\(\int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) [546]

Optimal result

Integrand size = 23, antiderivative size = 291 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3646, 3726, 3728, 3711, 3620, 3618, 65, 214} \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 d \left (a^2+b^2\right )^2}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 d \left (a^2+b^2\right )^2}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \]

[In]

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

[Out]

-(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt
[a + I*b]]/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a^
2*(a^2 + 2*b^2)*Tan[c + d*x]^2)/(b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) - (4*a*(8*a^4 + 15*a^2*b^2 + 4*
b^4)*Sqrt[a + b*Tan[c + d*x]])/(3*b^4*(a^2 + b^2)^2*d) + (2*(8*a^4 + 15*a^2*b^2 + b^4)*Tan[c + d*x]*Sqrt[a + b
*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)^2*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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3726

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*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((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 - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 \int \frac {\tan ^2(c+d x) \left (3 a^2-\frac {3}{2} a b \tan (c+d x)+\frac {3}{2} \left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )} \\ & = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {4 \int \frac {\tan (c+d x) \left (6 a^2 \left (a^2+2 b^2\right )-\frac {3}{2} a b^3 \tan (c+d x)+\frac {3}{4} \left (8 a^4+15 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d}+\frac {8 \int \frac {-\frac {3}{4} a \left (8 a^4+15 a^2 b^2+b^4\right )+\frac {9}{8} b^3 \left (a^2-b^2\right ) \tan (c+d x)-\frac {3}{4} a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{9 b^3 \left (a^2+b^2\right )^2} \\ & = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d}+\frac {8 \int \frac {\frac {9 a b^4}{4}+\frac {9}{8} b^3 \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{9 b^3 \left (a^2+b^2\right )^2} \\ & = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d}-\frac {i \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}+\frac {i \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2} \\ & = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 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)^2 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)^2 d} \\ & = -\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d}+\frac {i \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 )}{(a-i b)^2 b d}-\frac {i \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 )}{(a+i b)^2 b d} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.97 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {2 \left (-\frac {3 b \left (-2 a b^2+a^2 \sqrt {-b^2}+\left (-b^2\right )^{3/2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{2 \sqrt {-b^2} \left (a^2+b^2\right )^2 \sqrt {a-\sqrt {-b^2}}}-\frac {3 b \left (2 a b^2+a^2 \sqrt {-b^2}+\left (-b^2\right )^{3/2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{2 \sqrt {-b^2} \left (a^2+b^2\right )^2 \sqrt {a+\sqrt {-b^2}}}+\frac {a^3 \left (8 a^2+7 b^2\right )}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {6 a \tan ^2(c+d x)}{b (a+b \tan (c+d x))^{3/2}}+\frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}}-\frac {3 a^2 \left (8 a^4+15 a^2 b^2+5 b^4\right )}{b^3 \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}\right )}{3 b d} \]

[In]

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

[Out]

(2*((-3*b*(-2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(
2*Sqrt[-b^2]*(a^2 + b^2)^2*Sqrt[a - Sqrt[-b^2]]) - (3*b*(2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt
[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*Sqrt[-b^2]*(a^2 + b^2)^2*Sqrt[a + Sqrt[-b^2]]) + (a^3*(8*a^2 +
7*b^2))/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) - (6*a*Tan[c + d*x]^2)/(b*(a + b*Tan[c + d*x])^(3/2)) + T
an[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2) - (3*a^2*(8*a^4 + 15*a^2*b^2 + 5*b^4))/(b^3*(a^2 + b^2)^2*Sqrt[a + b*
Tan[c + d*x]])))/(3*b*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2208\) vs. \(2(261)=522\).

Time = 0.09 (sec) , antiderivative size = 2209, normalized size of antiderivative = 7.59

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2209\)
default	\(\text {Expression too large to display}\)	\(2209\)

[In]

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

[Out]

-6/d/b^4*a*(a+b*tan(d*x+c))^(1/2)+2/d*b^2/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(
1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-1/2/d*b^2/(a^2+b^2)^(7/2)*ln(b*tan(
d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a
^2-1/2/d*b^2/(a^2+b^2)^3*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2
))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/2/d*b^2/(a^2+b^2)^3*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)
^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/2/d*b^2/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c
))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-6/d*b
^2/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2
))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3-5/d*b^4/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^
2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-2/d*b^2/(a^2+b^2)^(5/2)/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)
^(1/2))*a+6/d*b^2/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^
(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3+5/d*b^4/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arc
tan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-2/d/(a^2+b^2)^(5
/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)
^(1/2)-2*a)^(1/2))*a^3+1/d*b^4/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)
-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))-1/d*b^4/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*ar
ctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))-1/4/d*b^4/(a^2+b^
2)^(7/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^
(1/2)+2*a)^(1/2)+1/4/d*b^4/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-10/d/b^2*a^4/(a^2+b^2)^2/(a+b*tan(d*x+c))^(1/2)-6/d/b^4*a^6
/(a^2+b^2)^2/(a+b*tan(d*x+c))^(1/2)+2/3/d/b^4*a^5/(a^2+b^2)/(a+b*tan(d*x+c))^(3/2)+1/d/(a^2+b^2)^(7/2)/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)
^(1/2))*a^5-3/4/d/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+
b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4-1/2/d/(a^2+b^2)^3*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2
*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/d/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)
-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^4
+1/2/d/(a^2+b^2)^3*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/d/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+
(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^4+3/4/d/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c))^(1
/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4-1/d/(a^2+b
^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^
2+b^2)^(1/2)-2*a)^(1/2))*a^5+2/d/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)
^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3+2/3/d/b^4*(a+b*tan(d*x+c))^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3401 vs. \(2 (257) = 514\).

