\(\int \frac {\sec ^2(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [402]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[In]

Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

Defer[Int][Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.21 (sec) , antiderivative size = 845, normalized size of antiderivative = 36.74 \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-\frac {i b \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {16 a^2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-8 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+20 i a^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+16 i a b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+10 a^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+8 a b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-120 a^2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+60 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-20 i a^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-16 i a b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-10 a^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-8 a b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+16 a^2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+2 b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-8 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{a \left (a^2-b^2\right )^2}+\frac {18 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {18 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 b \cos (c+d x) \left (-2 a^3-7 a b^2+3 a b^2 \cos (2 (c+d x))+2 b \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{a (a-b)^2 (a+b)^2 (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 d} \]

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

(((-I)*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (16*a^2*b*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)] + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2] - I*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (
16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 8*a*b^
2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 120*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^3*ArcT
an[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (60*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^3*Lo
g[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (16*I)*a*b^2
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 8*a*b^2*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^3*ArcTan[Sin[
c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^3*Log[1 - 2*Cos[
c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a*(a^2 - b^2)^2) + (18*Sin[(c + d*x)
/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (18*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])) + (12*b*Cos[c + d*x]*(-2*a^3 - 7*a*b^2 + 3*a*b^2*Cos[2*(c + d*x)] + 2*b*(2*a^2 + b^2)*Sin[c
+ d*x]))/(a*(a - b)^2*(a + b)^2*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(18*d)

Maple [N/A] (verified)

Time = 6.54 (sec) , antiderivative size = 398, normalized size of antiderivative = 17.30

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {-\frac {\left (2 a^{2}+b^{2}\right ) b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\left (-\frac {a^{2}}{3}+\frac {4 b^{2}}{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 b \left (a^{2}+2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\left (-\frac {2 a^{2}}{3}-\frac {10 b^{2}}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (2 a^{2}+b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}-\frac {a^{2}}{3}-\frac {2 b^{2}}{3}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (b \left (-11 a^{2}+2 b^{2}\right ) \textit {\_R}^{4}+2 a \left (5 a^{2}+4 b^{2}\right ) \textit {\_R}^{3}-54 a^{2} b \,\textit {\_R}^{2}+2 a \left (5 a^{2}+4 b^{2}\right ) \textit {\_R} -11 a^{2} b +2 b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{18 a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(398\)
default	\(\frac {\frac {2 b \left (\frac {-\frac {\left (2 a^{2}+b^{2}\right ) b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\left (-\frac {a^{2}}{3}+\frac {4 b^{2}}{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 b \left (a^{2}+2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\left (-\frac {2 a^{2}}{3}-\frac {10 b^{2}}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (2 a^{2}+b^{2}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}-\frac {a^{2}}{3}-\frac {2 b^{2}}{3}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (b \left (-11 a^{2}+2 b^{2}\right ) \textit {\_R}^{4}+2 a \left (5 a^{2}+4 b^{2}\right ) \textit {\_R}^{3}-54 a^{2} b \,\textit {\_R}^{2}+2 a \left (5 a^{2}+4 b^{2}\right ) \textit {\_R} -11 a^{2} b +2 b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{18 a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{\left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(398\)
risch	\(\text {Expression too large to display}\)	\(5261\)

[In]

int(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*b/(a-b)^2/(a+b)^2*((-1/3*(2*a^2+b^2)*b/a*tan(1/2*d*x+1/2*c)^5+(-1/3*a^2+4/3*b^2)*tan(1/2*d*x+1/2*c)^4-4
/3*b*(a^2+2*b^2)/a*tan(1/2*d*x+1/2*c)^3+(-2/3*a^2-10/3*b^2)*tan(1/2*d*x+1/2*c)^2+1/3*(2*a^2+b^2)*b/a*tan(1/2*d
*x+1/2*c)-1/3*a^2-2/3*b^2)/(a*tan(1/2*d*x+1/2*c)^6+3*tan(1/2*d*x+1/2*c)^4*a+8*tan(1/2*d*x+1/2*c)^3*b+3*tan(1/2
*d*x+1/2*c)^2*a+a)+1/18/a*sum((b*(-11*a^2+2*b^2)*_R^4+2*a*(5*a^2+4*b^2)*_R^3-54*a^2*b*_R^2+2*a*(5*a^2+4*b^2)*_
R-11*a^2*b+2*b^3)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b
+3*_Z^2*a+a)))-1/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)-1/(a-b)^2/(tan(1/2*d*x+1/2*c)+1))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 43.88 (sec) , antiderivative size = 102913, normalized size of antiderivative = 4474.48 \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**2/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [N/A]

