\(\int \frac {\sec ^4(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [403]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^4/(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 = {} \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[In]

Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

Defer[Int][Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^4(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 = 2.25 (sec) , antiderivative size = 1158, normalized size of antiderivative = 50.35 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {4 i b^2 \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 {14 a^4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+74 a^2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-7 i a^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-37 i a^2 b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+144 i a^3 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+36 i a b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+72 a^3 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+18 a b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-180 a^4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-372 a^2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+90 i a^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+186 i a^2 b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-144 i a^3 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-36 i a b^3 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-72 a^3 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-18 a b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+14 a^4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+74 a^2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+2 b^4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-7 i a^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-37 i a^2 b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-i b^4 \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 ]+\frac {3 \sec ^3(c+d x) \left (48 a^5 b+568 a^3 b^3+14 a b^5+\left (78 a^5 b+606 a^3 b^3+81 a b^5\right ) \cos (2 (c+d x))+18 a b^3 \left (4 a^2+b^2\right ) \cos (4 (c+d x))+2 a^5 b \cos (6 (c+d x))-30 a^3 b^3 \cos (6 (c+d x))-17 a b^5 \cos (6 (c+d x))+48 a^6 \sin (c+d x)-244 a^4 b^2 \sin (c+d x)+20 a^2 b^4 \sin (c+d x)-4 b^6 \sin (c+d x)+16 a^6 \sin (3 (c+d x))-194 a^4 b^2 \sin (3 (c+d x))-86 a^2 b^4 \sin (3 (c+d x))-6 b^6 \sin (3 (c+d x))-14 a^4 b^2 \sin (5 (c+d x))-74 a^2 b^4 \sin (5 (c+d x))-2 b^6 \sin (5 (c+d x))\right )}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{72 a \left (a^2-b^2\right )^3 d} \]

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (14*a^4*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)] + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c
+ d*x] - #1)] - (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]
- I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (36*I
)*a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 18*a*b^3
*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 180*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 372*a^2*b^2*Ar
cTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (90*I)*a^
4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (186*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^4*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (36*I)*
a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 18*a*b