Time = 0.33 (sec) , antiderivative size = 3401, normalized size of antiderivative = 11.69 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/6*(3*((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*d*tan(d*x + c) + (a^6*
b^4 + 2*a^4*b^6 + a^2*b^8)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 +
5*a^2*b^8 + b^10)*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)/((a^20 + 10*a^18*b^2
+ 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*
a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2))*log((5*a^4 - 10
*a^2*b^2 + b^4)*sqrt(b*tan(d*x + c) + a) + ((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^
2*b^10 - b^12)*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)/((a^20 + 10*a^18*b^2 + 4
5*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*
b^18 + b^20)*d^4)) - (5*a^7 - 25*a^5*b^2 + 31*a^3*b^4 - 3*a*b^6)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^10 +
5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 2
0*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b
^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5
*a^2*b^8 + b^10)*d^2))) - 3*((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*d
*tan(d*x + c) + (a^6*b^4 + 2*a^4*b^6 + a^2*b^8)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a
^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*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)
/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^1
4 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*
d^2))*log((5*a^4 - 10*a^2*b^2 + b^4)*sqrt(b*tan(d*x + c) + a) - ((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b
^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*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)/((a
^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 +
45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - (5*a^7 - 25*a^5*b^2 + 31*a^3*b^4 - 3*a*b^6)*d)*sqrt((a^5 - 10*a^3*b^
2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*
a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6
*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2))) - 3*((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x + c)^2 + 2*(a^5*b^5 +
2*a^3*b^7 + a*b^9)*d*tan(d*x + c) + (a^6*b^4 + 2*a^4*b^6 + a^2*b^8)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - (a^
10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a
^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6
+ 5*a^2*b^8 + b^10)*d^2))*log((5*a^4 - 10*a^2*b^2 + b^4)*sqrt(b*tan(d*x + c) + a) + ((3*a^12 + 14*a^10*b^2 +
25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 2
0*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b
^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) + (5*a^7 - 25*a^5*b^2 + 31*a^3*b^4 - 3*a*b^6)*d)*
sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6
+ 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10
+ 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2))) + 3*((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x
+ c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*d*tan(d*x + c) + (a^6*b^4 + 2*a^4*b^6 + a^2*b^8)*d)*sqrt((a^5 - 10*a^
3*b^2 + 5*a*b^4 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 +
252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10
*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2))*log((5*a^4 - 10*a^2*b^2 + b^4)*sqrt(b*tan(d*x + c) + a) - ((3*
a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*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)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*
a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) + (5*a^7 - 25*a^5*b^2 + 31*a
^3*b^4 - 3*a*b^6)*d)*sqrt((a^5 - 10*a^3*b^2 + 5*a*b^4 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^
8 + b^10)*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)/((a^20 + 10*a^18*b^2 + 45*a^1
6*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18
+ b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^2))) - 4*(16*a^7 + 30*a^5*b^
2 + 8*a^3*b^4 - (a^4*b^3 + 2*a^2*b^5 + b^7)*tan(d*x + c)^3 + 6*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*tan(d*x + c)^2 +
3*(8*a^6*b + 15*a^4*b^3 + 5*a^2*b^5)*tan(d*x + c))*sqrt(b*tan(d*x + c) + a))/((a^4*b^6 + 2*a^2*b^8 + b^10)*d*t
an(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*d*tan(d*x + c) + (a^6*b^4 + 2*a^4*b^6 + a^2*b^8)*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 16.33 (sec) , antiderivative size = 4681, normalized size of antiderivative = 16.09 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(48*a
*b^20*d^4 - ((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*
(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a
^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4
*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18*d^4 + 1216*a^5*b^16*d^4 + 2240*a^7*b^14*d^4 + 2464*a^9*b^12*d^4 + 15
68*a^11*b^10*d^4 + 448*a^13*b^8*d^4 - 64*a^15*b^6*d^4 - 80*a^17*b^4*d^4 - 16*a^19*b^2*d^4))/2 - ((a + b*tan(c
+ d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 32
0*a^12*b^6*d^3 - 16*a^16*b^2*d^3))/2)*(1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i
- 10*a^3*b^2*d^2))^(1/2)*1i - (((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a
^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 + ((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i -
