\(\int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [545]

Optimal result

Integrand size = 23, antiderivative size = 241 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}} \]

[Out]

1/4*(8*a^2-15*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(7/2)/d-arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1
/2))/(a-I*b)^(3/2)/d-arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d+1/4*b^2*(7*a^2+15*b^2)/a^3/
(a^2+b^2)/d/(a+b*tan(d*x+c))^(1/2)+5/4*b*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))^(1/2)-1/2*cot(d*x+c)^2/a/d/(a+b*tan
(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3650, 3730, 3731, 3734, 3620, 3618, 65, 214, 3715} \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d}+\frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}} \]

[In]

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

[Out]

((8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(7/2)*d) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/
Sqrt[a - I*b]]/((a - I*b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(3/2)*d) + (b^
2*(7*a^2 + 15*b^2))/(4*a^3*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (5*b*Cot[c + d*x])/(4*a^2*d*Sqrt[a + b*Ta
n[c + d*x]]) - Cot[c + d*x]^2/(2*a*d*Sqrt[a + b*Tan[c + d*x]])

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 3650

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 + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*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] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !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 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, 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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3731

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*t
an[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x]
)^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[
e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1)
+ a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^
2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

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 {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {\int \frac {\cot ^2(c+d x) \left (\frac {5 b}{2}+2 a \tan (c+d x)+\frac {5}{2} b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{2 a} \\ & = \frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {\cot (c+d x) \left (\frac {1}{4} \left (-8 a^2+15 b^2\right )+\frac {15}{4} b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{2 a^2} \\ & = \frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {\cot (c+d x) \left (-\frac {1}{8} \left (8 a^2-15 b^2\right ) \left (a^2+b^2\right )+a^3 b \tan (c+d x)+\frac {1}{8} b^2 \left (7 a^2+15 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )} \\ & = \frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{8 a^3}+\frac {\int \frac {a^3 b+a^4 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )} \\ & = \frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a-b)}+\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a+b)}-\frac {\left (8 a^2-15 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 a^3 d} \\ & = \frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b) d}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b) d}-\frac {\left (8 a^2-15 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{4 a^3 b d} \\ & = \frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d}+\frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}+\frac {\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 )}{(i a-b) b d}-\frac {\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 (i a+b) d} \\ & = \frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.85 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {-\frac {4 a^3 \left (a+\sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}+\frac {4 a^3 \left (-a+\sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}+\frac {7 a^2 b^2+15 b^4+5 a b \left (a^2+b^2\right ) \cot (c+d x)-2 a^2 \left (a^2+b^2\right ) \cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}}{4 a^2 d} \]

[In]

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

[Out]

(((8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + ((-4*a^3*(a + Sqrt[-b^2])*ArcTanh[Sqrt
[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/Sqrt[a - Sqrt[-b^2]] + (4*a^3*(-a + Sqrt[-b^2])*ArcTanh[Sqrt[a + b
*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + (7*a^2*b^2 + 15*b^4 + 5*a*b*(a^2 + b^2)*Cot[c + d
*x] - 2*a^2*(a^2 + b^2)*Cot[c + d*x]^2)/Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)))/(4*a^2*d)

Maple [F(-1)]

Timed out.

hanged

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2182 vs. \(2 (201) = 402\).

Time = 0.55 (sec) , antiderivative size = 4381, normalized size of antiderivative = 18.18 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/8*(4*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^
8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sq
rt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^
12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*a^4 - 4*a^2*b^2 + b^4)*
d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*
a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^2))) - 4*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(((a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15
*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 -
b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^
6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*a^4 - 4*a^2*b^2
+ b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^
2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2
*b^4 + b^6)*d^2))) - 4*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(-((a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*
b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(
3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*
b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*a^4 - 4*
a^2*b^2 + b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 +
6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6)*d^2))) + 4*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(-
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4
+ 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2
))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2
- 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3
*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/
((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + ((8*a^4*b - 7*a^2*b^3 - 15*b^5)*tan(d*x + c)^3 + (8*a^5 - 7*a^3*b^2 - 1
5*a*b^4)*tan(d*x + c)^2)*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c))
+ 2*(2*a^5 + 2*a^3*b^2 - (7*a^3*b^2 + 15*a*b^4)*tan(d*x + c)^2 - 5*(a^4*b + a^2*b^3)*tan(d*x + c))*sqrt(b*tan
(d*x + c) + a))/((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2), -1/4*(((8*a^4*b - 7*a
^2*b^3 - 15*b^5)*tan(d*x + c)^3 + (8*a^5 - 7*a^3*b^2 - 15*a*b^4)*tan(d*x + c)^2)*sqrt(-a)*arctan(sqrt(b*tan(d*
x + c) + a)*sqrt(-a)/a) + 2*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*
a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*lo
g(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*
a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*a^4
- 4*a^2*b^2 + b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12
+ 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^2))) - 2*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqr
t(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^
4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d
^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b
^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) -
(3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)
/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - 2*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)
^2)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 1
5*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-
(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*
d^4)) + (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b
^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^
2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + 2*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan
(d*x + c)^2)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^1
0*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3
*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d
^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4)) + (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2
- 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3
+ 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + (2*a^5 + 2*a^3*b^2 - (7*a^3*b^2 + 15*a*b^4)*tan(d*x
+ c)^2 - 5*(a^4*b + a^2*b^3)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^
7 + a^5*b^2)*d*tan(d*x + c)^2)]

