\(\int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx\) [554]

Optimal result

Integrand size = 14, antiderivative size = 194 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{7/2} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{7/2} d}-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3564, 3610, 3620, 3618, 65, 214} \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 \sqrt {a+b \tan (c+d x)}}-\frac {4 a b}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{3/2}}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{7/2}}+\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{7/2}} \]

[In]

Int[(a + b*Tan[c + d*x])^(-7/2),x]

[Out]

((-I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(7/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a + I*b]])/((a + I*b)^(7/2)*d) - (2*b)/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2)) - (4*a*b)/(3*(a^2
+ b^2)^2*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/((a^2 + b^2)^3*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 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3610

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

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]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\frac {i (a+i b) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )+(-i a-b) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}} \]

[In]

Integrate[(a + b*Tan[c + d*x])^(-7/2),x]

[Out]

(I*(a + I*b)*Hypergeometric2F1[-5/2, 1, -3/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1
[-5/2, 1, -3/2, (a + b*Tan[c + d*x])/(a + I*b)])/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3081\) vs. \(2(166)=332\).

Time = 0.11 (sec) , antiderivative size = 3082, normalized size of antiderivative = 15.89

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3082\)
default	\(\text {Expression too large to display}\)	\(3082\)

[In]

int(1/(a+b*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-4/3*a*b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(3/2)-2/5*b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(5/2)+2/d*b^3/(a^2+b^2)^3/(a+
b*tan(d*x+c))^(1/2)-1/4/d/b/(a^2+b^2)^(9/2)*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^6-5/d*b^3/(a^2+b^2)^(9/2)/(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+7/d*b^5/(a^
2+b^2)^(9/2)/(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+1/4/d/b/(a^2+b^2)^4*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^5-3/4/d*b^3/(a^2+b^2)^4*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/d*b^3/(a^2+b^2
)^4/(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^2-1/4/d/b/(a^2+b^2)^4*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^5+3/4/d*b^3/(a^2+b^2)^4*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/d*b^3/(a^2+b^2)^4/(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^2-1/d/b/(a^2+b^2)^(9/2)/(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^7-3/d*b^3/(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/d*b/(a^2+b
^2)^(9/2)/(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+2/d*b/(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^3-2/d*b/(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
-1/2/d*b/(a^2+b^2)^4*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+5/4/d*b/(a^2+b^2)^(9/2)*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/d/b/(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+
1/4/d/b/(a^2+b^2)^(9/2)*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^6-5/4/d*b^3/(a^2+b^2)^(9/2)*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+5/4/d*b^3/(a^2+b^2)^(9/2)*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^
2-7/d*b^5/(a^2+b^2)^(9/2)/(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-1/d/b/(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/d*b^3/(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+1/d/b/(a^2+b^2)^(9/2)/(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^7+5/d*b^3/(a^2+b^2)^(9/2)/(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^3-5/4/d*b/(a^
2+b^2)^(9/2)*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^4-3/d*b/(a^2+b^2)^(9/2)/(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^5-3/d*b/(a^2+b^2)^4/(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+1/2/d*b
/(a^2+b^2)^4*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+3/d*b/(a^2+b^2)^4/(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^4-1/4/d*b^5/(a^2+b^2)^(9/2)*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)+1/4/d*b^5/(a^2
+b^2)^(9/2)*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/d*b^5/(a^2+b^2)^4/(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))-1/d*b^5/(a^2+b^2)^4/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n(((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))-6/d*b/(a^2+b^2)^3/(a
+b*tan(d*x+c))^(1/2)*a^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4720 vs. \(2 (160) = 320\).

Time = 0.33 (sec) , antiderivative size = 4720, normalized size of antiderivative = 24.33 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*tan(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

