*b^4 - 3*a^2*b^6)/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(a*tan(1/2*d*x + 1/2*
c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2) + log(abs(tan(1/2*d*x + 1/2*c)))/a^3)/d

Mupad [B] (verification not implemented)

Time = 17.19 (sec) , antiderivative size = 2999, normalized size of antiderivative = 7.46 \[ \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^2*sin(c + d*x)*(a + b*sin(c + d*x))^3),x)

[Out]

log(tan(c/2 + (d*x)/2))/(a^3*d) - ((2*a^6 - 3*b^6 + 10*a^2*b^4 + 6*a^4*b^2)/(a*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*
b^2)) + (tan(c/2 + (d*x)/2)*(2*a^6*b - 8*b^7 + 29*a^2*b^5 + 22*a^4*b^3))/(a^2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b
^2)) - (2*tan(c/2 + (d*x)/2)^3*(2*a^6*b - 2*b^7 + 13*a^2*b^5 + 2*a^4*b^3))/(a^2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4
*b^2)) - (tan(c/2 + (d*x)/2)^5*(6*a^6*b - 4*b^7 + 11*a^2*b^5 + 2*a^4*b^3))/(a^2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4
*b^2)) + (2*tan(c/2 + (d*x)/2)^2*(2*a^8 - 3*b^8 + 10*a^2*b^6 + 8*a^4*b^4 - 2*a^6*b^2))/(a^3*(a^6 - b^6 + 3*a^2
*b^4 - 3*a^4*b^2)) - (tan(c/2 + (d*x)/2)^4*(17*a^2*b^6 - 6*b^8 - 2*a^8 + 18*a^4*b^4 + 18*a^6*b^2))/(a^3*(a^6 -
b^6 + 3*a^2*b^4 - 3*a^4*b^2)))/(d*(a^2*tan(c/2 + (d*x)/2)^6 - a^2 - tan(c/2 + (d*x)/2)^2*(a^2 + 4*b^2) + tan(
c/2 + (d*x)/2)^4*(a^2 + 4*b^2) + 4*a*b*tan(c/2 + (d*x)/2)^5 - 4*a*b*tan(c/2 + (d*x)/2))) + (b^3*atan(((b^3*(-(
a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)*(tan(c/2 + (d*x)/2)*(2*a^24 + 8*a^4*b^20 - 76*a^6*b^1
8 + 346*a^8*b^16 - 938*a^10*b^14 + 1612*a^12*b^12 - 1790*a^14*b^10 + 1276*a^16*b^8 - 566*a^18*b^6 + 148*a^20*b
^4 - 22*a^22*b^2) - 2*a^23*b + 4*a^5*b^19 - 37*a^7*b^17 + 164*a^9*b^15 - 433*a^11*b^13 + 722*a^13*b^11 - 769*a
^15*b^9 + 512*a^17*b^7 - 199*a^19*b^5 + 38*a^21*b^3 + (b^3*(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2
*b^2)/2)*(tan(c/2 + (d*x)/2)*(6*a^27 + 8*a^7*b^20 - 78*a^9*b^18 + 342*a^11*b^16 - 888*a^13*b^14 + 1512*a^15*b^
12 - 1764*a^17*b^10 + 1428*a^19*b^8 - 792*a^21*b^6 + 288*a^23*b^4 - 62*a^25*b^2) - 2*a^26*b + 2*a^8*b^19 - 18*
a^10*b^17 + 72*a^12*b^15 - 168*a^14*b^13 + 252*a^16*b^11 - 252*a^18*b^9 + 168*a^20*b^7 - 72*a^22*b^5 + 18*a^24
*b^3))/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2))*1i)
/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2) + (b^3*(-(
a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)*(tan(c/2 + (d*x)/2)*(2*a^24 + 8*a^4*b^20 - 76*a^6*b^1
8 + 346*a^8*b^16 - 938*a^10*b^14 + 1612*a^12*b^12 - 1790*a^14*b^10 + 1276*a^16*b^8 - 566*a^18*b^6 + 148*a^20*b
^4 - 22*a^22*b^2) - 2*a^23*b + 4*a^5*b^19 - 37*a^7*b^17 + 164*a^9*b^15 - 433*a^11*b^13 + 722*a^13*b^11 - 769*a
^15*b^9 + 512*a^17*b^7 - 199*a^19*b^5 + 38*a^21*b^3 - (b^3*(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2
*b^2)/2)*(tan(c/2 + (d*x)/2)*(6*a^27 + 8*a^7*b^20 - 78*a^9*b^18 + 342*a^11*b^16 - 888*a^13*b^14 + 1512*a^15*b^
