3.6.82 \(\int \frac {\tan ^{\frac {9}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx\) [582]

Optimal. Leaf size=300 \[ -\frac {(a+b) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a^{9/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right ) d}+\frac {(a-b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d} \]

[Out]

-2*a^(9/2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(7/2)/(a^2+b^2)/d+1/2*(a+b)*arctan(-1+2^(1/2)*tan(d*x+c)
^(1/2))/(a^2+b^2)/d*2^(1/2)+1/2*(a+b)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)+1/4*(a-b)*ln(1-2^
(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-1/4*(a-b)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^
2+b^2)/d*2^(1/2)+2*(a^2-b^2)*tan(d*x+c)^(1/2)/b^3/d-2/3*a*tan(d*x+c)^(3/2)/b^2/d+2/5*tan(d*x+c)^(5/2)/b/d

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Rubi [A]
time = 0.56, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3647, 3728, 3729, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {(a+b) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a^{9/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} d \left (a^2+b^2\right )}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x]),x]

[Out]

-(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt
[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(7/2)*(
a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a -
b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*(a^2 - b^2)*Sqrt[Tan[c
+ d*x]])/(b^3*d) - (2*a*Tan[c + d*x]^(3/2))/(3*b^2*d) + (2*Tan[c + d*x]^(5/2))/(5*b*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 210

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

Rule 211

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

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 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])))

Rule 3729

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (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 - b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !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*} \int \frac {\tan ^{\frac {9}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {2 \int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\frac {5 a}{2}-\frac {5}{2} b \tan (c+d x)-\frac {5}{2} a \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{5 b}\\ &=-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {4 \int \frac {\sqrt {\tan (c+d x)} \left (\frac {15 a^2}{4}+\frac {15}{4} \left (a^2-b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{15 b^2}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {8 \int \frac {-\frac {15}{8} a \left (a^2-b^2\right )+\frac {15}{8} b^3 \tan (c+d x)-\frac {15}{8} a \left (a^2-b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{15 b^3}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {8 \int \frac {\frac {15 b^4}{8}+\frac {15}{8} a b^3 \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{15 b^3 \left (a^2+b^2\right )}-\frac {a^5 \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b^3 \left (a^2+b^2\right )}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {16 \text {Subst}\left (\int \frac {\frac {15 b^4}{8}+\frac {15}{8} a b^3 x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{15 b^3 \left (a^2+b^2\right ) d}-\frac {a^5 \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{b^3 \left (a^2+b^2\right ) d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}-\frac {(a-b) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {\left (2 a^5\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac {2 a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right ) d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {(a-b) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right ) d}+\frac {(a-b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}\\ &=-\frac {(a+b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{7/2} \left (a^2+b^2\right ) d}+\frac {(a-b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{b^3 d}-\frac {2 a \tan ^{\frac {3}{2}}(c+d x)}{3 b^2 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 b d}\\ \end {align*}

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Mathematica [A]
time = 1.86, size = 243, normalized size = 0.81 \begin {gather*} \frac {-\frac {15 \left (2 \sqrt {2} b^{7/2} (a+b) \left (\text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )+8 a^{9/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {2} b^{7/2} (-a+b) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+8 \sqrt {b} \left (-a^2+b^2\right ) \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}\right )}{b^{3/2} \left (a^2+b^2\right )}-40 a \tan ^{\frac {3}{2}}(c+d x)+24 b \tan ^{\frac {5}{2}}(c+d x)}{60 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x]),x]

[Out]

((-15*(2*Sqrt[2]*b^(7/2)*(a + b)*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x
]]]) + 8*a^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*b^(7/2)*(-a + b)*(Log[1 - Sqrt[2]*Sqrt
[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*Sqrt[b]*(-a^2 + b^2)*
(a^2 + b^2)*Sqrt[Tan[c + d*x]]))/(b^(3/2)*(a^2 + b^2)) - 40*a*Tan[c + d*x]^(3/2) + 24*b*Tan[c + d*x]^(5/2))/(6
0*b^2*d)