-1/30*(15*((a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b
^8)*d*tan(d*x + c)^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*d*tan(d*x + c) + (a^9 + 3*a^7*b^2 + 3*a^5*b
^4 + a^3*b^6)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 + (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^
6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*
a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8
+ 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^
6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)))/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8
+ 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2))*log(-(7*a^6*b - 35*a^4*b^3 + 21*a^2*b^5 - b^7)*sqrt(b*tan(d*x + c) +
a) + ((a^18 + a^16*b^2 - 20*a^14*b^4 - 84*a^12*b^6 - 154*a^10*b^8 - 154*a^8*b^10 - 84*a^6*b^12 - 20*a^4*b^14 +
a^2*b^16 + b^18)*d^3*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*
b^12 + b^14)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^
12 + 3432*a^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^
26 + b^28)*d^4)) + 4*(7*a^9*b^2 - 42*a^7*b^4 + 56*a^5*b^6 - 22*a^3*b^8 + a*b^10)*d)*sqrt(-(a^7 - 21*a^5*b^2 +
35*a^3*b^4 - 7*a*b^6 + (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 +
b^14)*d^2*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14
)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a
^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*
d^4)))/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2))) -
15*((a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*d*
tan(d*x + c)^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*d*tan(d*x + c) + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a
^3*b^6)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 + (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35
*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^
8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 200
2*a^18*b^10 + 3003*a^16*b^12 + 3432*a^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22
+ 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)))/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a
^4*b^10 + 7*a^2*b^12 + b^14)*d^2))*log(-(7*a^6*b - 35*a^4*b^3 + 21*a^2*b^5 - b^7)*sqrt(b*tan(d*x + c) + a) - (
(a^18 + a^16*b^2 - 20*a^14*b^4 - 84*a^12*b^6 - 154*a^10*b^8 - 154*a^8*b^10 - 84*a^6*b^12 - 20*a^4*b^14 + a^2*b
^16 + b^18)*d^3*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 +
b^14)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3
432*a^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b
^28)*d^4)) + 4*(7*a^9*b^2 - 42*a^7*b^4 + 56*a^5*b^6 - 22*a^3*b^8 + a*b^10)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3
*b^4 - 7*a*b^6 + (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)
*d^2*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^
28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a^14*b^
14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)))
/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2))) - 15*((
a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*d*tan(d*
x + c)^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*d*tan(d*x + c) + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6
)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b
^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 51
1*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18
*b^10 + 3003*a^16*b^12 + 3432*a^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*
a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)))/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^1
0 + 7*a^2*b^12 + b^14)*d^2))*log(-(7*a^6*b - 35*a^4*b^3 + 21*a^2*b^5 - b^7)*sqrt(b*tan(d*x + c) + a) + ((a^18
+ a^16*b^2 - 20*a^14*b^4 - 84*a^12*b^6 - 154*a^10*b^8 - 154*a^8*b^10 - 84*a^6*b^12 - 20*a^4*b^14 + a^2*b^16 +
b^18)*d^3*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)
/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a^
14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d
^4)) - 4*(7*a^9*b^2 - 42*a^7*b^4 + 56*a^5*b^6 - 22*a^3*b^8 + a*b^10)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 -
7*a*b^6 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2*s
qrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^28 + 1
4*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a^14*b^14 + 3
003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)))/((a^1
4 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2))) + 15*((a^6*b^
3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*d*tan(d*x + c)
^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*d*tan(d*x + c) + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d)*s
qrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 2
1*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*
b^10 - 42*a^2*b^12 + b^14)/((a^28 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10
+ 3003*a^16*b^12 + 3432*a^14*b^14 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^
24 + 14*a^2*b^26 + b^28)*d^4)))/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*
a^2*b^12 + b^14)*d^2))*log(-(7*a^6*b - 35*a^4*b^3 + 21*a^2*b^5 - b^7)*sqrt(b*tan(d*x + c) + a) - ((a^18 + a^16
*b^2 - 20*a^14*b^4 - 84*a^12*b^6 - 154*a^10*b^8 - 154*a^8*b^10 - 84*a^6*b^12 - 20*a^4*b^14 + a^2*b^16 + b^18)*
d^3*sqrt(-(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^2
8 + 14*a^26*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a^14*b^1
4 + 3003*a^12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)) -
4*(7*a^9*b^2 - 42*a^7*b^4 + 56*a^5*b^6 - 22*a^3*b^8 + a*b^10)*d)*sqrt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b
^6 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2*sqrt(-(
49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/((a^28 + 14*a^26
*b^2 + 91*a^24*b^4 + 364*a^22*b^6 + 1001*a^20*b^8 + 2002*a^18*b^10 + 3003*a^16*b^12 + 3432*a^14*b^14 + 3003*a^
12*b^16 + 2002*a^10*b^18 + 1001*a^8*b^20 + 364*a^6*b^22 + 91*a^4*b^24 + 14*a^2*b^26 + b^28)*d^4)))/((a^14 + 7*
a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d^2))) + 4*(58*a^4*b + a^2
*b^3 + 3*b^5 + 15*(3*a^2*b^3 - b^5)*tan(d*x + c)^2 + 20*(5*a^3*b^2 - a*b^4)*tan(d*x + c))*sqrt(b*tan(d*x + c)
+ a))/((a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*
d*tan(d*x + c)^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*d*tan(d*x + c) + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 +
a^3*b^6)*d)