10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 768
0*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^1
7*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18*d^4 + 1216*a^5*b^16*d^4 + 2240*a^7*b^14*d^4
+ 2464*a^9*b^12*d^4 + 1568*a^11*b^10*d^4 + 448*a^13*b^8*d^4 - 64*a^15*b^6*d^4 - 80*a^17*b^4*d^4 - 16*a^19*b^2*
d^4))/2 + ((a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3
+ 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))/2)*(1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d
^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*1i)/((((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i
+ a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 - ((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^
2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 +
2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680
*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18*d^4 + 1216*a^5*b^16*
d^4 + 2240*a^7*b^14*d^4 + 2464*a^9*b^12*d^4 + 1568*a^11*b^10*d^4 + 448*a^13*b^8*d^4 - 64*a^15*b^6*d^4 - 80*a^1
7*b^4*d^4 - 16*a^19*b^2*d^4))/2 - ((a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*
d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))/2)*(1/(a^5*d^2 - b^5*d^2*1i
+ 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2) + (((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^
4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 + ((1/(a^5*d^2 - b^5*d^2*1i + 5
*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5
+ 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440
*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18
*d^4 + 1216*a^5*b^16*d^4 + 2240*a^7*b^14*d^4 + 2464*a^9*b^12*d^4 + 1568*a^11*b^10*d^4 + 448*a^13*b^8*d^4 - 64*
a^15*b^6*d^4 - 80*a^17*b^4*d^4 - 16*a^19*b^2*d^4))/2 + ((a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18
*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))/2)*(1/
(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2) - 16*b^16*d^2 -
80*a^2*b^14*d^2 - 144*a^4*b^12*d^2 - 80*a^6*b^10*d^2 + 80*a^8*b^8*d^2 + 144*a^10*b^6*d^2 + 80*a^12*b^4*d^2 + 1
6*a^14*b^2*d^2))*(1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1
/2)*1i - atan(((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))
^(1/2)*((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*
(96*a*b^20*d^4 + 736*a^3*b^18*d^4 + 2432*a^5*b^16*d^4 + 4480*a^7*b^14*d^4 + 4928*a^9*b^12*d^4 + 3136*a^11*b^10
*d^4 + 896*a^13*b^8*d^4 - 128*a^15*b^6*d^4 - 160*a^17*b^4*d^4 - 32*a^19*b^2*d^4 + (1i/(4*(a^5*d^2*1i - b^5*d^2
+ a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^2
2*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 +
13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) + (a + b*ta
n(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3
+ 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))*1i - (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2
*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^
2 - a^3*b^2*d^2*10i)))^(1/2)*(96*a*b^20*d^4 + 736*a^3*b^18*d^4 + 2432*a^5*b^16*d^4 + 4480*a^7*b^14*d^4 + 4928*
a^9*b^12*d^4 + 3136*a^11*b^10*d^4 + 896*a^13*b^8*d^4 - 128*a^15*b^6*d^4 - 160*a^17*b^4*d^4 - 32*a^19*b^2*d^4 -
(1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*
tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14
*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 +
64*a^21*b^2*d^5)) - (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*
b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))*1i)/((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*
d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i
- 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(96*a*b^20*d^4 + 736*a^3*b^18*d^4 + 2432*a^5*b^16*d^
4 + 4480*a^7*b^14*d^4 + 4928*a^9*b^12*d^4 + 3136*a^11*b^10*d^4 + 896*a^13*b^8*d^4 - 128*a^15*b^6*d^4 - 160*a^1
7*b^4*d^4 - 32*a^19*b^2*d^4 + (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3
*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*
a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*
b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) + (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 +
1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3)) + (1i/(4*(a^5
*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*((1i/(4*(a^5*d^2*1i
- b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(96*a*b^20*d^4 + 736*a^3*b
^18*d^4 + 2432*a^5*b^16*d^4 + 4480*a^7*b^14*d^4 + 4928*a^9*b^12*d^4 + 3136*a^11*b^10*d^4 + 896*a^13*b^8*d^4 -
128*a^15*b^6*d^4 - 160*a^17*b^4*d^4 - 32*a^19*b^2*d^4 - (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*
d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 +
2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680
*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) - (a + b*tan(c + d*x))^(1/2)*(320*a^4
*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^
16*b^2*d^3)) - 16*b^16*d^2 - 80*a^2*b^14*d^2 - 144*a^4*b^12*d^2 - 80*a^6*b^10*d^2 + 80*a^8*b^8*d^2 + 144*a^10*
b^6*d^2 + 80*a^12*b^4*d^2 + 16*a^14*b^2*d^2))*(1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a
^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*2i + (2*(a + b*tan(c + d*x))^(3/2))/(3*b^4*d) - (6*a*(a + b*tan(c + d*x)
)^(1/2))/(b^4*d) + ((2*a^5)/(3*(a^2 + b^2)) - (2*(3*a^6 + 5*a^4*b^2)*(a + b*tan(c + d*x)))/(a^2 + b^2)^2)/(b^4
*d*(a + b*tan(c + d*x))^(3/2))