\(\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [523]

Optimal result

Integrand size = 21, antiderivative size = 138 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

-2*a^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+(a-I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2
))/d+(a+I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d+2*b^2*(a+b*tan(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3647, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b^2 \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Int[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(-2*a^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + ((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/S
qrt[a - I*b]])/d + ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b^2*Sqrt[a + b*Tan
[c + d*x]])/d

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

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

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*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[n, 0] &&  !LeQ[n, -
1]

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.59 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\frac {-2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+(a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+a^2 \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 i a \sqrt {a+i b} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-\sqrt {a+i b} b^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 b^2 \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(-2*a^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[
a - I*b]] + a^2*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + (2*I)*a*Sqrt[a + I*b]*b*ArcTan
h[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - Sqrt[a + I*b]*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]
+ 2*b^2*Sqrt[a + b*Tan[c + d*x]])/d

Maple [F(-1)]

Timed out.

hanged

[In]

int(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2),x)

[Out]

int(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1121 vs. \(2 (110) = 220\).

Time = 0.36 (sec) , antiderivative size = 2255, normalized size of antiderivative = 16.34 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[1/2*(2*a^(5/2)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c)) + 4*sqrt(b*tan(d
*x + c) + a)*b^2 - d*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20
*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) + (2*a*d^3*sqr
t(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)
*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/
d^4))/d^2)) + d*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*
b^8 + b^10)/d^4))/d^2)*log((5*a^8 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) - (2*a*d^3*sqrt(-(2
5*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*s
qrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))
/d^2)) + d*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 +
b^10)/d^4))/d^2)*log((5*a^8 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) + (2*a*d^3*sqrt(-(25*a^8
*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((
a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)
) - d*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10
)/d^4))/d^2)*log((5*a^8 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) - (2*a*d^3*sqrt(-(25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((a^5 -
10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)))/d,
1/2*(4*sqrt(-a)*a^2*arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a) + 4*sqrt(b*tan(d*x + c) + a)*b^2 - d*sqrt((a^
5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*l
og((5*a^8 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) + (2*a*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4
+ 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((a^5 - 10*a^3*b^2 +
5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)) + d*sqrt((a^5 - 1
0*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5
*a^8 - 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) - (2*a*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110
*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b
^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)) + d*sqrt((a^5 - 10*a^3
*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8
- 14*a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) + (2*a*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*
b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 -
d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)) - d*sqrt((a^5 - 10*a^3*b^2
+ 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8 - 14*
a^4*b^4 - 8*a^2*b^6 + b^8)*sqrt(b*tan(d*x + c) + a) - (2*a*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 -
20*a^2*b^8 + b^10)/d^4) + (5*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - b^6)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*s
qrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4))/d^2)))/d]

Sympy [F]

\[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(5/2)*cot(c + d*x), x)

Maxima [F]

\[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right ) \,d x } \]

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(5/2)*cot(d*x + c), x)

Giac [F(-1)]

Timed out. \[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.24 (sec) , antiderivative size = 3333, normalized size of antiderivative = 24.15 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)*(a + b*tan(c + d*x))^(5/2),x)

[Out]

