Integrand size = 27, antiderivative size = 174 \[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {(i a-b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {(i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}+\frac {4 a b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]
(I*a-b)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d-(I*a +b)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d+2*b*(3*a ^2-b^2)/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)+4/3*a*b/(a^2+b^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 = 156, normalized size of antiderivative = 0.90 \[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {i \cos (c+d x) \left ((a+i b)^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b)^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )\right ) (a-b \tan (c+d x))}{3 (a-i b) (a+i b) d (a \cos (c+d x)-b \sin (c+d x)) (a+b \tan (c+d x))^{3/2}} \]
((-1/3*I)*Cos[c + d*x]*((a + I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a - b*Tan[c + d*x]))/((a - I*b)*(a + I*b )*d*(a*Cos[c + d*x] - b*Sin[c + d*x])*(a + b*Tan[c + d*x])^(3/2))
Time = 0.86 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 4012, 25, 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.
| \(\displaystyle \int \frac {b \tan (c+d x)-a}{(a+b \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {b \tan (c+d x)-a}{(a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int -\frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}+\frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a^2+b^2}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {1}{2} (a-i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {1}{2} (a-i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^3 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {i (a+i b)^3 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i (a-i b)^3 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {i (a-i b)^3 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i (a+i b)^3 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {(a-i b)^3 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {(a+i b)^3 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 a b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {(a-i b)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}+\frac {(a+i b)^3 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}}{a^2+b^2}}{a^2+b^2}\) |
(4*a*b)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - ((((a + I*b)^3*ArcT an[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + ((a - I*b)^3*ArcTan[Ta n[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)
3.4.72.3.1 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[(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. \(3054\) vs. \(2(150)=300\).
Time = 0.10 (sec) , antiderivative size = 3055, normalized size of antiderivative = 17.56
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(3055\) |
| default | \(\text {Expression too large to display}\) | \(3055\) |
| parts | \(\text {Expression too large to display}\) | \(4474\) |
-1/2/d*b/(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+2/d*b ^3/(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^2+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^7-5 /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+1/4/d/b/(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^ 6-5/4/d*b^3/(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+7/d*b^5/(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)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*ta n(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+4/3*a*b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)-2/d*b^3/(a^2+b^2)^2/(a+ b*tan(d*x+c))^(1/2)+1/4/d*b^5/(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)+1/d*b^5/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta...
Leaf count of result is larger than twice the leaf count of optimal. 3695 vs. \(2 (144) = 288\).
Time = 0.34 (sec) , antiderivative size = 3695, normalized size of antiderivative = 21.24 \[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
1/6*(3*((a^4*b^2 + 2*a^2*b^4 + b^6)*d*tan(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^ 3 + a*b^5)*d*tan(d*x + c) + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*sqrt(-(a^7 - 21 *a^5*b^2 + 35*a^3*b^4 - 7*a*b^6 + (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(-(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^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(-(7*a^8*b - 28*a^6*b^3 - 14*a^4*b^5 + 20*a^2*b^7 - b^9)*sqrt(b*tan(d*x + c) + a) + ((a ^14 - a^12*b^2 - 19*a^10*b^4 - 45*a^8*b^6 - 45*a^6*b^8 - 19*a^4*b^10 - a^2 *b^12 + b^14)*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^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)) + 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^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(-(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^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^...
\[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=- \int \frac {a}{a^{2} \sqrt {a + b \tan {\left (c + d x \right )}} + 2 a b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )} + b^{2} \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}}\, dx - \int \left (- \frac {b \tan {\left (c + d x \right )}}{a^{2} \sqrt {a + b \tan {\left (c + d x \right )}} + 2 a b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )} + b^{2} \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}}\right )\, dx \]
-Integral(a/(a**2*sqrt(a + b*tan(c + d*x)) + 2*a*b*sqrt(a + b*tan(c + d*x) )*tan(c + d*x) + b**2*sqrt(a + b*tan(c + d*x))*tan(c + d*x)**2), x) - Inte gral(-b*tan(c + d*x)/(a**2*sqrt(a + b*tan(c + d*x)) + 2*a*b*sqrt(a + b*tan (c + d*x))*tan(c + d*x) + b**2*sqrt(a + b*tan(c + d*x))*tan(c + d*x)**2), x)
Exception generated. \[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 24.86 (sec) , antiderivative size = 8437, normalized size of antiderivative = 48.49 \[ \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
(log(((((-(4*a^7*d^2 + (320*a^6*b^8*d^4 - 16*a^4*b^10*d^4 - 1760*a^8*b^6*d ^4 + 1600*a^10*b^4*d^4 - 400*a^12*b^2*d^4)^(1/2) + 20*a^3*b^4*d^2 - 40*a^5 *b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b ^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(896*a^7*b^15*d^4 - 32*a*b^21*d^4 - 160*a^3 *b^19*d^4 - 128*a^5*b^17*d^4 - ((a + b*tan(c + d*x))^(1/2)*(-(4*a^7*d^2 + (320*a^6*b^8*d^4 - 16*a^4*b^10*d^4 - 1760*a^8*b^6*d^4 + 1600*a^10*b^4*d^4 - 400*a^12*b^2*d^4)^(1/2) + 20*a^3*b^4*d^2 - 40*a^5*b^2*d^2)/(a^10*d^4 + b ^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4) )^(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 + 3136*a^9*b^13*d^4 + 4928*a^11*b^11*d^4 + 4480*a^13*b^9*d^4 + 2432* a^15*b^7*d^4 + 736*a^17*b^5*d^4 + 96*a^19*b^3*d^4))/4 + (a + b*tan(c + d*x ))^(1/2)*(320*a^6*b^14*d^3 - 16*a^2*b^18*d^3 + 1024*a^8*b^12*d^3 + 1440*a^ 10*b^10*d^3 + 1024*a^12*b^8*d^3 + 320*a^14*b^6*d^3 - 16*a^18*b^2*d^3))*(-( 4*a^7*d^2 + (320*a^6*b^8*d^4 - 16*a^4*b^10*d^4 - 1760*a^8*b^6*d^4 + 1600*a ^10*b^4*d^4 - 400*a^12*b^2*d^4)^(1/2) + 20*a^3*b^4*d^2 - 40*a^5*b^2*d^2)/( a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5* a^8*b^2*d^4))^(1/2))/4 - 16*a^4*b^15*d^2 - 96*a^6*b^13*d^2 - 240*a^8*b^11* d^2 - 320*a^10*b^9*d^2 - 240*a^12*b^7*d^2 - 96*a^14*b^5*d^2 - 16*a^16*b...