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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A] Result contains complex when optimal does not.
time = 1.26, size = 1158, normalized size = 44.54 \begin {gather*} \frac {4 i b^2 \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {14 a^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+74 a^2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-7 i a^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-37 i a^2 b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+144 i a^3 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+36 i a b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+72 a^3 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+18 a b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-180 a^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-372 a^2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+90 i a^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+186 i a^2 b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-144 i a^3 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-36 i a b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-72 a^3 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-18 a b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+14 a^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+74 a^2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+2 b^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-7 i a^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-37 i a^2 b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-i b^4 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$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} \end {gather*}

Warning: Unable to verify antiderivative.

[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)

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Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(524\) vs. \(2(25)=50\).
time = 8.93, size = 525, normalized size = 20.19

method	result	size
derivativedivides	\(\frac {-\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 {4 b +a}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\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 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\munderset {\textit {\_R} =\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}}}{d}\)	\(525\)
default	\(\frac {-\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 {4 b +a}{\left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\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 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\munderset {\textit {\_R} =\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}}}{d}\)	\(525\)
risch	\(\text {Expression too large to display}\)	\(8211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-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-(4*b+a)/(a+b)^3/(tan(1/2*d*
x+1/2*c)-1)+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*a*tan(1/2*d*x+1/2*c)^4+8*b*tan(1/2*d*x+1/2*c)^3+3*a*tan(1/
2*d*x+1/2*c)^2+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=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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.

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Fricas [A] Result contains complex when optimal does not.
time = 104.28, size = 133123, normalized size = 5120.12 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(36*(2*a^5*b - 30*a^3*b^3 - 17*a*b^5)*cos(d*x + c)^6 - 36*a^5*b + 72*a^3*b^3 - 36*a*b^5 - 108*(a^5*b - 2
1*a^3*b^3 - 10*a*b^5)*cos(d*x + c)^4 + sqrt(2/3)*sqrt(1/6)*((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*cos(d*x
+ c)^3 - ((a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c)^5 - (a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d
*cos(d*x + c)^3)*sin(d*x + c))*sqrt(-(573480*a^8*b^4 + 4293324*a^6*b^6 + 3847662*a^4*b^8 + 159894*a^2*b^10 - 1
7010*b^12 - (a^16 - 7*a^14*b^2 + 21*a^12*b^4 - 35*a^10*b^6 + 35*a^8*b^8 - 21*a^6*b^10 + 7*a^4*b^12 - a^2*b^14)
*((-I*sqrt(3) + 1)*((1180*a^8*b^4 + 8834*a^6*b^6 + 7917*a^4*b^8 + 329*a^2*b^10 - 35*b^12)^2/(a^16*d^2 - 7*a^14
*b^2*d^2 + 21*a^12*b^4*d^2 - 35*a^10*b^6*d^2 + 35*a^8*b^8*d^2 - 21*a^6*b^10*d^2 + 7*a^4*b^12*d^2 - a^2*b^14*d^
2)^2 + 15*(1029*a^4*b^6 - 3173*a^2*b^8 + 119*b^10)/(a^18*d^4 - 7*a^16*b^2*d^4 + 21*a^14*b^4*d^4 - 35*a^12*b^6*
d^4 + 35*a^10*b^8*d^4 - 21*a^8*b^10*d^4 + 7*a^6*b^12*d^4 - a^4*b^14*d^4))/(-1/531441*(1180*a^8*b^4 + 8834*a^6*
b^6 + 7917 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 25.44, size = 2500, normalized size = 96.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^...

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