Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]
Result contains complex when optimal does not.
Time = 2.25 (sec) , antiderivative size = 1158, normalized size of antiderivative = 50.35 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx =\text {Too large to display} \]
((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*ArcT an[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 - 18 0*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 372*a^2*b^2*ArcTan[S in[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...
Not integrable
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {3042, 3707}
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.
\(\displaystyle \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (c+d x)^4 \left (a+b \sin (c+d x)^3\right )^2}dx\) |
\(\Big \downarrow \) 3707 |
\(\displaystyle \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2}dx\) |
3.5.3.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Unintegrable[(d*Cos[e + f*x])^m*(a + b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
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 | \(\text {Expression too large to display}\) | \(8211\) |
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=RootOf(_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))
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 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
Not integrable
Time = 4.53 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]
Time = 23.83 (sec) , antiderivative size = 4657, normalized size of antiderivative = 202.48 \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
symsum(log(26838024192*a^8*b^54 - tan(c/2 + (d*x)/2)*(7962624000*a^7*b^55 - 508612608000*a^9*b^53 + 8841498624000*a^11*b^51 - 82283765760000*a^13*b^ 49 + 501714984960000*a^15*b^47 - 2205295497216000*a^17*b^45 + 737918163763 2000*a^19*b^43 - 19451488075776000*a^21*b^41 + 41318016122880000*a^23*b^39 - 71811432161280000*a^25*b^37 + 103155513237504000*a^27*b^35 - 1232249069 07648000*a^29*b^33 + 122756816093184000*a^31*b^31 - 101967282708480000*a^3 3*b^29 + 70396872007680000*a^35*b^27 - 40129785593856000*a^37*b^25 + 18687 625592832000*a^39*b^23 - 6994754113536000*a^41*b^21 + 2053854351360000*a^4 3*b^19 - 455730831360000*a^45*b^17 + 71860690944000*a^47*b^15 - 7177310208 000*a^49*b^13 + 341397504000*a^51*b^11) - 392822784*a^6*b^56 - root(186004 35*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^1 0*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 - 117 649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x)/2)*(76441190 4*a^6*b^58 - 61439606784*a^8*b^56 + 2110475575296*a^10*b^54 - 336436371210 24*a^12*b^52 + 319697763065856*a^14*b^50 - 2067381036048384*a^16*b^48 + 98 10082122817536*a^18*b^46 - 35797302942326784*a^20*b^44 + 10361376601303449 6*a^22*b^42 - 243004699498881024*a^24*b^40 + 468678655511248896*a^26*b^...