Sympy [F]

\[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 6216, normalized size of antiderivative = 25.79 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*b^4)/(a*(a^2 + b^2)) - ((25*b^4 + 9*a^2*b^2)*(a + b*tan(c + d*x)))/(4*a^2*(a^2 + b^2)) + (b^2*(7*a^2 + 15*
b^2)*(a + b*tan(c + d*x))^2)/(4*a^3*(a^2 + b^2)))/(d*(a + b*tan(c + d*x))^(5/2) - 2*a*d*(a + b*tan(c + d*x))^(
3/2) + a^2*d*(a + b*tan(c + d*x))^(1/2)) + log((((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2
*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4
)))^(1/2)*((((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^
3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(251658240*a^24*b^30*d^8 -
(a + b*tan(c + d*x))^(1/2)*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2
*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(134217728
*a^27*b^28*d^9 + 1409286144*a^29*b^26*d^9 + 6643777536*a^31*b^24*d^9 + 18522046464*a^33*b^22*d^9 + 33822867456
*a^35*b^20*d^9 + 42278584320*a^37*b^18*d^9 + 36641439744*a^39*b^16*d^9 + 21743271936*a^41*b^14*d^9 + 845571686
4*a^43*b^12*d^9 + 1946157056*a^45*b^10*d^9 + 201326592*a^47*b^8*d^9) + 2382364672*a^26*b^28*d^8 + 9948889088*a
^28*b^26*d^8 + 23924310016*a^30*b^24*d^8 + 36071014400*a^32*b^22*d^8 + 34292629504*a^34*b^20*d^8 + 18555600896
*a^36*b^18*d^8 + 2483027968*a^38*b^16*d^8 - 3841982464*a^40*b^14*d^8 - 2852126720*a^42*b^12*d^8 - 855638016*a^
44*b^10*d^8 - 100663296*a^46*b^8*d^8) + (a + b*tan(c + d*x))^(1/2)*(235929600*a^22*b^30*d^7 + 1871708160*a^24*
b^28*d^7 + 6295650304*a^26*b^26*d^7 + 11144265728*a^28*b^24*d^7 + 9560915968*a^30*b^22*d^7 - 337641472*a^32*b^
20*d^7 - 9307160576*a^34*b^18*d^7 - 8887730176*a^36*b^16*d^7 - 2943352832*a^38*b^14*d^7 + 621805568*a^40*b^12*
d^7 + 721420288*a^42*b^10*d^7 + 150994944*a^44*b^8*d^7))*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^
4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^
4*b^2*d^4)))^(1/2) + 117964800*a^21*b^30*d^6 + 699924480*a^23*b^28*d^6 + 1889533952*a^25*b^26*d^6 + 3336568832
*a^27*b^24*d^6 + 4495245312*a^29*b^22*d^6 + 4279238656*a^31*b^20*d^6 + 1923088384*a^33*b^18*d^6 - 773849088*a^
35*b^16*d^6 - 1421344768*a^37*b^14*d^6 - 587726848*a^39*b^12*d^6 - 25165824*a^41*b^10*d^6 + 25165824*a^43*b^8*
d^6) - (a + b*tan(c + d*x))^(1/2)*(704643072*a^29*b^20*d^5 - 290979840*a^23*b^26*d^5 - 465043456*a^25*b^24*d^5
- 37224448*a^27*b^22*d^5 - 58982400*a^21*b^28*d^5 + 767033344*a^31*b^18*d^5 + 238551040*a^33*b^16*d^5 + 15728
64*a^35*b^14*d^5 + 92536832*a^37*b^12*d^5 + 96468992*a^39*b^10*d^5 + 25165824*a^41*b^8*d^5))*((((8*a^3*d^2 - 2
4*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*
d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + 29491200*a^22*b^26*d^4 + 190709760*a^24*b^24*d^4 + 50
9214720*a^26*b^22*d^4 + 701890560*a^28*b^20*d^4 + 481689600*a^30*b^18*d^4 + 68812800*a^32*b^16*d^4 - 123863040