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Maple [A]
time = 0.15, size = 284, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )-2 b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )}{b^{3}}-\frac {2 a^{5} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b^{3} \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(284\)
default	\(\frac {\frac {\frac {2 b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )-2 b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )}{b^{3}}-\frac {2 a^{5} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b^{3} \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(284\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(9/2)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^3*(1/5*b^2*tan(d*x+c)^(5/2)-1/3*a*b*tan(d*x+c)^(3/2)+a^2*tan(d*x+c)^(1/2)-b^2*tan(d*x+c)^(1/2))-2/b^3
*a^5/(a^2+b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))+2/(a^2+b^2)*(1/8*b*2^(1/2)*(ln((1+2^(1/2)*ta
n(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arc
tan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*a*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x
+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))))

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Maxima [A]
time = 0.49, size = 225, normalized size = 0.75 \begin {gather*} -\frac {\frac {120 \, a^{5} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} b^{3} + b^{5}\right )} \sqrt {a b}} - \frac {15 \, {\left (2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{2} + b^{2}} - \frac {8 \, {\left (3 \, b^{2} \tan \left (d x + c\right )^{\frac {5}{2}} - 5 \, a b \tan \left (d x + c\right )^{\frac {3}{2}} + 15 \, {\left (a^{2} - b^{2}\right )} \sqrt {\tan \left (d x + c\right )}\right )}}{b^{3}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(9/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(120*a^5*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2*b^3 + b^5)*sqrt(a*b)) - 15*(2*sqrt(2)*(a + b)*arct
an(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a + b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan
(d*x + c)))) - sqrt(2)*(a - b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a - b)*log(-sqrt(
2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^2 + b^2) - 8*(3*b^2*tan(d*x + c)^(5/2) - 5*a*b*tan(d*x + c)^(3/2
) + 15*(a^2 - b^2)*sqrt(tan(d*x + c)))/b^3)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3679 vs. \(2 (252) = 504\).
time = 9.64, size = 7470, normalized size = 24.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(9/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

[-1/60*(60*sqrt(2)*(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d^5*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*
b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/
((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(-((a^8 +
2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7
*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^
4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 +
2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a
^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b
^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*
a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x
+ c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^10*b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*
sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4
*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 +
2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d
^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4))*cos(d*x + c)^2 + 60*sqrt(2)*(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d^5*s
qrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2
*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((
a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^8*b +
4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a
^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((a^
4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(
a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^
4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7 - a^5*b^2 - a^3*
b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c)
)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4
- 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2
+ b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((a^10*b + 3*a^8*b^3 + 2
*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a^4
- 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a
^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c
)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4))*cos(d*x + c)^2 - 30*a^4*sqrt
(-a/b)*cos(d*x + c)^2*log(-(6*a*b*cos(d*x + c)*sin(d*x + c) - (a^2 - b^2)*cos(d*x + c)^2 - b^2 - 4*(a*b*cos(d*
x + c)^2 - b^2*cos(d*x + c)*sin(d*x + c))*sqrt(-a/b)*sqrt(sin(d*x + c)/cos(d*x + c)))/(2*a*b*cos(d*x + c)*sin(
d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) - 15*sqrt(2)*(2*(a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4))*cos(d*x + c)^2 - (a^2*b^3 + b^5)*d*cos(d*x + c)^2)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*
b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d
^4))^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(
2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*
b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^
2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sq...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{\frac {9}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(9/2)/(a+b*tan(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**(9/2)/(a + b*tan(c + d*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(9/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 10.44, size = 2500, normalized size = 8.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^(9/2)/(a + b*tan(c + d*x)),x)

[Out]