log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((((
(-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d
^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a
^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((128*a*b^8*(3*a^4 + b^4
+ 4*a^2*b^2))/d - 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2
*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 + (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(9*a^6
+ 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))/2 - (96*a^2*b^8*(a^8 + b^8 + 16*a^2*b^6 - 10*a^4*b^4 - 24*a^6*b^2))/
d^3))/2 - (32*b^8*(a + b*tan(c + d*x))^(1/2)*(3*a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 18*a^6*b^6 + 45*a^8*b^
4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*(a^2 + b^2)^3*(6*a^4 + b^4 + 3*a^2*b^2))/d^5)*((20*a^2*b^8*d^4 - b^10
*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4) + a^5/(4*d^2) + (5*a*b^4)/(4*d^2) - (
5*a^3*b^2)/(2*d^2))^(1/2) - log(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*
a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2
*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d
^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/
2)*((128*a*b^8*(3*a^4 + b^4 + 4*a^2*b^2))/d + 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^
2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 - (64*a*b^8*(a + b
*tan(c + d*x))^(1/2)*(9*a^6 + 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))/2 - (96*a^2*b^8*(a^8 + b^8 + 16*a^2*b^6
- 10*a^4*b^4 - 24*a^6*b^2))/d^3))/2 + (32*b^8*(a + b*tan(c + d*x))^(1/2)*(3*a^12 + b^12 + 6*a^2*b^10 + 15*a^4*
b^8 + 18*a^6*b^6 + 45*a^8*b^4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*(a^2 + b^2)^3*(6*a^4 + b^4 + 3*a^2*b^2))/
d^5)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*
b^4*d^2 - 10*a^3*b^2*d^2)/(4*d^4))^(1/2) - log(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5
*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4
*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 +
10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^
3*b^2*d^2)/d^4)^(1/2)*((128*a*b^8*(3*a^4 + b^4 + 4*a^2*b^2))/d + 128*b^8*(-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^
2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/
2 - (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(9*a^6 + 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))/2 - (96*a^2*b^8*(a^8
+ b^8 + 16*a^2*b^6 - 10*a^4*b^4 - 24*a^6*b^2))/d^3))/2 + (32*b^8*(a + b*tan(c + d*x))^(1/2)*(3*a^12 + b^12 +
6*a^2*b^10 + 15*a^4*b^8 + 18*a^6*b^6 + 45*a^8*b^4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*(a^2 + b^2)^3*(6*a^4
+ b^4 + 3*a^2*b^2))/d^5)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(
1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*d^4))^(1/2) + log(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2
)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2
) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^
5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2
- 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((128*a*b^8*(3*a^4 + b^4 + 4*a^2*b^2))/d - 128*b^8*(-((-b^2*d^4*(5*
a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*t
an(c + d*x))^(1/2)))/2 + (64*a*b^8*(a + b*tan(c + d*x))^(1/2)*(9*a^6 + 19*b^6 - 5*a^2*b^4 - 51*a^4*b^2))/d^2))
/2 - (96*a^2*b^8*(a^8 + b^8 + 16*a^2*b^6 - 10*a^4*b^4 - 24*a^6*b^2))/d^3))/2 - (32*b^8*(a + b*tan(c + d*x))^(1
/2)*(3*a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 18*a^6*b^6 + 45*a^8*b^4 - 24*a^10*b^2))/d^4))/2 + (32*a^3*b^10*
(a^2 + b^2)^3*(6*a^4 + b^4 + 3*a^2*b^2))/d^5)*(a^5/(4*d^2) - (20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 10
0*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4) + (5*a*b^4)/(4*d^2) - (5*a^3*b^2)/(2*d^2))^(1/2) + (2*b^2*(a + b
*tan(c + d*x))^(1/2))/d + (atan((b^20*(a^5)^(1/2)*(a + b*tan(c + d*x))^(1/2)*64i)/(64*a^3*b^20 + 384*a^5*b^18
+ 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016*a^13*b^10 + 576*a^15*b^8) + (a^2*b^18*(a^5)^(1/2)*(a +
b*tan(c + d*x))^(1/2)*384i)/(64*a^3*b^20 + 384*a^5*b^18 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016
*a^13*b^10 + 576*a^15*b^8) + (a^4*b^16*(a^5)^(1/2)*(a + b*tan(c + d*x))^(1/2)*960i)/(64*a^3*b^20 + 384*a^5*b^1
8 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016*a^13*b^10 + 576*a^15*b^8) - (a^6*b^14*(a^5)^(1/2)*(a
+ b*tan(c + d*x))^(1/2)*1280i)/(64*a^3*b^20 + 384*a^5*b^18 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6
016*a^13*b^10 + 576*a^15*b^8) + (a^8*b^12*(a^5)^(1/2)*(a + b*tan(c + d*x))^(1/2)*3520i)/(64*a^3*b^20 + 384*a^5
*b^18 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016*a^13*b^10 + 576*a^15*b^8) + (a^10*b^10*(a^5)^(1/2
)*(a + b*tan(c + d*x))^(1/2)*6016i)/(64*a^3*b^20 + 384*a^5*b^18 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^1
2 + 6016*a^13*b^10 + 576*a^15*b^8) + (a^12*b^8*(a^5)^(1/2)*(a + b*tan(c + d*x))^(1/2)*576i)/(64*a^3*b^20 + 384
*a^5*b^18 + 960*a^7*b^16 - 1280*a^9*b^14 + 3520*a^11*b^12 + 6016*a^13*b^10 + 576*a^15*b^8))*(a^5)^(1/2)*2i)/d