^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 74*a^2*b^2*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (7*I)*a^
4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^4*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) & ] + (3*Sec[c + d*x]^3*(48*a^5*b
+ 568*a^3*b^3 + 14*a*b^5 + (78*a^5*b + 606*a^3*b^3 + 81*a*b^5)*Cos[2*(c + d*x)] + 18*a*b^3*(4*a^2 + b^2)*Cos[
4*(c + d*x)] + 2*a^5*b*Cos[6*(c + d*x)] - 30*a^3*b^3*Cos[6*(c + d*x)] - 17*a*b^5*Cos[6*(c + d*x)] + 48*a^6*Sin
[c + d*x] - 244*a^4*b^2*Sin[c + d*x] + 20*a^2*b^4*Sin[c + d*x] - 4*b^6*Sin[c + d*x] + 16*a^6*Sin[3*(c + d*x)]
- 194*a^4*b^2*Sin[3*(c + d*x)] - 86*a^2*b^4*Sin[3*(c + d*x)] - 6*b^6*Sin[3*(c + d*x)] - 14*a^4*b^2*Sin[5*(c +
d*x)] - 74*a^2*b^4*Sin[5*(c + d*x)] - 2*b^6*Sin[5*(c + d*x)]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/
(72*a*(a^2 - b^2)^3*d)

Maple [N/A] (verified)

Time = 10.45 (sec) , antiderivative size = 525, normalized size of antiderivative = 22.83

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \left (\frac {\frac {\left (a^{4}+7 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}-3 b^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 b^{2} \left (2 a^{2}+b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\left (6 a^{2} b +6 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (a^{4}+7 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+2 a^{2} b +b^{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 (\left (19 a^{4}+28 a^{2} b^{2}-2 b^{4}\right ) \textit {\_R}^{4}+18 a b \left (-4 a^{2}-b^{2}\right ) \textit {\_R}^{3}+6 a^{2} \left (11 a^{2}+34 b^{2}\right ) \textit {\_R}^{2}+18 a b \left (-4 a^{2}-b^{2}\right ) \textit {\_R} +19 a^{4}+28 a^{2} b^{2}-2 b^{4}\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 )^{3} \left (a +b \right )^{3}}-\frac {1}{3 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -4 b}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +4 b}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(525\)
default	\(\frac {\frac {2 b^{2} \left (\frac {\frac {\left (a^{4}+7 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}-3 b^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 b^{2} \left (2 a^{2}+b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\left (6 a^{2} b +6 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (a^{4}+7 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+2 a^{2} b +b^{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 (\left (19 a^{4}+28 a^{2} b^{2}-2 b^{4}\right ) \textit {\_R}^{4}+18 a b \left (-4 a^{2}-b^{2}\right ) \textit {\_R}^{3}+6 a^{2} \left (11 a^{2}+34 b^{2}\right ) \textit {\_R}^{2}+18 a b \left (-4 a^{2}-b^{2}\right ) \textit {\_R} +19 a^{4}+28 a^{2} b^{2}-2 b^{4}\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 )^{3} \left (a +b \right )^{3}}-\frac {1}{3 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -4 b}{\left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{3 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +4 b}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(525\)
risch	\(\text {Expression too large to display}\)	\(8211\)

[In]

int(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*b^2/(a-b)^3/(a+b)^3*((1/3*(a^4+7*a^2*b^2+b^4)/a*tan(1/2*d*x+1/2*c)^5-3*b^3*tan(1/2*d*x+1/2*c)^4+4*b^2*(
2*a^2+b^2)/a*tan(1/2*d*x+1/2*c)^3+(6*a^2*b+6*b^3)*tan(1/2*d*x+1/2*c)^2-1/3*(a^4+7*a^2*b^2+b^4)/a*tan(1/2*d*x+1
