Integrand size = 14, antiderivative size = 194 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{7/2} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{7/2} d}-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}} \]
-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)
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 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\frac {i (a+i b) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )+(-i a-b) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}} \]
(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))
Time = 1.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3964, 3042, 4012, 3042, 4012, 3042, 4022, 3042, 4020, 25, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \tan (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}}dx}{a^2+b^2}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}}dx}{a^2+b^2}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a^2+b^2}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a^2+b^2}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a^2+b^2}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a^2+b^2}-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a^2+b^2}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}+\frac {-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {1}{2} (a-i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}+\frac {-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {1}{2} (a-i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}+\frac {-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {i (a+i b)^3 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i (a-i b)^3 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}+\frac {-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {i (a-i b)^3 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i (a+i b)^3 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}+\frac {-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {(a-i b)^3 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {(a+i b)^3 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}+\frac {-\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {(a-i b)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}+\frac {(a+i b)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
(-2*b)/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2)) + ((-4*a*b)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + ((((a + I*b)^3*ArcTan[Tan[c + d*x]/Sq rt[a - I*b]])/(Sqrt[a - I*b]*d) + ((a - I*b)^3*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d))/(a^2 + b^2) - (2*b*(3*a^2 - b^2))/((a^2 + b^2) *d*Sqrt[a + b*Tan[c + d*x]]))/(a^2 + b^2))/(a^2 + b^2)
3.6.54.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
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] + Simp[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]
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] + Simp[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 ]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(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]
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 | \(\text {Expression too large to display}\) | \(3082\) |
-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...
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 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
-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^1 2 + 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^1 4)/((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)) + 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)*sq rt(-(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 + (a^14 + 7*a^12*b^2 + 21*...
\[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
Timed out. \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Time = 24.33 (sec) , antiderivative size = 5307, normalized size of antiderivative = 27.36 \[ \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
(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 - 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) - ((-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 + 19219 2*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^1 7*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 - 19 712*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*...