*a^34*b^14*d^4 - 80609280*a^36*b^12*d^4 - 15728640*a^38*b^10*d^4)*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4
- a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d
^4 + 3*a^4*b^2*d^4)))^(1/2) + log(((-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3
*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*((
(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a
*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(251658240*a^24*b^30*d^8 - (a + b*tan
(c + d*x))^(1/2)*(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2
) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(134217728*a^27*b^28
*d^9 + 1409286144*a^29*b^26*d^9 + 6643777536*a^31*b^24*d^9 + 18522046464*a^33*b^22*d^9 + 33822867456*a^35*b^20
*d^9 + 42278584320*a^37*b^18*d^9 + 36641439744*a^39*b^16*d^9 + 21743271936*a^41*b^14*d^9 + 8455716864*a^43*b^1
2*d^9 + 1946157056*a^45*b^10*d^9 + 201326592*a^47*b^8*d^9) + 2382364672*a^26*b^28*d^8 + 9948889088*a^28*b^26*d
^8 + 23924310016*a^30*b^24*d^8 + 36071014400*a^32*b^22*d^8 + 34292629504*a^34*b^20*d^8 + 18555600896*a^36*b^18
*d^8 + 2483027968*a^38*b^16*d^8 - 3841982464*a^40*b^14*d^8 - 2852126720*a^42*b^12*d^8 - 855638016*a^44*b^10*d^
8 - 100663296*a^46*b^8*d^8) + (a + b*tan(c + d*x))^(1/2)*(235929600*a^22*b^30*d^7 + 1871708160*a^24*b^28*d^7 +
6295650304*a^26*b^26*d^7 + 11144265728*a^28*b^24*d^7 + 9560915968*a^30*b^22*d^7 - 337641472*a^32*b^20*d^7 - 9
307160576*a^34*b^18*d^7 - 8887730176*a^36*b^16*d^7 - 2943352832*a^38*b^14*d^7 + 621805568*a^40*b^12*d^7 + 7214
20288*a^42*b^10*d^7 + 150994944*a^44*b^8*d^7))*(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2
*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4
)))^(1/2) + 117964800*a^21*b^30*d^6 + 699924480*a^23*b^28*d^6 + 1889533952*a^25*b^26*d^6 + 3336568832*a^27*b^2
4*d^6 + 4495245312*a^29*b^22*d^6 + 4279238656*a^31*b^20*d^6 + 1923088384*a^33*b^18*d^6 - 773849088*a^35*b^16*d
^6 - 1421344768*a^37*b^14*d^6 - 587726848*a^39*b^12*d^6 - 25165824*a^41*b^10*d^6 + 25165824*a^43*b^8*d^6) - (a
+ b*tan(c + d*x))^(1/2)*(704643072*a^29*b^20*d^5 - 290979840*a^23*b^26*d^5 - 465043456*a^25*b^24*d^5 - 372244
48*a^27*b^22*d^5 - 58982400*a^21*b^28*d^5 + 767033344*a^31*b^18*d^5 + 238551040*a^33*b^16*d^5 + 1572864*a^35*b
^14*d^5 + 92536832*a^37*b^12*d^5 + 96468992*a^39*b^10*d^5 + 25165824*a^41*b^8*d^5))*(-(((8*a^3*d^2 - 24*a*b^2*
d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^
6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + 29491200*a^22*b^26*d^4 + 190709760*a^24*b^24*d^4 + 509214720*
a^26*b^22*d^4 + 701890560*a^28*b^20*d^4 + 481689600*a^30*b^18*d^4 + 68812800*a^32*b^16*d^4 - 123863040*a^34*b^
14*d^4 - 80609280*a^36*b^12*d^4 - 15728640*a^38*b^10*d^4)*(-(((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*
d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*