/2*c)+2*a^2*b+b^3)/(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(((19*a^4+28*a^2*b^2-2*b^4)*_R^4+18*a*b*(-4*a^2-b^2)*_R^3+6*a^2*(11*a^2+34*b^2)*_R^2+18*a
*b*(-4*a^2-b^2)*_R+19*a^4+28*a^2*b^2-2*b^4)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootO
f(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a)))-1/3/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^3+1/2/(a-b)^2/(tan(1/2*d*x+1/2*c)+
1)^2-(a-4*b)/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)-1/3/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^2/(tan(1/2*d*x+1/2*
c)-1)^2-(a+4*b)/(a+b)^3/(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 = 140.12 (sec) , antiderivative size = 133123, normalized size of antiderivative = 5787.96 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**4/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(sec(d*x+c)^4/(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 = 4.53 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 23.83 (sec) , antiderivative size = 4657, normalized size of antiderivative = 202.48 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x)^3)^2),x)

[Out]

symsum(log(26838024192*a^8*b^54 - tan(c/2 + (d*x)/2)*(7962624000*a^7*b^55 - 508612608000*a^9*b^53 + 8841498624
000*a^11*b^51 - 82283765760000*a^13*b^49 + 501714984960000*a^15*b^47 - 2205295497216000*a^17*b^45 + 7379181637
632000*a^19*b^43 - 19451488075776000*a^21*b^41 + 41318016122880000*a^23*b^39 - 71811432161280000*a^25*b^37 + 1
03155513237504000*a^27*b^35 - 123224906907648000*a^29*b^33 + 122756816093184000*a^31*b^31 - 101967282708480000
*a^33*b^29 + 70396872007680000*a^35*b^27 - 40129785593856000*a^37*b^25 + 18687625592832000*a^39*b^23 - 6994754
113536000*a^41*b^21 + 2053854351360000*a^43*b^19 - 455730831360000*a^45*b^17 + 71860690944000*a^47*b^15 - 7177
310208000*a^49*b^13 + 341397504000*a^51*b^11) - 392822784*a^6*b^56 - root(18600435*a^18*b^6*d^6 - 18600435*a^1
6*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 3720087*a^12*b^12*d^6 + 53
1441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4
- 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b^6*d^2 + 433755*a^6*b^10
*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x)/2)*(764411904*a^6*b^58 - 61439606784*a
^8*b^56 + 2110475575296*a^10*b^54 - 33643637121024*a^12*b^52 + 319697763065856*a^14*b^50 - 2067381036048384*a^
16*b^48 + 9810082122817536*a^18*b^46 - 35797302942326784*a^20*b^44 + 103613766013034496*a^22*b^42 - 2430046994
98881024*a^24*b^40 + 468678655511248896*a^26*b^38 - 750973819695611904*a^28*b^36 + 1006348379003928576*a^30*b^
34 - 1132028278205497344*a^32*b^32 + 1070100496146087936*a^34*b^30 - 848821864657895424*a^36*b^28 + 5626355927
01198336*a^38*b^26 - 309384400894377984*a^40*b^24 + 139566181489975296*a^42*b^22 - 50807786761396224*a^44*b^20
+ 14569217952178176*a^46*b^18 - 3172130021597184*a^48*b^16 + 494158536400896*a^50*b^14 - 49418889191424*a^52*
b^12 + 2463538323456*a^54*b^10 - 14338695168*a^56*b^8) + 95551488*a^7*b^57 + 35879583744*a^9*b^55 - 1812522147
840*a^11*b^53 + 29896430247936*a^13*b^51 - 273690491977728*a^15*b^49 + 1665068560662528*a^17*b^47 - 7358934856
605696*a^19*b^45 + 24887080515133440*a^21*b^43 - 66575487905316864*a^23*b^41 + 144045035942510592*a^25*b^39 -
255939373888192512*a^27*b^37 + 377317716543258624*a^29*b^35 - 464589495171809280*a^31*b^33 + 47947008416012697
6*a^33*b^31 - 415092174607761408*a^35*b^29 + 300910589340991488*a^37*b^27 - 181823043267035136*a^39*b^25 + 908
