Integrand size = 23, antiderivative size = 344 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
-3/8*arctanh(cos(d*x+c))/a/d+b*cot(d*x+c)/a^2/d-3/8*cot(d*x+c)*csc(d*x+c)/ a/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d-2/3*b^(5/3)*arctan((b^(1/3)+a^(1/3)*ta n(1/2*d*x+1/2*c))/(a^(2/3)-b^(2/3))^(1/2))/a^(7/3)/d/(a^(2/3)-b^(2/3))^(1/ 2)+2/3*(-1)^(1/3)*b^(5/3)*arctan(((-1)^(2/3)*b^(1/3)+a^(1/3)*tan(1/2*d*x+1 /2*c))/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2))/a^(7/3)/d/(a^(2/3)+(-1)^(1/3)*b ^(2/3))^(1/2)+2/3*(-1)^(2/3)*b^(5/3)*arctan(((-1)^(1/3)*b^(1/3)-a^(1/3)*ta n(1/2*d*x+1/2*c))/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2))/a^(7/3)/d/(a^(2/3)-( -1)^(2/3)*b^(2/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.84 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {-64 b^2 \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (32 b \cot \left (\frac {1}{2} (c+d x)\right )-6 a \csc ^2\left (\frac {1}{2} (c+d x)\right )-a \csc ^4\left (\frac {1}{2} (c+d x)\right )-24 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \sec ^4\left (\frac {1}{2} (c+d x)\right )-32 b \tan \left (\frac {1}{2} (c+d x)\right )\right )}{192 a^2 d} \]
(-64*b^2*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-2* ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + # 1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*(32* b*Cot[(c + d*x)/2] - 6*a*Csc[(c + d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a* Log[Cos[(c + d*x)/2]] + 24*a*Log[Sin[(c + d*x)/2]] + 6*a*Sec[(c + d*x)/2]^ 2 + a*Sec[(c + d*x)/2]^4 - 32*b*Tan[(c + d*x)/2]))/(192*a^2*d)
Time = 0.73 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 3699, 2009}
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.
| \(\displaystyle \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^5 \left (a+b \sin (c+d x)^3\right )}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle \int \left (\frac {b^2 \sin (c+d x)}{a^2 \left (a+b \sin ^3(c+d x)\right )}-\frac {b \csc ^2(c+d x)}{a^2}+\frac {\csc ^5(c+d x)}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}\) |
(-2*b^(5/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^( 2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*(-1)^(1/3)*b^(5/3)*ArcT an[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/ 3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) + (2*(-1)^( 2/3)*b^(5/3)*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x )/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - (-1) ^(2/3)*b^(2/3)]*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + (b*Cot[c + d*x])/ (a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d* x]^3)/(4*a*d)
3.2.88.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.57
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{4}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -8 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{2}}+\frac {2 b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(196\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{4}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -8 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{2}}+\frac {2 b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(196\) |
| risch | \(\frac {3 a \,{\mathrm e}^{7 i \left (d x +c \right )}-11 a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i b \,{\mathrm e}^{6 i \left (d x +c \right )}-11 a \,{\mathrm e}^{3 i \left (d x +c \right )}-24 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a \,{\mathrm e}^{i \left (d x +c \right )}+24 i b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i b}{4 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+32 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (782757789696 a^{16} d^{6}-782757789696 a^{14} b^{2} d^{6}\right ) \textit {\_Z}^{6}-254803968 a^{10} b^{4} d^{4} \textit {\_Z}^{4}-b^{10}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {8153726976 i d^{5} a^{15}}{a^{2} b^{8}+b^{10}}-\frac {8153726976 i d^{5} a^{13} b^{2}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{5}+\left (\frac {84934656 i d^{4} a^{13} b}{a^{2} b^{8}+b^{10}}-\frac {84934656 i d^{4} a^{11} b^{3}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{4}+\left (-\frac {1769472 i d^{3} a^{9} b^{4}}{a^{2} b^{8}+b^{10}}-\frac {884736 i d^{3} a^{7} b^{6}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{3}+\left (-\frac {18432 i d^{2} a^{7} b^{5}}{a^{2} b^{8}+b^{10}}-\frac {9216 i d^{2} a^{5} b^{7}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{2}+\left (-\frac {96 i d \,a^{5} b^{6}}{a^{2} b^{8}+b^{10}}-\frac {192 i d \,a^{3} b^{8}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R} -\frac {i b^{9} a}{a^{2} b^{8}+b^{10}}\right )\right )-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}\) | \(503\) |
1/d*(1/16/a^2*(1/4*tan(1/2*d*x+1/2*c)^4*a+2*tan(1/2*d*x+1/2*c)^2*a-8*b*tan (1/2*d*x+1/2*c))+2/3*b^2/a^2*sum((_R^3+_R)/(_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)) -1/64/a/tan(1/2*d*x+1/2*c)^4-1/8/a/tan(1/2*d*x+1/2*c)^2+3/8/a*ln(tan(1/2*d *x+1/2*c))+1/2/a^2*b/tan(1/2*d*x+1/2*c))
Result contains complex when optimal does not.
Time = 131.01 (sec) , antiderivative size = 21564, normalized size of antiderivative = 62.69 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
\[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
Time = 15.22 (sec) , antiderivative size = 1560, normalized size of antiderivative = 4.53 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
symsum(log((262144*b^14*tan(c/2 + (d*x)/2) - 3072*a^3*b^11 + 155648*root(7 29*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)*a^4*b^11 - 393216*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^2*a^5*b^11 + 774144*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^ 4*z^4 - b^10, z, k)^2*a^7*b^9 - 2064384*root(729*a^14*b^2*z^6 - 729*a^16*z ^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^8*b^9 + 2073600*root(729*a^14*b^2* z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^10*b^7 - 9510912*r oot(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^4*a^1 1*b^7 + 2737152*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^4*a^13*b^5 + 10616832*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 2 43*a^10*b^4*z^4 - b^10, z, k)^5*a^12*b^7 - 10285056*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^5*a^14*b^5 + 3732480*root( 729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^5*a^16*b^ 3 + 7962624*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10 , z, k)^6*a^15*b^5 - 9953280*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^ 10*b^4*z^4 - b^10, z, k)^6*a^17*b^3 + 98304*a^2*b^12*tan(c/2 + (d*x)/2) - 262144*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)*a^3*b^12*tan(c/2 + (d*x)/2) + 165888*root(729*a^14*b^2*z^6 - 729*a^16*z ^6 - 243*a^10*b^4*z^4 - b^10, z, k)*a^5*b^10*tan(c/2 + (d*x)/2) - 1327104* root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^2...