12 - 1764*a^17*b^10 + 1428*a^19*b^8 - 792*a^21*b^6 + 288*a^23*b^4 - 62*a^25*b^2) - 2*a^26*b + 2*a^8*b^19 - 18*
a^10*b^17 + 72*a^12*b^15 - 168*a^14*b^13 + 252*a^16*b^11 - 252*a^18*b^9 + 168*a^20*b^7 - 72*a^22*b^5 + 18*a^24
*b^3))/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2))*1i)
/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2))/(2*tan(c/
2 + (d*x)/2)*(2*a^3*b^18 - 41*a^5*b^16 + 225*a^7*b^14 - 715*a^9*b^12 + 1135*a^11*b^10 - 792*a^13*b^8 + 146*a^1
5*b^6 + 40*a^17*b^4) + 4*a^2*b^19 - 38*a^4*b^17 + 184*a^6*b^15 - 530*a^8*b^13 + 940*a^10*b^11 - 1034*a^12*b^9
+ 688*a^14*b^7 - 254*a^16*b^5 + 40*a^18*b^3 + (b^3*(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)
*(tan(c/2 + (d*x)/2)*(2*a^24 + 8*a^4*b^20 - 76*a^6*b^18 + 346*a^8*b^16 - 938*a^10*b^14 + 1612*a^12*b^12 - 1790
*a^14*b^10 + 1276*a^16*b^8 - 566*a^18*b^6 + 148*a^20*b^4 - 22*a^22*b^2) - 2*a^23*b + 4*a^5*b^19 - 37*a^7*b^17
+ 164*a^9*b^15 - 433*a^11*b^13 + 722*a^13*b^11 - 769*a^15*b^9 + 512*a^17*b^7 - 199*a^19*b^5 + 38*a^21*b^3 + (b
^3*(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)*(tan(c/2 + (d*x)/2)*(6*a^27 + 8*a^7*b^20 - 78*a
^9*b^18 + 342*a^11*b^16 - 888*a^13*b^14 + 1512*a^15*b^12 - 1764*a^17*b^10 + 1428*a^19*b^8 - 792*a^21*b^6 + 288
*a^23*b^4 - 62*a^25*b^2) - 2*a^26*b + 2*a^8*b^19 - 18*a^10*b^17 + 72*a^12*b^15 - 168*a^14*b^13 + 252*a^16*b^11
- 252*a^18*b^9 + 168*a^20*b^7 - 72*a^22*b^5 + 18*a^24*b^3))/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*
a^9*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2)))/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 -
35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2) - (b^3*(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)*(t
an(c/2 + (d*x)/2)*(2*a^24 + 8*a^4*b^20 - 76*a^6*b^18 + 346*a^8*b^16 - 938*a^10*b^14 + 1612*a^12*b^12 - 1790*a^
14*b^10 + 1276*a^16*b^8 - 566*a^18*b^6 + 148*a^20*b^4 - 22*a^22*b^2) - 2*a^23*b + 4*a^5*b^19 - 37*a^7*b^17 + 1
64*a^9*b^15 - 433*a^11*b^13 + 722*a^13*b^11 - 769*a^15*b^9 + 512*a^17*b^7 - 199*a^19*b^5 + 38*a^21*b^3 - (b^3*
(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)*(tan(c/2 + (d*x)/2)*(6*a^27 + 8*a^7*b^20 - 78*a^9*
b^18 + 342*a^11*b^16 - 888*a^13*b^14 + 1512*a^15*b^12 - 1764*a^17*b^10 + 1428*a^19*b^8 - 792*a^21*b^6 + 288*a^
23*b^4 - 62*a^25*b^2) - 2*a^26*b + 2*a^8*b^19 - 18*a^10*b^17 + 72*a^12*b^15 - 168*a^14*b^13 + 252*a^16*b^11 -
252*a^18*b^9 + 168*a^20*b^7 - 72*a^22*b^5 + 18*a^24*b^3))/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9
*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2)))/(a^17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 - 35
*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(10*a^4 + b^4 - (7*a^2*b^2)/2)*2i)/(d*(a^
17 - a^3*b^14 + 7*a^5*b^12 - 21*a^7*b^10 + 35*a^9*b^8 - 35*a^11*b^6 + 21*a^13*b^4 - 7*a^15*b^2))