63416678809600*a^41*b^23 - 37111903240495104*a^43*b^21 + 12175612162301952*a^45*b^19 - 3127996467412992*a^47*b
^17 + 605418993598464*a^49*b^15 - 82897275985920*a^51*b^13 + 7145262637056*a^53*b^11 - 290870673408*a^55*b^9 +
root(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087
*a^22*b^2*d^6 - 3720087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 1558
30311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^
2 + 3750705*a^10*b^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d
*x)/2)*(45578059776*a^9*b^57 - 1988020371456*a^11*b^55 + 21725255172096*a^13*b^53 - 78629462802432*a^15*b^51 -
330769869373440*a^17*b^49 + 5337288405614592*a^19*b^47 - 32144913894998016*a^21*b^45 + 126404118900965376*a^2
3*b^43 - 367050326151462912*a^25*b^41 + 829818883454238720*a^27*b^39 - 1502808604998893568*a^29*b^37 + 2216700
870917750784*a^31*b^35 - 2688523449382600704*a^33*b^33 + 2692902186903011328*a^35*b^31 - 2227622993351147520*a
^37*b^29 + 1515332894269243392*a^39*b^27 - 839694861496221696*a^41*b^25 + 372789943915216896*a^43*b^23 - 12885
4679612424192*a^45*b^21 + 32863270985072640*a^47*b^19 - 5445156193763328*a^49*b^17 + 316457498640384*a^51*b^15
+ 91463986446336*a^53*b^13 - 25165538721792*a^55*b^11 + 2461645209600*a^57*b^9 - 73741860864*a^59*b^7) + root
(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22
*b^2*d^6 - 3720087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311
*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3
750705*a^10*b^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(root(18600435*a^1
8*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 37
20087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311*a^12*b^8*d^4
- 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b
^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x)/2)*(6965703475
2*a^11*b^59 - 2855938424832*a^13*b^57 + 46200028299264*a^15*b^55 - 432918470983680*a^17*b^53 + 273299375849472
0*a^19*b^51 - 12560556506480640*a^21*b^49 + 43925900257198080*a^23*b^47 - 119837962587340800*a^25*b^45 + 25765
1619562782720*a^27*b^43 - 433619569038458880*a^29*b^41 + 549558392034263040*a^31*b^39 - 452796847276032000*a^3
3*b^37 + 36223747782082560*a^35*b^35 + 641677817854033920*a^37*b^33 - 1337691257381191680*a^39*b^31 + 17594391
77986867200*a^41*b^29 - 1756851767431004160*a^43*b^27 + 1404659530591764480*a^45*b^25 - 917046791277281280*a^4
7*b^23 + 491599995054981120*a^49*b^21 - 215796448806174720*a^51*b^19 + 76837281894236160*a^53*b^17 - 218247679
85909760*a^55*b^15 + 4817480523448320*a^57*b^13 - 793393625825280*a^59*b^11 + 91181058490368*a^61*b^9 - 646068
9973248*a^63*b^7 + 208971104256*a^65*b^5) + root(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20
*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 3720087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 5314
41*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4
+ 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5
488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x)/2)*(39182082048*a^14*b^58 - 1057916215296*a^16*b^56 + 137529107
98848*a^18*b^54 - 114607589990400*a^20*b^52 + 687645539942400*a^22*b^50 - 3163169483735040*a^24*b^48 + 1159828
8107028480*a^26*b^46 - 34794864321085440*a^28*b^44 + 86987160802713600*a^30*b^42 - 183639561694617600*a^32*b^4
0 + 330551211050311680*a^34*b^38 - 510851871623208960*a^36*b^36 + 681135828830945280*a^38*b^34 - 7859259563433
98400*a^40*b^32 + 785925956343398400*a^42*b^30 - 681135828830945280*a^44*b^28 + 510851871623208960*a^46*b^26 -
