3.5.20 \(\int \frac {a B+b B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx\) [420]

Optimal. Leaf size=154 \[ \frac {B \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 B}{d \sqrt {\tan (c+d x)}} \]

[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*B/d/tan(d*x+c)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 154, 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, 3555, 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}{d \sqrt {\tan (c+d x)}}-\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[(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(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)/(d*Sqrt[Tan[c + d*x]])

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 3555

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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 {a B+b B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx &=B \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {2 B}{d \sqrt {\tan (c+d x)}}-B \int \sqrt {\tan (c+d x)} \, dx\\ &=-\frac {2 B}{d \sqrt {\tan (c+d x)}}-\frac {B \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {2 B}{d \sqrt {\tan (c+d x)}}-\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}{d \sqrt {\tan (c+d x)}}+\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}{d \sqrt {\tan (c+d x)}}-\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}{d \sqrt {\tan (c+d x)}}-\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}{d \sqrt {\tan (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 34, normalized size = 0.22 \begin {gather*} -\frac {2 B \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(c+d x)\right )}{d \sqrt {\tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B*Hypergeometric2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2])/(d*Sqrt[Tan[c + d*x]])

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Maple [A]
time = 0.05, size = 102, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {B \left (-\frac {2}{\sqrt {\tan \left (d x +c \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(102\)
default	\(\frac {B \left (-\frac {2}{\sqrt {\tan \left (d x +c \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*B*(-2/tan(d*x+c)^(1/2)-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 = 0.52, size = 122, normalized size = 0.79 \begin {gather*} -\frac {{\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 + \frac {8 \, B}{\sqrt {\tan \left (d x + c\right )}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*B+b*B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/4*((2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*arctan(-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 + 8*B/sqrt(tan(d*x + c)))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (124) = 248\).
time = 1.56, size = 642, normalized size = 4.17 \begin {gather*} \frac {8 \, B \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \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 ) + 4 \, {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \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 ) + {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \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 ) - {\left (\sqrt {2} d \cos \left (d x + c\right )^{2} - \sqrt {2} d\right )} \left (\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \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 )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*B*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 4*(sqrt(2)*d*cos(d*x + c)^2 - 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) + 4*(sqrt(2)*d*cos(d*x + c)^2 - 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) + (sqrt(2)*d*cos(d*x + c)^2 - sqrt(2)*d)*(B^4/d^4)^(1/
4)*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)*c
os(d*x + c) + B^6*sin(d*x + c))/cos(d*x + c)) - (sqrt(2)*d*cos(d*x + c)^2 - sqrt(2)*d)*(B^4/d^4)^(1/4)*log(-(s
qrt(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)))/(d*cos(d*x + c)^2 - d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} B \int \frac {1}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*B+b*B*tan(d*x+c))/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c)),x)

[Out]

B*Integral(tan(c + d*x)**(-3/2), 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((a*B+b*B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 10.89, size = 2500, normalized size = 16.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*a + B*b*tan(c + d*x))/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))),x)

[Out]

atan(((tan(c + d*x)^(1/2)*(64*B^4*a^2*b^7*d^5 - 32*B^4*a^4*b^5*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)*(
(tan(c + d*x)^(1/2)*(128*B^2*a^5*b^4*d^7 - 448*B^2*a^3*b^6*d^7 + 64*B^2*a^7*b^2*d^7 + 512*B^2*a*b^8*d^7) - (((
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)*(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)*(512*b^9*d^9 +
512*a^2*b^7*d^9 - 512*a^4*b^5*d^9 - 512*a^6*b^3*d^9) - 512*B*b^9*d^8 - 640*B*a^2*b^7*d^8 + 256*B*a^4*b^5*d^8 +
384*B*a^6*b^3*d^8))*(((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^4*d^6 - 32*B^3*a^7*b^2*d^6 + 128*B^
3*a*b^8*d^6))*(((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 + (tan(c + d*x)^(1/2)*(64*B^4*a^2*b^7*d^5 - 32*B^4*a^4
*b^5*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)*((tan(c + d*x)^(1/2)*(128*B^2*a^5*b^4*d^7 - 448*B^2*a^3*b^6
*d^7 + 64*B^2*a^7*b^2*d^7 + 512*B^2*a*b^8*d^7) - (((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)*(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)*(512*b^9*d^9 + 512*a^2*b^7*d^9 - 512*a^4*b^5*d^9 - 512*a^6*b^3*d^9) + 512
*B*b^9*d^8 + 640*B*a^2*b^7*d^8 - 256*B*a^4*b^5*d^8 - 384*B*a^6*b^3*d^8))*(((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^4*d^6 + 32*B^3*a^7*b^2*d^6 - 128*B^3*a*b^8*d^6))*(((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)/(
(tan(c + d*x)^(1/2)*(64*B^4*a^2*b^7*d^5 - 32*B^4*a^4*b^5*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)*((tan(c
+ d*x)^(1/2)*(128*B^2*a^5*b^4*d^7 - 448*B^2*a^3*b^6*d^7 + 64*B^2*a^7*b^2*d^7 + 512*B^2*a*b^8*d^7) - (((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)*(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 + 3
2*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)*(512*b^9*d^9 + 512*a^
2*b^7*d^9 - 512*a^4*b^5*d^9 - 512*a^6*b^3*d^9) - 512*B*b^9*d^8 - 640*B*a^2*b^7*d^8 + 256*B*a^4*b^5*d^8 + 384*B
*a^6*b^3*d^8))*(((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^4*d^6 - 32*B^3*a^7*b^2*d^6 + 128*B^3*a*b^
8*d^6))*(((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) - (tan(c + d*x)^(1/2)*(64*B^4*a^2*b^7*d^5 - 32*B^4*a^4*b^5*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)*((tan(c + d*x)^(1/2)*(128*B^2*a^5*b^4*d^7 - 448*B^2*a^3*b^6*d^7 + 64
*B^2*a^7*b^2*d^7 + 512*B^2*a*b^8*d^7) - (((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)*(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)*(512*b^9*d^9 + 512*a^2*b^7*d^9 - 512*a^4*b^5*d^9 - 512*a^6*b^3*d^9) + 512*B*b^9*d^
8 + 640*B*a^2*b^7*d^8 - 256*B*a^4*b^5*d^8 - 384*B*a^6*b^3*d^8))*(((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^4*d^6 + 32*B^3*a^7*b^2*d^6 - 128*B^3*a*b^8*d^6))*(((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^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*(16*B*b^9*d^2 + 28*B*a^2*b^7*d^2 + 8*B*a^4*b^5*d^2 - 4*B*a^6*b^3
*d^2))/d^3 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4)
)^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d...

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