3.5.16 \(\int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [416]
Optimal. Leaf size=156 \[ \frac {B \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d} \]
[Out]
-1/2*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/4*B*
ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)+1/4*B*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)+
2/3*B*tan(d*x+c)^(3/2)/d
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Rubi [A]
time = 0.07, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {21, 3554,
3557, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {B \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {B \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {B \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(Tan[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]
[Out]
(B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d
) - (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x
]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*B*Tan[c + d*x]^(3/2))/(3*d)
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 303
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))
Rule 335
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 631
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1176
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1179
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 3554
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 3557
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=B \int \tan ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}-B \int \sqrt {\tan (c+d x)} \, dx\\ &=\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {B \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {(2 B) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {B \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {B \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {B \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {B \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {B \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {B \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}\\ &=-\frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {B \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {B \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}\\ &=\frac {B \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 B \tan ^{\frac {3}{2}}(c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 38, normalized size = 0.24 \begin {gather*} -\frac {2 B \left (-1+\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right )\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Tan[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]
[Out]
(-2*B*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2])*Tan[c + d*x]^(3/2))/(3*d)
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Maple [A]
time = 0.05, size = 102, normalized size = 0.65
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method |
result |
size |
| | |
derivativedivides |
\(\frac {B \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) |
\(102\) |
default |
\(\frac {B \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) |
\(102\) |
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^(5/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
1/d*B*(2/3*tan(d*x+c)^(3/2)-1/4*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2
)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))
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Maxima [A]
time = 0.51, size = 123, normalized size = 0.79 \begin {gather*} \frac {8 \, B \tan \left (d x + c\right )^{\frac {3}{2}} - 3 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} B}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(5/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/12*(8*B*tan(d*x + c)^(3/2) - 3*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*a
rctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) +
1) + sqrt(2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))*B)/d
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 592 vs.
\(2 (124) = 248\).
time = 1.73, size = 592, normalized size = 3.79 \begin {gather*} \frac {12 \, \sqrt {2} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} B^{3} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} + B^{4} - \sqrt {2} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sqrt {2} B^{3} d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + B^{4} d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) + B^{6} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}}}{B^{4}}\right ) \cos \left (d x + c\right ) + 12 \, \sqrt {2} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} B^{3} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - B^{4} - \sqrt {2} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \sqrt {-\frac {\sqrt {2} B^{3} d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - B^{4} d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) - B^{6} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}}}{B^{4}}\right ) \cos \left (d x + c\right ) + 3 \, \sqrt {2} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \cos \left (d x + c\right ) \log \left (\frac {\sqrt {2} B^{3} d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + B^{4} d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) + B^{6} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) - 3 \, \sqrt {2} d \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \cos \left (d x + c\right ) \log \left (-\frac {\sqrt {2} B^{3} d^{3} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - B^{4} d^{2} \sqrt {\frac {B^{4}}{d^{4}}} \cos \left (d x + c\right ) - B^{6} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + 8 \, B \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(5/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/12*(12*sqrt(2)*d*(B^4/d^4)^(1/4)*arctan(-(sqrt(2)*B^3*d*(B^4/d^4)^(1/4)*sqrt(sin(d*x + c)/cos(d*x + c)) + B^