330551211050311680*a^48*b^24 + 183639561694617600*a^50*b^22 - 86987160802713600*a^52*b^20 + 34794864321085440
*a^54*b^18 - 11598288107028480*a^56*b^16 + 3163169483735040*a^58*b^14 - 687645539942400*a^60*b^12 + 1146075899
90400*a^62*b^10 - 13752910798848*a^64*b^8 + 1057916215296*a^66*b^6 - 39182082048*a^68*b^4) + 156728328192*a^13
*b^59 - 4349211107328*a^15*b^57 + 58185391841280*a^17*b^55 - 499689092358144*a^19*b^53 + 3094404929740800*a^21
*b^51 - 14715614554767360*a^23*b^49 + 55882660879319040*a^25*b^47 - 173974321605427200*a^27*b^45 + 45233323617
4110720*a^29*b^43 - 995519729186611200*a^31*b^41 + 1873123529285099520*a^33*b^39 - 3035061119643770880*a^35*b^
37 + 4257098930193408000*a^37*b^35 - 5187111311866429440*a^39*b^33 + 5501481694403788800*a^41*b^31 - 508232118
4353976320*a^43*b^29 + 4086814972985671680*a^45*b^27 - 2854760459070873600*a^47*b^25 + 1726211879929405440*a^4
9*b^23 - 898867328294707200*a^51*b^21 + 400140939692482560*a^53*b^19 - 150777745391370240*a^55*b^17 + 47447542
256025600*a^57*b^15 - 12240090610974720*a^59*b^13 + 2521366979788800*a^61*b^11 - 398834413166592*a^63*b^9 + 45
490397257728*a^65*b^7 - 3330476974080*a^67*b^5 + 117546246144*a^69*b^3) + 8707129344*a^12*b^58 - 1332190789632
*a^14*b^56 + 28681284059136*a^16*b^54 - 311301641871360*a^18*b^52 + 2177740120227840*a^20*b^50 - 1092239719170
0480*a^22*b^48 + 41634880384204800*a^24*b^46 - 125003771820195840*a^26*b^44 + 302447666790973440*a^28*b^42 - 5
98319665965711360*a^30*b^40 + 975644030336532480*a^32*b^38 - 1314242849218682880*a^34*b^36 + 14554184376729600
00*a^36*b^34 - 1304054920154972160*a^38*b^32 + 908181105107927040*a^40*b^30 - 436625531301888000*a^42*b^28 + 6
6949248132956160*a^44*b^26 + 118659409094983680*a^46*b^24 - 149422959601090560*a^48*b^22 + 105921118310768640*
a^50*b^20 - 54125344499466240*a^52*b^18 + 21015701527265280*a^54*b^16 - 6236220178759680*a^56*b^14 + 138822116
6960640*a^58*b^12 - 222162405212160*a^60*b^10 + 23587613392896*a^62*b^8 - 1410554953728*a^64*b^6 + 30474952704
*a^66*b^4) - tan(c/2 + (d*x)/2)*(505980960768*a^12*b^56 - 28050984640512*a^14*b^54 + 435764251090944*a^16*b^52
- 3575718109347840*a^18*b^50 + 18730264859099136*a^20*b^48 - 67896173119315968*a^22*b^46 + 175151109969174528
*a^24*b^44 - 313178493592682496*a^26*b^42 + 322543721316925440*a^28*b^40 + 87817901724942336*a^30*b^38 - 11411
07740572336128*a^32*b^36 + 2683287241504063488*a^34*b^34 - 4099946394045874176*a^36*b^32 + 4680202272693534720
*a^38*b^30 - 4159807137221197824*a^40*b^28 + 2907691359083200512*a^42*b^26 - 1583635567837888512*a^44*b^24 + 6
50291463103832064*a^46*b^22 - 184497987902054400*a^48*b^20 + 25459845498372096*a^50*b^18 + 4948055537467392*a^
52*b^16 - 3746991697108992*a^54*b^14 + 988831432433664*a^56*b^12 - 136164991057920*a^58*b^10 + 8069573984256*a
^60*b^8 + 13544423424*a^62*b^6) + 137379151872*a^11*b^57 - 4254400143360*a^13*b^55 + 29689859874816*a^15*b^53
+ 87020018122752*a^17*b^51 - 2614627107274752*a^19*b^49 + 20133104812498944*a^21*b^47 - 94005764925972480*a^23
*b^45 + 309275227789295616*a^25*b^43 - 759972938071523328*a^27*b^41 + 1428994663615807488*a^29*b^39 - 20578779
23764617216*a^31*b^37 + 2199908326418841600*a^33*b^35 - 1543980376177311744*a^35*b^33 + 260078196862697472*a^3
7*b^31 + 1033592707257090048*a^39*b^29 - 1728050263069556736*a^41*b^27 + 1665648670228807680*a^43*b^25 - 11485
76443783962624*a^45*b^23 + 593098899751084032*a^47*b^21 - 228687912023703552*a^49*b^19 + 63216104157609984*a^5