Not integrable

Time = 4.34 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 19.15 (sec) , antiderivative size = 3148, normalized size of antiderivative = 136.87 \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^2*(a + b*sin(c + d*x)^3)^2),x)

[Out]

symsum(log(5479612416*a^8*b^36 - 180486144*a^6*b^38 - root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 26572
05*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066
715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*
a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*(764411904*a^6*b^40 - 27805483008*a^8*b^38 + 437297
356800*a^10*b^36 - 3672461721600*a^12*b^34 + 19250011791360*a^14*b^32 - 69150635753472*a^16*b^30 + 18016587200
1024*a^18*b^28 - 352655758540800*a^20*b^26 + 529923028377600*a^22*b^24 - 618699706859520*a^24*b^22 + 563713761
042432*a^26*b^20 - 399760062234624*a^28*b^18 + 218398602240000*a^30*b^16 - 90108039168000*a^32*b^14 + 27130620
764160*a^34*b^12 - 5617221156864*a^36*b^10 + 713536708608*a^38*b^8 - 41803776000*a^40*b^6) - root(5314410*a^16
*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*
a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a
^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(root(5314410*a^16*b^4*d^6 - 531
4410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 115
14555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 9
8415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*(157695787008*a^12*b^38 -
4039140556800*a^14*b^36 + 39183049506816*a^16*b^34 - 212750482120704*a^18*b^32 + 750889290203136*a^20*b^30 - 1
854140141887488*a^22*b^28 + 3327952874029056*a^24*b^26 - 4413464400863232*a^26*b^24 + 4311710468702208*a^28*b^
22 - 3009938035433472*a^30*b^20 + 1359808836452352*a^32*b^18 - 238981192998912*a^34*b^16 - 150898421366784*a^3
6*b^14 + 136937506922496*a^38*b^12 - 52028967665664*a^40*b^10 + 10565134000128*a^42*b^8 - 976165945344*a^44*b^
6 + 12093235200*a^46*b^4) - root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*
a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882
*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 6
4*b^8, d, k)*(tan(c/2 + (d*x)/2)*(69657034752*a^11*b^41 - 1619526057984*a^13*b^39 + 16404231684096*a^15*b^37 -
99052303417344*a^17*b^35 + 405403942256640*a^19*b^33 - 1203882531618816*a^21*b^31 + 2700324609196032*a^23*b^2
9 - 4688893637296128*a^25*b^27 + 6394933732442112*a^27*b^25 - 6897962008903680*a^29*b^23 + 5886924977995776*a^
31*b^21 - 3949971812646912*a^33*b^19 + 2053768012627968*a^35*b^17 - 806001549115392*a^37*b^15 + 22777850363904
0*a^39*b^13 - 42212163059712*a^41*b^11 + 3970450980864*a^43*b^9 + 52242776064*a^45*b^7 - 34828517376*a^47*b^5)