4 - sqrt(2)*d*(B^4/d^4)^(1/4)*sqrt((sqrt(2)*B^3*d^3*(B^4/d^4)^(3/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x +
c) + B^4*d^2*sqrt(B^4/d^4)*cos(d*x + c) + B^6*sin(d*x + c))/cos(d*x + c)))/B^4)*cos(d*x + c) + 12*sqrt(2)*d*(B
^4/d^4)^(1/4)*arctan(-(sqrt(2)*B^3*d*(B^4/d^4)^(1/4)*sqrt(sin(d*x + c)/cos(d*x + c)) - B^4 - sqrt(2)*d*(B^4/d^
4)^(1/4)*sqrt(-(sqrt(2)*B^3*d^3*(B^4/d^4)^(3/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) - B^4*d^2*sqrt(B^
4/d^4)*cos(d*x + c) - B^6*sin(d*x + c))/cos(d*x + c)))/B^4)*cos(d*x + c) + 3*sqrt(2)*d*(B^4/d^4)^(1/4)*cos(d*x
+ c)*log((sqrt(2)*B^3*d^3*(B^4/d^4)^(3/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) + B^4*d^2*sqrt(B^4/d^4
)*cos(d*x + c) + B^6*sin(d*x + c))/cos(d*x + c)) - 3*sqrt(2)*d*(B^4/d^4)^(1/4)*cos(d*x + c)*log(-(sqrt(2)*B^3*
d^3*(B^4/d^4)^(3/4)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c) - B^4*d^2*sqrt(B^4/d^4)*cos(d*x + c) - B^6*si
n(d*x + c))/cos(d*x + c)) + 8*B*sqrt(sin(d*x + c)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} B \int \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)**(5/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)
[Out]
B*Integral(tan(c + d*x)**(5/2), x)
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(5/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="giac")
[Out]
Timed out
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Mupad [B]
time = 11.61, size = 2500, normalized size = 16.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((tan(c + d*x)^(5/2)*(B*a + B*b*tan(c + d*x)))/(a + b*tan(c + d*x)),x)
[Out]
atan(((((((32*(4*B*a^2*b^8*d^4 + 8*B*a^4*b^6*d^4 + 4*B*a^6*b^4*d^4))/(b*d^5) - (32*tan(c + d*x)^(1/2)*(((64*B^
4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^
4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4))*(((6
4*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(4*B^2*a^5*b^5*d^2 - 14*B^2*a^3*b^7*d^2 + 2*B^2*a^
7*b^3*d^2 + 16*B^2*a^9*b*d^2))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*(B^3*a^5*b^5*d^2 - 15*B^3
*a^7*b^3*d^2 + 12*B^3*a^9*b*d^2))/(b*d^5))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b
^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*
B^4*a^10 - B^4*a^4*b^6))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^
(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((((32*(4*B*a^2*b^8*d^4 + 8*B*a
^4*b^6*d^4 + 4*B*a^6*b^4*d^4))/(b*d^5) + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^1
0*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^
4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (
32*tan(c + d*x)^(1/2)*(4*B^2*a^5*b^5*d^2 - 14*B^2*a^3*b^7*d^2 + 2*B^2*a^7*b^3*d^2 + 16*B^2*a^9*b*d^2))/(b*d^4)
)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^
4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*(B^3*a^5*b^5*d^2 - 15*B^3*a^7*b^3*d^2 + 12*B^3*a^9*b*d^2))/(b*d
^5))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(2*B^4*a^10 - B^4*a^4*b^6))/(b*d^4))*(((6
4*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((((32*(4*B*a^2*b^8*d^4 + 8*B*a^4*b^6*d^4 + 4*B*a^6*b^4*d^4))/(b*d^5)
- (32*tan(c + d*x)^(1/2)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) -
8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^
4 - 16*a^6*b^4*d^4))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2
) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(4*B^2*a^5*b^5*d
^2 - 14*B^2*a^3*b^7*d^2 + 2*B^2*a^7*b^3*d^2 + 16*B^2*a^9*b*d^2))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*
a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/
2) - (32*(B^3*a^5*b^5*d^2 - 15*B^3*a^7*b^3*d^2 + 12*B^3*a^9*b*d^2))/(b*d^5))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(
16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^
(1/2) + (32*tan(c + d*x)^(1/2)*(2*B^4*a^10 - B^4*a^4*b^6))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^
4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (
((((32*(4*B*a^2*b^8*d^4 + 8*B*a^4*b^6*d^4 + 4*B*a^6*b^4*d^4))/(b*d^5) + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^6*b
^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 +
2*a^2*b^2*d^4)))^(1/2)*(16*b^10*d^4 + 16*a^2*b^8*d^4 - 16*a^4*b^6*d^4 - 16*a^6*b^4*d^4))/(b*d^4))*(((64*B^4*a
^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d
^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*B^2*a^5*b^5*d^2 - 14*B^2*a^3*b^7*d^2 + 2*B^2*a^7*b^3*d
^2 + 16*B^2*a^9*b*d^2))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(
1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*(B^3*a^5*b^5*d^2 - 15*B^3*a^7*b^
3*d^2 + 12*B^3*a^9*b*d^2))/(b*d^5))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4)
)^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(2*B^4*a^1
0 - B^4*a^4*b^6))/(b*d^4))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) -
8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (64*(B^5*a^10 - B^5*a^8*b^2))/(b*d^5)))*((
(64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^
4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^
4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*...
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