Sympy [F]

\[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(1/(a+b*tan(d*x+c))**(7/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(-7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(a+b*tan(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*tan(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 24.33 (sec) , antiderivative size = 5307, normalized size of antiderivative = 27.36 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

int(1/(a + b*tan(c + d*x))^(7/2),x)

[Out]

(log((((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 964
8*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*
b^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - ((-1/(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b
^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2))^(1/2)*(((-1/(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d
^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2))^(1/2)*(a + b*tan(c +
d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*b^24*d^5 +
192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*a^19*b^14*
d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29*b^4*d^5 +
64*a^31*b^2*d^5))/2 - 128*a*b^29*d^4 - 1408*a^3*b^27*d^4 - 6912*a^5*b^25*d^4 - 19712*a^7*b^23*d^4 - 35200*a^9
*b^21*d^4 - 38016*a^11*b^19*d^4 - 16896*a^13*b^17*d^4 + 16896*a^15*b^15*d^4 + 38016*a^17*b^13*d^4 + 35200*a^19
*b^11*d^4 + 19712*a^21*b^9*d^4 + 6912*a^23*b^7*d^4 + 1408*a^25*b^5*d^4 + 128*a^27*b^3*d^4))/2)*(-1/(a^7*d^2 +
b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2)
)^(1/2))/2 - 8*b^23*d^2 - 48*a^2*b^21*d^2 - 72*a^4*b^19*d^2 + 192*a^6*b^17*d^2 + 1008*a^8*b^15*d^2 + 2016*a^10
*b^13*d^2 + 2352*a^12*b^11*d^2 + 1728*a^14*b^9*d^2 + 792*a^16*b^7*d^2 + 208*a^18*b^5*d^2 + 24*a^20*b^3*d^2)*(-
1/(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21
*a^5*b^2*d^2))^(1/2))/2 - log(192*a^6*b^17*d^2 - 8*b^23*d^2 - 48*a^2*b^21*d^2 - 72*a^4*b^19*d^2 - (-1/(4*(a^7*
d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^
2*d^2)))^(1/2)*((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*
d^3 - 9648*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1
344*a^20*b^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1/(4*(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2
*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2)))^(1/2)*(128*a*b^29*d^4 + (-1/(4*(a
^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5
*b^2*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7
*b^26*d^5 + 87360*a^9*b^24*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a
^17*b^16*d^5 + 320320*a^19*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a
^27*b^6*d^5 + 960*a^29*b^4*d^5 + 64*a^31*b^2*d^5) + 1408*a^3*b^27*d^4 + 6912*a^5*b^25*d^4 + 19712*a^7*b^23*d^4
+ 35200*a^9*b^21*d^4 + 38016*a^11*b^19*d^4 + 16896*a^13*b^17*d^4 - 16896*a^15*b^15*d^4 - 38016*a^17*b^13*d^4
- 35200*a^19*b^11*d^4 - 19712*a^21*b^9*d^4 - 6912*a^23*b^7*d^4 - 1408*a^25*b^5*d^4 - 128*a^27*b^3*d^4)) + 1008
*a^8*b^15*d^2 + 2016*a^10*b^13*d^2 + 2352*a^12*b^11*d^2 + 1728*a^14*b^9*d^2 + 792*a^16*b^7*d^2 + 208*a^18*b^5*
d^2 + 24*a^20*b^3*d^2)*(-1/(4*(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^
4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2)))^(1/2) + atan(-((-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^
6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((a + b*tan(c + d*x))^(
1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 9648*a^8*b^18*d^3 - 8640*a^10*b^
16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*b^6*d^3 + 96*a^22*b^4*d^3 - 1
6*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35
i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21
*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^
32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*b^24*d^5 + 192192*a^11*b^22*d^5
