Integrand size = 11, antiderivative size = 32 \[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=-\frac {1}{8} \text {arctanh}(\cos (x))+\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{2 \sqrt {2}}-\frac {\sec (x)}{8} \] Output:
-1/8*arctanh(cos(x))+1/4*arctanh(cos(x)*2^(1/2))*2^(1/2)-1/8*sec(x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=\frac {1}{9} \left (-6 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )-9 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-2 \log (1-2 \sin (x))+192 \text {RootSum}\left [1-27648 \text {$\#$1}^2+884736 \text {$\#$1}^3\&,\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-1-4 \sin (x)-384 \sin (x) \text {$\#$1}+18432 \sin (x) \text {$\#$1}^2\right )\right ) \text {$\#$1}\&\right ]\right ) \] Input:
Integrate[(Cos[5*x] + Sin[4*x])^(-1),x]
Output:
(-6*Log[Sec[x/2]^2] - 9*Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2] ] - 2*Log[1 - 2*Sin[x]] + 192*RootSum[1 - 27648*#1^2 + 884736*#1^3 & , Log [-(Sec[x/2]^2*(-1 - 4*Sin[x] - 384*Sin[x]*#1 + 18432*Sin[x]*#1^2))]*#1 & ] )/9
Result contains complex when optimal does not.
Time = 2.35 (sec) , antiderivative size = 679, normalized size of antiderivative = 21.22, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4829, 2462, 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.
| \(\displaystyle \int \frac {1}{\sin (4 x)+\cos (5 x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (4 x)+\cos (5 x)}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right ) \left (16 \sin ^4(x)-8 \sin ^3(x)-12 \sin ^2(x)+4 \sin (x)+1\right )}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {16 \sin (x) (\sin (x)+1)}{3 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}-\frac {1}{2 (\sin (x)-1)}-\frac {4}{9 (2 \sin (x)-1)}+\frac {1}{18 (\sin (x)+1)}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2\ 2^{2/3} \left (-\sqrt {3}+i\right ) \left (2^{2/3}-\left (1+i \sqrt {3}\right )^{2/3}\right ) \text {arctanh}\left (\frac {8 \left (1+i \sqrt {3}\right )^{2/3} \sin (x)+2 \sqrt [3]{2+2 i \sqrt {3}}+2^{2/3} \left (1+i \sqrt {3}\right )}{2 \sqrt {3 \left (\sqrt [3]{2}-i \sqrt [3]{2} \sqrt {3}+2 \left (1+i \sqrt {3}\right )^{4/3}-\left (2+2 i \sqrt {3}\right )^{2/3}\right )}}\right )}{3 \left (2\ 2^{2/3}+2 \left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{4/3}\right ) \sqrt {\sqrt [3]{2}-i \sqrt [3]{2} \sqrt {3}+2 \left (1+i \sqrt {3}\right )^{4/3}-\left (2+2 i \sqrt {3}\right )^{2/3}}}+\frac {2}{9} \log \left (8 \sin ^3(x)-6 \sin (x)-1\right )-\frac {2 \sqrt [3]{1+i \sqrt {3}} \left (2 \sqrt [3]{2}+\sqrt [3]{1+i \sqrt {3}}+\left (2+2 i \sqrt {3}\right )^{2/3}\right ) \log \left (8 \left (1+i \sqrt {3}\right )^{2/3} \sin ^2(x)+2\ 2^{2/3} \left (1+i \sqrt {3}+2^{2/3} \sqrt [3]{1+i \sqrt {3}}\right ) \sin (x)+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{4/3}-2 \left (1+i \sqrt {3}\right )^{2/3}+2\ 2^{2/3}\right )}{9 \left (2\ 2^{2/3}+2 \left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{4/3}\right )}-\frac {2}{9} \log (1-2 \sin (x))-\frac {1}{2} \log (1-\sin (x))+\frac {1}{18} \log (\sin (x)+1)+\frac {4 \sqrt [3]{1+i \sqrt {3}} \left (2 \sqrt [3]{2}+\sqrt [3]{1+i \sqrt {3}}+\left (2+2 i \sqrt {3}\right )^{2/3}\right ) \log \left (-4 \sqrt [3]{1+i \sqrt {3}} \sin (x)+\left (2+2 i \sqrt {3}\right )^{2/3}+2 \sqrt [3]{2}\right )}{9 \left (2\ 2^{2/3}+2 \left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{4/3}\right )}\) |