a^4*b^2*d^4)))^(1/2) - log(((a + b*tan(c + d*x))^(1/2)*(704643072*a^29*b^20*d^5 - 290979840*a^23*b^26*d^5 - 46
5043456*a^25*b^24*d^5 - 37224448*a^27*b^22*d^5 - 58982400*a^21*b^28*d^5 + 767033344*a^31*b^18*d^5 + 238551040*
a^33*b^16*d^5 + 1572864*a^35*b^14*d^5 + 92536832*a^37*b^12*d^5 + 96468992*a^39*b^10*d^5 + 25165824*a^41*b^8*d^
5) + ((a^3*d^2 + (6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^
2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2)*((((a^3*d^2 + (6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - 3*a*b^2*d^2
)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2)*(((a^3*d^2 + (6*a^2*b^4*d^4 - b^6*d^4 - 9*a
^4*b^2*d^4)^(1/2) - 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2)*(a + b*tan(c
+ d*x))^(1/2)*(134217728*a^27*b^28*d^9 + 1409286144*a^29*b^26*d^9 + 6643777536*a^31*b^24*d^9 + 18522046464*a^
33*b^22*d^9 + 33822867456*a^35*b^20*d^9 + 42278584320*a^37*b^18*d^9 + 36641439744*a^39*b^16*d^9 + 21743271936*
a^41*b^14*d^9 + 8455716864*a^43*b^12*d^9 + 1946157056*a^45*b^10*d^9 + 201326592*a^47*b^8*d^9) + 251658240*a^24
*b^30*d^8 + 2382364672*a^26*b^28*d^8 + 9948889088*a^28*b^26*d^8 + 23924310016*a^30*b^24*d^8 + 36071014400*a^32
*b^22*d^8 + 34292629504*a^34*b^20*d^8 + 18555600896*a^36*b^18*d^8 + 2483027968*a^38*b^16*d^8 - 3841982464*a^40
*b^14*d^8 - 2852126720*a^42*b^12*d^8 - 855638016*a^44*b^10*d^8 - 100663296*a^46*b^8*d^8) - (a + b*tan(c + d*x)
)^(1/2)*(235929600*a^22*b^30*d^7 + 1871708160*a^24*b^28*d^7 + 6295650304*a^26*b^26*d^7 + 11144265728*a^28*b^24
*d^7 + 9560915968*a^30*b^22*d^7 - 337641472*a^32*b^20*d^7 - 9307160576*a^34*b^18*d^7 - 8887730176*a^36*b^16*d^
7 - 2943352832*a^38*b^14*d^7 + 621805568*a^40*b^12*d^7 + 721420288*a^42*b^10*d^7 + 150994944*a^44*b^8*d^7))*((
a^3*d^2 + (6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d
^4 + 12*a^4*b^2*d^4))^(1/2) + 117964800*a^21*b^30*d^6 + 699924480*a^23*b^28*d^6 + 1889533952*a^25*b^26*d^6 + 3
336568832*a^27*b^24*d^6 + 4495245312*a^29*b^22*d^6 + 4279238656*a^31*b^20*d^6 + 1923088384*a^33*b^18*d^6 - 773
849088*a^35*b^16*d^6 - 1421344768*a^37*b^14*d^6 - 587726848*a^39*b^12*d^6 - 25165824*a^41*b^10*d^6 + 25165824*
a^43*b^8*d^6))*((a^3*d^2 + (6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d
^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2) + 29491200*a^22*b^26*d^4 + 190709760*a^24*b^24*d^4 + 509214720*a^
26*b^22*d^4 + 701890560*a^28*b^20*d^4 + 481689600*a^30*b^18*d^4 + 68812800*a^32*b^16*d^4 - 123863040*a^34*b^14
*d^4 - 80609280*a^36*b^12*d^4 - 15728640*a^38*b^10*d^4)*((a^3*d^2 + (6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^
(1/2) - 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2) - log(((a + b*tan(c + d*
x))^(1/2)*(704643072*a^29*b^20*d^5 - 290979840*a^23*b^26*d^5 - 465043456*a^25*b^24*d^5 - 37224448*a^27*b^22*d^
5 - 58982400*a^21*b^28*d^5 + 767033344*a^31*b^18*d^5 + 238551040*a^33*b^16*d^5 + 1572864*a^35*b^14*d^5 + 92536