+ 8707129344*a^12*b^40 - 470184984576*a^14*b^38 + 6308315209728*a^16*b^36 - 44092902998016*a^18*b^34 + 197477
693521920*a^20*b^32 - 623151832891392*a^22*b^30 + 1459506434899968*a^24*b^28 - 2616109254180864*a^26*b^26 + 36
53180601827328*a^28*b^24 - 4009284777738240*a^30*b^22 + 3462677318909952*a^32*b^20 - 2339013569937408*a^34*b^1
8 + 1217047711186944*a^36*b^16 - 473946464452608*a^38*b^14 + 130868154040320*a^40*b^12 - 22777850363904*a^42*b
^10 + 1645647446016*a^44*b^8 + 156728328192*a^46*b^6 - 30474952704*a^48*b^4 + root(5314410*a^16*b^4*d^6 - 5314
410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 1151
4555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98
415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*(39182082048*a^14*b^40 - 70
5277476864*a^16*b^38 + 5994858553344*a^18*b^36 - 31972578951168*a^20*b^34 + 119897171066880*a^22*b^32 - 335712
078987264*a^24*b^30 + 727376171139072*a^26*b^28 - 1246930579095552*a^28*b^26 + 1714529546256384*a^30*b^24 - 19
05032829173760*a^32*b^22 + 1714529546256384*a^34*b^20 - 1246930579095552*a^36*b^18 + 727376171139072*a^38*b^16
- 335712078987264*a^40*b^14 + 119897171066880*a^42*b^12 - 31972578951168*a^44*b^10 + 5994858553344*a^46*b^8 -
705277476864*a^48*b^6 + 39182082048*a^50*b^4) + 156728328192*a^13*b^41 - 2938656153600*a^15*b^39 + 2609526664
3968*a^17*b^37 - 145874891464704*a^19*b^35 + 575506421121024*a^21*b^33 - 1702539829149696*a^23*b^31 + 39166409
21518080*a^25*b^29 - 7169850829799424*a^27*b^27 + 10598909922312192*a^29*b^25 - 12763719955464192*a^31*b^23 +
12573216672546816*a^33*b^21 - 10131310955151360*a^35*b^19 + 6650296421842944*a^37*b^17 - 3524976829366272*a^39
*b^15 + 1486724921229312*a^41*b^13 - 487581829005312*a^43*b^11 + 119897171066880*a^45*b^9 - 20805685567488*a^4
7*b^7 + 2272560758784*a^49*b^5 - 117546246144*a^51*b^3)) - 59982446592*a^11*b^39 + 1080651497472*a^13*b^37 - 6
860250464256*a^15*b^35 + 16482112118784*a^17*b^33 + 27170113388544*a^19*b^31 - 327284061511680*a^21*b^29 + 119
4949984370688*a^23*b^27 - 2698934854606848*a^25*b^25 + 4276847122808832*a^27*b^23 - 4968511002943488*a^29*b^21
+ 4288329891495936*a^31*b^19 - 2730918075604992*a^33*b^17 + 1245220111908864*a^35*b^15 - 377418744815616*a^37
*b^13 + 60571629010944*a^39*b^11 + 1483598094336*a^41*b^9 - 2465085063168*a^43*b^7 + 316842762240*a^45*b^5) -
1719926784*a^8*b^40 + 52457766912*a^10*b^38 - 657657004032*a^12*b^36 + 4778655326208*a^14*b^34 - 2313011286835
2*a^16*b^32 + 80237540597760*a^18*b^30 - 208280123670528*a^20*b^28 + 415493301510144*a^22*b^26 - 6473545351004
16*a^24*b^24 + 794486155567104*a^26*b^22 - 769729798176768*a^28*b^20 + 586362545233920*a^30*b^18 - 34739113431
8592*a^32*b^16 + 156884680286208*a^34*b^14 - 52204937674752*a^36*b^12 + 12071252385792*a^38*b^10 - 17329337303