1*b^17 - 11132817065533440*a^53*b^15 + 707704347303936*a^55*b^13 + 175924646019072*a^57*b^11 - 46657636319232*
a^59*b^9 + 3600881713152*a^61*b^7) + 1719926784*a^8*b^58 - 109860323328*a^10*b^56 + 2586984873984*a^12*b^54 -
35812476739584*a^14*b^52 + 329722810195968*a^16*b^50 - 2157051013447680*a^18*b^48 + 10507597396918272*a^20*b^4
6 - 39457190948069376*a^22*b^44 + 117177686419562496*a^24*b^42 - 280405445559386112*a^26*b^40 + 54797133496909
8240*a^28*b^38 - 882457306853326848*a^30*b^36 + 1177391139070132224*a^32*b^34 - 1303949437690281984*a^34*b^32
+ 1196629258750230528*a^36*b^30 - 904425852978708480*a^38*b^28 + 556165530870792192*a^40*b^26 - 27208276349475
2256*a^42*b^24 + 101333478214434816*a^44*b^22 - 25813305663086592*a^46*b^20 + 2756171653079040*a^48*b^18 + 957
737252339712*a^50*b^16 - 557094927384576*a^52*b^14 + 135955536224256*a^54*b^12 - 17862568353792*a^56*b^10 + 10
32386052096*a^58*b^8)) - 547736297472*a^10*b^52 + 5998567809024*a^12*b^50 - 42798845214720*a^14*b^48 + 2188373
97897216*a^16*b^46 - 847734439845888*a^18*b^44 + 2578107250925568*a^20*b^42 - 6304715180015616*a^22*b^40 + 126
05115522908160*a^24*b^38 - 20839646107090944*a^26*b^36 + 28704537977536512*a^28*b^34 - 33083332509007872*a^30*
b^32 + 31955047610056704*a^32*b^30 - 25837736359772160*a^34*b^28 + 17420116682981376*a^36*b^26 - 9723722502832
128*a^38*b^24 + 4443893749628928*a^40*b^22 - 1635506216902656*a^42*b^20 + 472961442078720*a^44*b^18 - 10350208
9764864*a^46*b^16 + 16115525517312*a^48*b^14 - 1591065649152*a^50*b^12 + 74879852544*a^52*b^10)*root(18600435*
a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 -
3720087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311*a^12*b^8*
d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^1
0*b^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k), k, 1, 6)/d + ((2*(4*a^4*b +
3*b^5 + 38*a^2*b^3))/(3*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (2*tan(c/2 + (d*x)/2)^8*(47*b^5 - 4*a^4*b + 62
*a^2*b^3))/((a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (4*tan(c/2 + (d*x)/2)^6*(119*b^5 - 24*a^4*b + 220*a^2*b^3))
/(3*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (6*tan(c/2 + (d*x)/2)^2*(b^5 + 4*a^2*b^3))/((a^2 - b^2)*(a^4 + b^4
- 2*a^2*b^2)) - (100*tan(c/2 + (d*x)/2)^4*(b^5 + 2*a^2*b^3))/((a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (6*tan(c/
2 + (d*x)/2)^10*(b^5 + 4*a^2*b^3))/((a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (2*tan(c/2 + (d*x)/2)^11*(b^6 - 3*a
^6 + 19*a^2*b^4 + 28*a^4*b^2))/(3*a*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (2*tan(c/2 + (d*x)/2)^9*(9*b^6 - 7*
a^6 + 19*a^2*b^4 + 24*a^4*b^2))/(3*a*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (4*tan(c/2 + (d*x)/2)^7*(3*a^6 + 1
7*b^6 + 179*a^2*b^4 + 26*a^4*b^2))/(3*a*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (2*tan(c/2 + (d*x)/2)^3*(7*a^6
+ 15*b^6 + 285*a^2*b^4 + 8*a^4*b^2))/(3*a*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (4*tan(c/2 + (d*x)/2)^5*(19*b
^6 - 3*a^6 + 277*a^2*b^4 + 22*a^4*b^2))/(3*a*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (2*tan(c/2 + (d*x)/2)*(b^6
- 3*a^6 + 19*a^2*b^4 + 28*a^4*b^2))/(3*a*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)))/(d*(a - 3*a*tan(c/2 + (d*x)/2)
^4 + 3*a*tan(c/2 + (d*x)/2)^8 - a*tan(c/2 + (d*x)/2)^12 + 8*b*tan(c/2 + (d*x)/2)^3 - 24*b*tan(c/2 + (d*x)/2)^5
+ 24*b*tan(c/2 + (d*x)/2)^7 - 8*b*tan(c/2 + (d*x)/2)^9))