(log(((((((((256*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(1i/(d^2*(a*1i - b)^2))^(1/2) - (128*a*b*(4*
a^2 - b^2)*(a^2 + b^2)^2)/d)*(1i/(d^2*(a*1i - b)^2))^(1/2))/2 + (64*a*tan(c + d*x)^(1/2)*(8*a^10 - 7*b^10 + 2*
a^2*b^8 + a^4*b^6))/(b^4*d^2))*(1i/(d^2*(a*1i - b)^2))^(1/2))/2 - (32*a^2*(12*a^8 + b^8 + a^2*b^6 + 16*a^4*b^4
- 16*a^6*b^2))/(b^4*d^3))*(1i/(d^2*(a*1i - b)^2))^(1/2))/2 + (32*tan(c + d*x)^(1/2)*(2*a^10 - b^10))/(b^5*d^4
))*(1i/(d^2*(a*1i - b)^2))^(1/2))/2 - (32*a^5*(a^4 + b^4 - a^2*b^2))/(b^5*d^5))*(-1/(b^2*d^2*1i - a^2*d^2*1i +
2*a*b*d^2))^(1/2))/2 - tan(c + d*x)^(1/2)*(2/(b*d) - (2*a^2)/(b^3*d)) - log(- ((((((((256*b^3*tan(c + d*x)^(1
/2)*(a^2 - b^2)*(a^2 + b^2)^2*(1i/(d^2*(a*1i - b)^2))^(1/2) + (128*a*b*(4*a^2 - b^2)*(a^2 + b^2)^2)/d)*(1i/(d^
2*(a*1i - b)^2))^(1/2))/2 + (64*a*tan(c + d*x)^(1/2)*(8*a^10 - 7*b^10 + 2*a^2*b^8 + a^4*b^6))/(b^4*d^2))*(1i/(
d^2*(a*1i - b)^2))^(1/2))/2 + (32*a^2*(12*a^8 + b^8 + a^2*b^6 + 16*a^4*b^4 - 16*a^6*b^2))/(b^4*d^3))*(1i/(d^2*
(a*1i - b)^2))^(1/2))/2 + (32*tan(c + d*x)^(1/2)*(2*a^10 - b^10))/(b^5*d^4))*(1i/(d^2*(a*1i - b)^2))^(1/2))/2
- (32*a^5*(a^4 + b^4 - a^2*b^2))/(b^5*d^5))*(-1/(4*(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2)))^(1/2) + atan((((((-
1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((32*(8*a^3*b^10*d^4 - 4*a*b^12*d^4 + 28*a^5*b^8*d^4 + 16*a^7*b
^6*d^4))/(b^5*d^5) - (32*tan(c + d*x)^(1/2)*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^14*d^4 + 16
*a^2*b^12*d^4 - 16*a^4*b^10*d^4 - 16*a^6*b^8*d^4))/(b^5*d^4)) + (32*tan(c + d*x)^(1/2)*(16*a^11*b*d^2 - 14*a*b
^11*d^2 + 4*a^3*b^9*d^2 + 2*a^5*b^7*d^2))/(b^5*d^4))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*(1
2*a^10*b*d^2 + a^2*b^9*d^2 + a^4*b^7*d^2 + 16*a^6*b^5*d^2 - 16*a^8*b^3*d^2))/(b^5*d^5))*(-1i/(4*(b^2*d^2 - a^2
*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*a^10 - b^10))/(b^5*d^4))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a
*b*d^2*2i)))^(1/2)*1i - ((((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((32*(8*a^3*b^10*d^4 - 4*a*b^12*d^
4 + 28*a^5*b^8*d^4 + 16*a^7*b^6*d^4))/(b^5*d^5) + (32*tan(c + d*x)^(1/2)*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*
2i)))^(1/2)*(16*b^14*d^4 + 16*a^2*b^12*d^4 - 16*a^4*b^10*d^4 - 16*a^6*b^8*d^4))/(b^5*d^4)) - (32*tan(c + d*x)^
(1/2)*(16*a^11*b*d^2 - 14*a*b^11*d^2 + 4*a^3*b^9*d^2 + 2*a^5*b^7*d^2))/(b^5*d^4))*(-1i/(4*(b^2*d^2 - a^2*d^2 +
a*b*d^2*2i)))^(1/2) + (32*(12*a^10*b*d^2 + a^2*b^9*d^2 + a^4*b^7*d^2 + 16*a^6*b^5*d^2 - 16*a^8*b^3*d^2))/(b^5
*d^5))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*tan(c + d*x)^(1/2)*(2*a^10 - b^10))/(b^5*d^4))*(