832*a^37*b^12*d^5 + 96468992*a^39*b^10*d^5 + 25165824*a^41*b^8*d^5) + (-((6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*
d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2)*(((-((6*a
^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 +
12*a^4*b^2*d^4))^(1/2)*((-((6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*a^6*d^
4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(134217728*a^27*b^28*d^9 +
1409286144*a^29*b^26*d^9 + 6643777536*a^31*b^24*d^9 + 18522046464*a^33*b^22*d^9 + 33822867456*a^35*b^20*d^9 +
42278584320*a^37*b^18*d^9 + 36641439744*a^39*b^16*d^9 + 21743271936*a^41*b^14*d^9 + 8455716864*a^43*b^12*d^9 +
1946157056*a^45*b^10*d^9 + 201326592*a^47*b^8*d^9) + 251658240*a^24*b^30*d^8 + 2382364672*a^26*b^28*d^8 + 994
8889088*a^28*b^26*d^8 + 23924310016*a^30*b^24*d^8 + 36071014400*a^32*b^22*d^8 + 34292629504*a^34*b^20*d^8 + 18
555600896*a^36*b^18*d^8 + 2483027968*a^38*b^16*d^8 - 3841982464*a^40*b^14*d^8 - 2852126720*a^42*b^12*d^8 - 855
638016*a^44*b^10*d^8 - 100663296*a^46*b^8*d^8) - (a + b*tan(c + d*x))^(1/2)*(235929600*a^22*b^30*d^7 + 1871708
160*a^24*b^28*d^7 + 6295650304*a^26*b^26*d^7 + 11144265728*a^28*b^24*d^7 + 9560915968*a^30*b^22*d^7 - 33764147
2*a^32*b^20*d^7 - 9307160576*a^34*b^18*d^7 - 8887730176*a^36*b^16*d^7 - 2943352832*a^38*b^14*d^7 + 621805568*a
^40*b^12*d^7 + 721420288*a^42*b^10*d^7 + 150994944*a^44*b^8*d^7))*(-((6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)
^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2) + 117964800*a
^21*b^30*d^6 + 699924480*a^23*b^28*d^6 + 1889533952*a^25*b^26*d^6 + 3336568832*a^27*b^24*d^6 + 4495245312*a^29
*b^22*d^6 + 4279238656*a^31*b^20*d^6 + 1923088384*a^33*b^18*d^6 - 773849088*a^35*b^16*d^6 - 1421344768*a^37*b^
14*d^6 - 587726848*a^39*b^12*d^6 - 25165824*a^41*b^10*d^6 + 25165824*a^43*b^8*d^6))*(-((6*a^2*b^4*d^4 - b^6*d^
4 - 9*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(
1/2) + 29491200*a^22*b^26*d^4 + 190709760*a^24*b^24*d^4 + 509214720*a^26*b^22*d^4 + 701890560*a^28*b^20*d^4 +
481689600*a^30*b^18*d^4 + 68812800*a^32*b^16*d^4 - 123863040*a^34*b^14*d^4 - 80609280*a^36*b^12*d^4 - 15728640
*a^38*b^10*d^4)*(-((6*a^2*b^4*d^4 - b^6*d^4 - 9*a^4*b^2*d^4)^(1/2) - a^3*d^2 + 3*a*b^2*d^2)/(4*a^6*d^4 + 4*b^6
*d^4 + 12*a^2*b^4*d^4 + 12*a^4*b^2*d^4))^(1/2) - (atan((a^31*(a + b*tan(c + d*x))^(1/2)*4608i + a^17*b^14*(a +
b*tan(c + d*x))^(1/2)*101250i + a^19*b^12*(a + b*tan(c + d*x))^(1/2)*87750i - a^21*b^10*(a + b*tan(c + d*x))^
(1/2)*171450i - a^23*b^8*(a + b*tan(c + d*x))^(1/2)*89790i + a^25*b^6*(a + b*tan(c + d*x))^(1/2)*169472i - a^2
7*b^4*(a + b*tan(c + d*x))^(1/2)*47424i - a^29*b^2*(a + b*tan(c + d*x))^(1/2)*9216i)/(a^14*(a^7)^(1/2)*(101250
*b^14 + 87750*a^2*b^12 - 171450*a^4*b^10 - 89790*a^6*b^8 + a^7*(169472*a*b^6 + 4608*a^7 - 47424*a^3*b^4 - 9216
*a^5*b^2))))*(a^2 - (15*b^2)/8)*2i)/(d*(a^7)^(1/2))