04*a^40*b^8 + 116363796480*a^42*b^6 - tan(c/2 + (d*x)/2)*(19779158016*a^9*b^39 - 436216430592*a^11*b^37 + 3308
494159872*a^13*b^35 - 11619395371008*a^15*b^33 + 12486453460992*a^17*b^31 + 61196714901504*a^19*b^29 - 3343320
52733952*a^21*b^27 + 871706622099456*a^23*b^25 - 1507393926365184*a^25*b^23 + 1878255074082816*a^27*b^21 - 173
6372938899456*a^29*b^19 + 1197522672353280*a^31*b^17 - 608856446435328*a^33*b^15 + 221032950792192*a^35*b^13 -
53644731383808*a^37*b^11 + 7499310759936*a^39*b^9 - 345490292736*a^41*b^7 - 26873856000*a^43*b^5)) + 95551488
*a^7*b^39 + 6640828416*a^9*b^37 - 187507851264*a^11*b^35 + 1874314100736*a^13*b^33 - 10498349481984*a^15*b^31
+ 38554452099072*a^17*b^29 - 100273965023232*a^19*b^27 + 192807351779328*a^21*b^25 - 280858991542272*a^23*b^23
+ 313783776903168*a^25*b^21 - 269640960196608*a^27*b^19 + 177127448150016*a^29*b^17 - 87483347288064*a^31*b^1
5 + 31483928641536*a^33*b^13 - 7801408733184*a^35*b^11 + 1191025410048*a^37*b^9 - 84503347200*a^39*b^7) - 5983
7128704*a^10*b^34 + 363432738816*a^12*b^32 - 1444185759744*a^14*b^30 + 4071882866688*a^16*b^28 - 8529191903232
*a^18*b^26 + 13638053265408*a^20*b^24 - 16903052255232*a^22*b^22 + 16345206079488*a^24*b^20 - 12319205842944*a
^26*b^18 + 7172803362816*a^28*b^16 - 3166919368704*a^30*b^14 + 1026022588416*a^32*b^12 - 230217375744*a^34*b^1
0 + 31983206400*a^36*b^8 - 2073600000*a^38*b^6 - tan(c/2 + (d*x)/2)*(1911029760*a^7*b^37 - 56614256640*a^9*b^3
5 + 591941468160*a^11*b^33 - 3412860272640*a^13*b^31 + 12781922549760*a^15*b^29 - 33715581419520*a^17*b^27 + 6
5518222049280*a^19*b^25 - 96227753656320*a^21*b^23 + 108217793249280*a^23*b^21 - 93494981099520*a^25*b^19 + 61
692340469760*a^27*b^17 - 30585314672640*a^29*b^15 + 11042885468160*a^31*b^13 - 2743999856640*a^33*b^11 + 41994
8789760*a^35*b^9 - 29859840000*a^37*b^7))*root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*
d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2
*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 20
00*a^2*b^6 + 64*b^8, d, k), k, 1, 6)/d - ((2*(7*a^2*b + 2*b^3))/(3*(a^2 - b^2)^2) + (2*tan(c/2 + (d*x)/2)^6*(5
*a^2*b + 4*b^3))/(3*(a^2 - b^2)^2) + (2*tan(c/2 + (d*x)/2)^2*(19*a^2*b + 8*b^3))/(3*(a^2 - b^2)^2) - (2*tan(c/
2 + (d*x)/2)^4*(7*a^2*b + 38*b^3))/(3*(a^2 - b^2)^2) + (6*tan(c/2 + (d*x)/2)^3*(b^4 - a^4 + 5*a^2*b^2))/(a*(a^
2 - b^2)^2) - (2*tan(c/2 + (d*x)/2)^5*(9*a^4 + 7*b^4 + 11*a^2*b^2))/(3*a*(a^2 - b^2)^2) - (2*tan(c/2 + (d*x)/2
)^7*(3*a^4 + b^4 + 5*a^2*b^2))/(3*a*(a^4 + b^4 - 2*a^2*b^2)) - (2*tan(c/2 + (d*x)/2)*(3*a^4 + b^4 + 5*a^2*b^2)
)/(3*a*(a^2 - b^2)^2))/(d*(a + 2*a*tan(c/2 + (d*x)/2)^2 - 2*a*tan(c/2 + (d*x)/2)^6 - a*tan(c/2 + (d*x)/2)^8 +
8*b*tan(c/2 + (d*x)/2)^3 - 8*b*tan(c/2 + (d*x)/2)^5))