Input:
Int[(Cos[5*x] + Sin[4*x])^(-1),x]
Output:
(-2*2^(2/3)*(I - Sqrt[3])*(2^(2/3) - (1 + I*Sqrt[3])^(2/3))*ArcTanh[(2^(2/ 3)*(1 + I*Sqrt[3]) + 2*(2 + (2*I)*Sqrt[3])^(1/3) + 8*(1 + I*Sqrt[3])^(2/3) *Sin[x])/(2*Sqrt[3*(2^(1/3) - I*2^(1/3)*Sqrt[3] + 2*(1 + I*Sqrt[3])^(4/3) - (2 + (2*I)*Sqrt[3])^(2/3))])])/(3*(2*2^(2/3) + 2*(1 + I*Sqrt[3])^(2/3) + 2^(1/3)*(1 + I*Sqrt[3])^(4/3))*Sqrt[2^(1/3) - I*2^(1/3)*Sqrt[3] + 2*(1 + I*Sqrt[3])^(4/3) - (2 + (2*I)*Sqrt[3])^(2/3)]) - (2*Log[1 - 2*Sin[x]])/9 - Log[1 - Sin[x]]/2 + Log[1 + Sin[x]]/18 + (4*(1 + I*Sqrt[3])^(1/3)*(2*2^(1 /3) + (1 + I*Sqrt[3])^(1/3) + (2 + (2*I)*Sqrt[3])^(2/3))*Log[2*2^(1/3) + ( 2 + (2*I)*Sqrt[3])^(2/3) - 4*(1 + I*Sqrt[3])^(1/3)*Sin[x]])/(9*(2*2^(2/3) + 2*(1 + I*Sqrt[3])^(2/3) + 2^(1/3)*(1 + I*Sqrt[3])^(4/3))) - (2*(1 + I*Sq rt[3])^(1/3)*(2*2^(1/3) + (1 + I*Sqrt[3])^(1/3) + (2 + (2*I)*Sqrt[3])^(2/3 ))*Log[2*2^(2/3) - 2*(1 + I*Sqrt[3])^(2/3) + 2^(1/3)*(1 + I*Sqrt[3])^(4/3) + 2*2^(2/3)*(1 + I*Sqrt[3] + 2^(2/3)*(1 + I*Sqrt[3])^(1/3))*Sin[x] + 8*(1 + I*Sqrt[3])^(2/3)*Sin[x]^2])/(9*(2*2^(2/3) + 2*(1 + I*Sqrt[3])^(2/3) + 2 ^(1/3)*(1 + I*Sqrt[3])^(4/3))) + (2*Log[-1 - 6*Sin[x] + 8*Sin[x]^3])/9
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[(cos[(n_.)*((c_.) + (d_.)*(x_))]*(b_.) + (a_.)*sin[(m_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[1/d Subst[Int[Simplify[TrigExpand[a*Sin[ m*ArcSin[x]] + b*Cos[n*ArcSin[x]]]]^p/Sqrt[1 - x^2], x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[m/2] && Inte gerQ[(n - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.95 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94
| method | result | size |
| default | \(-\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\frac {2 \ln \left (2 \sin \left (x \right )-1\right )}{9}+\frac {\ln \left (1+\sin \left (x \right )\right )}{18}+\frac {8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{3}-6 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R} +1\right ) \textit {\_R} \ln \left (\sin \left (x \right )-\textit {\_R} \right )}{4 \textit {\_R}^{2}-1}\right )}{9}\) | \(62\) |
| risch | \(-\ln \left ({\mathrm e}^{i x}-i\right )+\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{9}-\frac {2 \ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{9}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 \textit {\_Z}^{3}-486 \textit {\_Z}^{2}+8\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {9}{2} i \textit {\_R} +i\right ) {\mathrm e}^{i x}-1\right )\right )\) | \(81\) |
Input:
int(1/(cos(5*x)+sin(4*x)),x,method=_RETURNVERBOSE)
Output:
-1/2*ln(sin(x)-1)-2/9*ln(2*sin(x)-1)+1/18*ln(1+sin(x))+8/9*sum((_R+1)*_R/( 4*_R^2-1)*ln(sin(x)-_R),_R=RootOf(8*_Z^3-6*_Z-1))
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 608, normalized size of antiderivative = 19.00 \[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x)),x, algorithm="fricas")
Output:
1/324*(81*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/3) + 54*sqrt(-1/972*(81*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/ 729)^(1/3) - 36)^2 - 6*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) - 8/27*(I*sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/3) + 20/3) + 72)*log(9/2 *(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) + 2/9*(I*sqrt(3) + 1)/(4 /729*I*sqrt(3) + 4/729)^(1/3) + 3*sqrt(-1/972*(81*(4/729*I*sqrt(3) + 4/729 )^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/ 3) - 36)^2 - 6*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) - 8/27*(I* sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/3) + 20/3) - 8*sin(x)) + 1/324*( 81*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4 /729*I*sqrt(3) + 4/729)^(1/3) - 54*sqrt(-1/972*(81*(4/729*I*sqrt(3) + 4/72 9)^(1/3)*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1 /3) - 36)^2 - 6*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) - 8/27*(I *sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/3) + 20/3) + 72)*log(-9/2*(4/72 9*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) - 2/9*(I*sqrt(3) + 1)/(4/729*I *sqrt(3) + 4/729)^(1/3) + 3*sqrt(-1/972*(81*(4/729*I*sqrt(3) + 4/729)^(1/3 )*(-I*sqrt(3) + 1) + 4*(I*sqrt(3) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/3) - 3 6)^2 - 6*(4/729*I*sqrt(3) + 4/729)^(1/3)*(-I*sqrt(3) + 1) - 8/27*(I*sqrt(3 ) + 1)/(4/729*I*sqrt(3) + 4/729)^(1/3) + 20/3) + 8*sin(x)) - 1/162*(81*...
\[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=\int \frac {1}{\sin {\left (4 x \right )} + \cos {\left (5 x \right )}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(4*x)),x)
Output:
Integral(1/(sin(4*x) + cos(5*x)), x)
\[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=\int { \frac {1}{\cos \left (5 \, x\right ) + \sin \left (4 \, x\right )} \,d x } \] Input:
integrate(1/(cos(5*x)+sin(4*x)),x, algorithm="maxima")
Output:
4/9*integrate((cos(6*x)*cos(3*x) - cos(3*x)*cos(8/3*arctan2(sin(3*x), cos( 3*x))) - cos(3*x)*cos(4/3*arctan2(sin(3*x), cos(3*x))) + cos(3*x)*cos(2/3* arctan2(sin(3*x), cos(3*x))) + cos(7/3*arctan2(sin(3*x), cos(3*x)))*sin(3* x) - cos(5/3*arctan2(sin(3*x), cos(3*x)))*sin(3*x) - cos(1/3*arctan2(sin(3 *x), cos(3*x)))*sin(3*x) + sin(6*x)*sin(3*x) - sin(3*x)*sin(8/3*arctan2(si n(3*x), cos(3*x))) - cos(3*x)*sin(7/3*arctan2(sin(3*x), cos(3*x))) + cos(3 *x)*sin(5/3*arctan2(sin(3*x), cos(3*x))) - sin(3*x)*sin(4/3*arctan2(sin(3* x), cos(3*x))) + sin(3*x)*sin(2/3*arctan2(sin(3*x), cos(3*x))) + cos(3*x)* sin(1/3*arctan2(sin(3*x), cos(3*x))) - cos(3*x))/(2*(sin(3*x) + 1)*cos(6*x ) - cos(6*x)^2 - cos(3*x)^2 + 2*(cos(6*x) - cos(4/3*arctan2(sin(3*x), cos( 3*x))) + cos(2/3*arctan2(sin(3*x), cos(3*x))) - sin(3*x) - sin(7/3*arctan2 (sin(3*x), cos(3*x))) + sin(5/3*arctan2(sin(3*x), cos(3*x))) + sin(1/3*arc tan2(sin(3*x), cos(3*x))) - 1)*cos(8/3*arctan2(sin(3*x), cos(3*x))) - cos( 8/3*arctan2(sin(3*x), cos(3*x)))^2 - 2*(cos(3*x) - cos(5/3*arctan2(sin(3*x ), cos(3*x))) - cos(1/3*arctan2(sin(3*x), cos(3*x))) + sin(6*x) - sin(4/3* arctan2(sin(3*x), cos(3*x))) + sin(2/3*arctan2(sin(3*x), cos(3*x))))*cos(7 /3*arctan2(sin(3*x), cos(3*x))) - cos(7/3*arctan2(sin(3*x), cos(3*x)))^2 + 2*(cos(3*x) - cos(1/3*arctan2(sin(3*x), cos(3*x))) + sin(6*x) - sin(4/3*a rctan2(sin(3*x), cos(3*x))) + sin(2/3*arctan2(sin(3*x), cos(3*x))))*cos(5/ 3*arctan2(sin(3*x), cos(3*x))) - cos(5/3*arctan2(sin(3*x), cos(3*x)))^2...
