3.262 \(\int \frac{\cos ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b d^{3/2}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{4 \sqrt{2} b d^{3/2}}-\frac{5 \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b d^{3/2}}+\frac{5 \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b d^{3/2}}-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}} \]

[Out]

(5*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(3/2)) - (5*ArcTan[1 + (Sqrt[2]*Sqrt[d*T
an[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(3/2)) - (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a +
b*x]]])/(8*Sqrt[2]*b*d^(3/2)) + (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqr
t[2]*b*d^(3/2)) - 5/(2*b*d*Sqrt[d*Tan[a + b*x]]) + Cos[a + b*x]^2/(2*b*d*Sqrt[d*Tan[a + b*x]])

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Rubi [A]  time = 0.183214, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {2607, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b d^{3/2}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{4 \sqrt{2} b d^{3/2}}-\frac{5 \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b d^{3/2}}+\frac{5 \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b d^{3/2}}-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^2/(d*Tan[a + b*x])^(3/2),x]

[Out]

(5*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(3/2)) - (5*ArcTan[1 + (Sqrt[2]*Sqrt[d*T
an[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(3/2)) - (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a +
b*x]]])/(8*Sqrt[2]*b*d^(3/2)) + (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqr
t[2]*b*d^(3/2)) - 5/(2*b*d*Sqrt[d*Tan[a + b*x]]) + Cos[a + b*x]^2/(2*b*d*Sqrt[d*Tan[a + b*x]])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

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 297

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 1162

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 617

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 204

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

Rule 1165

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 628

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]

Rubi steps

\begin{align*} \int \frac{\cos ^2(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(d x)^{3/2} \left (1+x^2\right )^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{(d x)^{3/2} \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{4 b}\\ &=-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}-\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{d x}}{1+x^2} \, dx,x,\tan (a+b x)\right )}{4 b d^2}\\ &=-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}-\frac{5 \operatorname{Subst}\left (\int \frac{x^2}{1+\frac{x^4}{d^2}} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{2 b d^3}\\ &=-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}+\frac{5 \operatorname{Subst}\left (\int \frac{d-x^2}{1+\frac{x^4}{d^2}} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{4 b d^3}-\frac{5 \operatorname{Subst}\left (\int \frac{d+x^2}{1+\frac{x^4}{d^2}} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{4 b d^3}\\ &=-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}-\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b d^{3/2}}-\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b d^{3/2}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 b d}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 b d}\\ &=-\frac{5 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b d^{3/2}}+\frac{5 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b d^{3/2}}-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b d^{3/2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b d^{3/2}}\\ &=\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b d^{3/2}}-\frac{5 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b d^{3/2}}-\frac{5 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b d^{3/2}}+\frac{5 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b d^{3/2}}-\frac{5}{2 b d \sqrt{d \tan (a+b x)}}+\frac{\cos ^2(a+b x)}{2 b d \sqrt{d \tan (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.282446, size = 115, normalized size = 0.46 \[ \frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left (-17 \cos (a+b x)+\cos (3 (a+b x))+5 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+5 \sqrt{\sin (2 (a+b x))} \log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{8 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x]*(-17*Cos[a + b*x] + Cos[3*(a + b*x)] + 5*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x
)]] + 5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]])*Sqrt[d*Tan[a + b*x]]
)/(8*b*d^2)

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Maple [C]  time = 0.144, size = 982, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/b*2^(1/2)*(5*I*cos(b*x+a)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*E
llipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(b*x+a)-1-sin(b*x+a))/sin
(b*x+a))^(1/2)-5*I*cos(b*x+a)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*(
-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2-1/2*
I,1/2*2^(1/2))+5*cos(b*x+a)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*Ell
ipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(b*x+a)-1-sin(b*x+a))/sin(b
*x+a))^(1/2)+5*cos(b*x+a)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*(-(co
s(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2-1/2*I,1/
2*2^(1/2))+5*I*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*EllipticPi((-(co
s(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)-
5*I*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*(-(cos(b*x+a)-1-sin(b*x+a))
/sin(b*x+a))^(1/2)*EllipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2-1/2*I,1/2*2^(1/2))+2*cos(b*x+a
)^3*2^(1/2)+5*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*(-(cos(b*x+a)-1-s
in(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))+5
*((cos(b*x+a)-1)/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a))/sin(b*x+a))^(1/2)*(-(cos(b*x+a)-1-sin(b*x+a))/si
n(b*x+a))^(1/2)*EllipticPi((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/2-1/2*I,1/2*2^(1/2))-10*cos(b*x+a)*
2^(1/2))*sin(b*x+a)/cos(b*x+a)^2/(d*sin(b*x+a)/cos(b*x+a))^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (a + b x \right )}}{\left (d \tan{\left (a + b x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x)**2/(d*tan(a + b*x))**(3/2), x)

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Giac [A]  time = 1.48193, size = 344, normalized size = 1.38 \begin{align*} -\frac{1}{16} \, d^{3}{\left (\frac{10 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{b d^{6}} + \frac{10 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{b d^{6}} - \frac{5 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt{2} \sqrt{d \tan \left (b x + a\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{b d^{6}} + \frac{5 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt{2} \sqrt{d \tan \left (b x + a\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{b d^{6}} + \frac{8 \,{\left (5 \, d^{2} \tan \left (b x + a\right )^{2} + 4 \, d^{2}\right )}}{{\left (\sqrt{d \tan \left (b x + a\right )} d^{2} \tan \left (b x + a\right )^{2} + \sqrt{d \tan \left (b x + a\right )} d^{2}\right )} b d^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2/(d*tan(b*x+a))^(3/2),x, algorithm="giac")

[Out]

-1/16*d^3*(10*sqrt(2)*abs(d)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(b*x + a)))/sqrt(abs
(d)))/(b*d^6) + 10*sqrt(2)*abs(d)^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sq
rt(abs(d)))/(b*d^6) - 5*sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) +
abs(d))/(b*d^6) + 5*sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(
d))/(b*d^6) + 8*(5*d^2*tan(b*x + a)^2 + 4*d^2)/((sqrt(d*tan(b*x + a))*d^2*tan(b*x + a)^2 + sqrt(d*tan(b*x + a)
)*d^2)*b*d^4))