-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i)/(((((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((32*
(8*a^3*b^10*d^4 - 4*a*b^12*d^4 + 28*a^5*b^8*d^4 + 16*a^7*b^6*d^4))/(b^5*d^5) - (32*tan(c + d*x)^(1/2)*(-1i/(4*
(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^14*d^4 + 16*a^2*b^12*d^4 - 16*a^4*b^10*d^4 - 16*a^6*b^8*d^4))/(
b^5*d^4)) + (32*tan(c + d*x)^(1/2)*(16*a^11*b*d^2 - 14*a*b^11*d^2 + 4*a^3*b^9*d^2 + 2*a^5*b^7*d^2))/(b^5*d^4))
*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*(12*a^10*b*d^2 + a^2*b^9*d^2 + a^4*b^7*d^2 + 16*a^6*b^
5*d^2 - 16*a^8*b^3*d^2))/(b^5*d^5))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*
(2*a^10 - b^10))/(b^5*d^4))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + ((((-1i/(4*(b^2*d^2 - a^2*d^2 +
a*b*d^2*2i)))^(1/2)*((32*(8*a^3*b^10*d^4 - 4*a*b^12*d^4 + 28*a^5*b^8*d^4 + 16*a^7*b^6*d^4))/(b^5*d^5) + (32*t
an(c + d*x)^(1/2)*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^14*d^4 + 16*a^2*b^12*d^4 - 16*a^4*b^1
0*d^4 - 16*a^6*b^8*d^4))/(b^5*d^4)) - (32*tan(c + d*x)^(1/2)*(16*a^11*b*d^2 - 14*a*b^11*d^2 + 4*a^3*b^9*d^2 +
2*a^5*b^7*d^2))/(b^5*d^4))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*(12*a^10*b*d^2 + a^2*b^9*d^2
+ a^4*b^7*d^2 + 16*a^6*b^5*d^2 - 16*a^8*b^3*d^2))/(b^5*d^5))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)
- (32*tan(c + d*x)^(1/2)*(2*a^10 - b^10))/(b^5*d^4))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (64*(
a^9 + a^5*b^4 - a^7*b^2))/(b^5*d^5)))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*2i + (2*tan(c + d*x)^(5
/2))/(5*b*d) - (2*a*tan(c + d*x)^(3/2))/(3*b^2*d) + (atan((((-a^9*b^7)^(1/2)*((32*tan(c + d*x)^(1/2)*(2*a^10 -
b^10))/(b^5*d^4) + (((32*(12*a^10*b*d^2 + a^2*b^9*d^2 + a^4*b^7*d^2 + 16*a^6*b^5*d^2 - 16*a^8*b^3*d^2))/(b^5*
d^5) + ((-a^9*b^7)^(1/2)*((32*tan(c + d*x)^(1/2)*(16*a^11*b*d^2 - 14*a*b^11*d^2 + 4*a^3*b^9*d^2 + 2*a^5*b^7*d^
2))/(b^5*d^4) + (((32*(8*a^3*b^10*d^4 - 4*a*b^12*d^4 + 28*a^5*b^8*d^4 + 16*a^7*b^6*d^4))/(b^5*d^5) - (32*tan(c
+ d*x)^(1/2)*(-a^9*b^7)^(1/2)*(16*b^14*d^4 + 16*a^2*b^12*d^4 - 16*a^4*b^10*d^4 - 16*a^6*b^8*d^4))/(b^12*d^5*(
a^2 + b^2)))*(-a^9*b^7)^(1/2))/(b^7*d*(a^2 + b^2))))/(b^7*d*(a^2 + b^2)))*(-a^9*b^7)^(1/2))/(b^7*d*(a^2 + b^2)
))*1i)/(b^7*d*(a^2 + b^2)) + ((-a^9*b^7)^(1/2)*((32*tan(c + d*x)^(1/2)*(2*a^10 - b^10))/(b^5*d^4) - (((32*(12*
a^10*b*d^2 + a^2*b^9*d^2 + a^4*b^7*d^2 + 16*a^6*b^5*d^2 - 16*a^8*b^3*d^2))/(b^5*d^5) - ((-a^9*b^7)^(1/2)*((32*
tan(c + d*x)^(1/2)*(16*a^11*b*d^2 - 14*a*b^11*d...

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