+ 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*a^19*b^14*d^5 + 192192*a^21*b^1
2*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29*b^4*d^5 + 64*a^31*b^2*d^5) - 1
28*a*b^29*d^4 - 1408*a^3*b^27*d^4 - 6912*a^5*b^25*d^4 - 19712*a^7*b^23*d^4 - 35200*a^9*b^21*d^4 - 38016*a^11*b
^19*d^4 - 16896*a^13*b^17*d^4 + 16896*a^15*b^15*d^4 + 38016*a^17*b^13*d^4 + 35200*a^19*b^11*d^4 + 19712*a^21*b
^9*d^4 + 6912*a^23*b^7*d^4 + 1408*a^25*b^5*d^4 + 128*a^27*b^3*d^4))*1i + (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6
*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((a + b*t
an(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 9648*a^8*b^18*d^3
- 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*b^6*d^3 + 96*a^
22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 +
a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(128*a*b^29*d^4 + (-1i/(4*(a^7*d^2*1i + b^7*d^2 -
a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(a +
b*tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*
b^24*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*
a^19*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29
*b^4*d^5 + 64*a^31*b^2*d^5) + 1408*a^3*b^27*d^4 + 6912*a^5*b^25*d^4 + 19712*a^7*b^23*d^4 + 35200*a^9*b^21*d^4
+ 38016*a^11*b^19*d^4 + 16896*a^13*b^17*d^4 - 16896*a^15*b^15*d^4 - 38016*a^17*b^13*d^4 - 35200*a^19*b^11*d^4
- 19712*a^21*b^9*d^4 - 6912*a^23*b^7*d^4 - 1408*a^25*b^5*d^4 - 128*a^27*b^3*d^4))*1i)/((-1i/(4*(a^7*d^2*1i + b
^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(
1/2)*((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 9648
*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*b
^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*
a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b
^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(a + b*
tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*b^2
4*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*a^1
9*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29*b^
4*d^5 + 64*a^31*b^2*d^5) - 128*a*b^29*d^4 - 1408*a^3*b^27*d^4 - 6912*a^5*b^25*d^4 - 19712*a^7*b^23*d^4 - 35200
*a^9*b^21*d^4 - 38016*a^11*b^19*d^4 - 16896*a^13*b^17*d^4 + 16896*a^15*b^15*d^4 + 38016*a^17*b^13*d^4 + 35200*
a^19*b^11*d^4 + 19712*a^21*b^9*d^4 + 6912*a^23*b^7*d^4 + 1408*a^25*b^5*d^4 + 128*a^27*b^3*d^4)) - (-1i/(4*(a^7
*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d
^2*21i)))^(1/2)*((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20
*d^3 - 9648*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 +
1344*a^20*b^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*
b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(128*a*b^29*d^4 + (-1i/(4
*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*
b^2*d^2*21i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*
a^7*b^26*d^5 + 87360*a^9*b^24*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 41184
0*a^17*b^16*d^5 + 320320*a^19*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 672
0*a^27*b^6*d^5 + 960*a^29*b^4*d^5 + 64*a^31*b^2*d^5) + 1408*a^3*b^27*d^4 + 6912*a^5*b^25*d^4 + 19712*a^7*b^23*
d^4 + 35200*a^9*b^21*d^4 + 38016*a^11*b^19*d^4 + 16896*a^13*b^17*d^4 - 16896*a^15*b^15*d^4 - 38016*a^17*b^13*d
^4 - 35200*a^19*b^11*d^4 - 19712*a^21*b^9*d^4 - 6912*a^23*b^7*d^4 - 1408*a^25*b^5*d^4 - 128*a^27*b^3*d^4)) - 1
6*b^23*d^2 - 96*a^2*b^21*d^2 - 144*a^4*b^19*d^2 + 384*a^6*b^17*d^2 + 2016*a^8*b^15*d^2 + 4032*a^10*b^13*d^2 +
4704*a^12*b^11*d^2 + 3456*a^14*b^9*d^2 + 1584*a^16*b^7*d^2 + 416*a^18*b^5*d^2 + 48*a^20*b^3*d^2))*(-1i/(4*(a^7
*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d
^2*21i)))^(1/2)*2i - ((2*b)/(5*(a^2 + b^2)) + (4*a*b*(a + b*tan(c + d*x)))/(3*(a^4 + b^4 + 2*a^2*b^2)) + (2*b*
(3*a^2 - b^2)*(a + b*tan(c + d*x))^2)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(d*(a + b*tan(c + d*x))^(5/2))