Exception generated. \[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(cos(5*x)+sin(4*x)),x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 2*(- 1/4*ln(-sin(sageVARx)+1)+1/36*ln(sin(sageVARx)+1)-1/9*ln(abs(2*sin(sageVAR x)-1))+((-1/98304*rootof([[-3,0,46080,0,-113246208],[1,0,-18432,0,84934656 ,0,-86973087744]])+
Time = 23.30 (sec) , antiderivative size = 507, normalized size of antiderivative = 15.84 \[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx =\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(4*x)),x)
Output:
log(tan(x/2) + 1)/9 - log(tan(x/2) - 1) - (2*log(tan(x/2)^2 - 4*tan(x/2) + 1))/9 + symsum(log((562949953421312*(2928640*root(z^3 - (2*z^2)/3 + 8/729 , z, k) - 179088*cos(x) - 23296*sin(x) - 3744936*root(z^3 - (2*z^2)/3 + 8/ 729, z, k)*cos(x) + 5983744*root(z^3 - (2*z^2)/3 + 8/729, z, k)*sin(x) - 1 40095872*root(z^3 - (2*z^2)/3 + 8/729, z, k)^2 - 109095660*root(z^3 - (2*z ^2)/3 + 8/729, z, k)^3 + 11742411969*root(z^3 - (2*z^2)/3 + 8/729, z, k)^4 - 7810568694*root(z^3 - (2*z^2)/3 + 8/729, z, k)^5 - 244216507572*root(z^ 3 - (2*z^2)/3 + 8/729, z, k)^6 + 20246839218*root(z^3 - (2*z^2)/3 + 8/729, z, k)^7 + 530216028495*root(z^3 - (2*z^2)/3 + 8/729, z, k)^8 + 140962140* root(z^3 - (2*z^2)/3 + 8/729, z, k)^2*cos(x) + 132253884*root(z^3 - (2*z^2 )/3 + 8/729, z, k)^3*cos(x) - 12897031329*root(z^3 - (2*z^2)/3 + 8/729, z, k)^4*cos(x) + 10366524342*root(z^3 - (2*z^2)/3 + 8/729, z, k)^5*cos(x) + 275492873304*root(z^3 - (2*z^2)/3 + 8/729, z, k)^6*cos(x) - 42376042458*ro ot(z^3 - (2*z^2)/3 + 8/729, z, k)^7*cos(x) - 572478342579*root(z^3 - (2*z^ 2)/3 + 8/729, z, k)^8*cos(x) + 14119866*root(z^3 - (2*z^2)/3 + 8/729, z, k )^2*sin(x) - 732819960*root(z^3 - (2*z^2)/3 + 8/729, z, k)^3*sin(x) - 2398 555071*root(z^3 - (2*z^2)/3 + 8/729, z, k)^4*sin(x) + 19796177250*root(z^3 - (2*z^2)/3 + 8/729, z, k)^5*sin(x) + 98538963534*root(z^3 - (2*z^2)/3 + 8/729, z, k)^6*sin(x) + 25019179398*root(z^3 - (2*z^2)/3 + 8/729, z, k)^7* sin(x) - 337338020601*root(z^3 - (2*z^2)/3 + 8/729, z, k)^8*sin(x) + 30...
\[ \int \frac {1}{\cos (5 x)+\sin (4 x)} \, dx=\int \frac {1}{\cos \left (5 x \right )+\sin \left (4 x \right )}d x \] Input:
int(1/(cos(5*x)+sin(4*x)),x)
Output:
int(1/(cos(5*x) + sin(4*x)),x)