Integrand size = 9, antiderivative size = 26 \[ \int \csc (4 x) \sin ^3(x) \, dx=-\frac {1}{4} \text {arctanh}(\sin (x))+\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{4 \sqrt {2}} \]
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(26)=52\).
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \csc (4 x) \sin ^3(x) \, dx=\frac {1}{16} \left (4 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-4 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\sqrt {2} \left (-\log \left (\sqrt {2}-2 \sin (x)\right )+\log \left (\sqrt {2}+2 \sin (x)\right )\right )\right ) \]
(4*Log[Cos[x/2] - Sin[x/2]] - 4*Log[Cos[x/2] + Sin[x/2]] + Sqrt[2]*(-Log[S qrt[2] - 2*Sin[x]] + Log[Sqrt[2] + 2*Sin[x]]))/16
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4878, 27, 1450, 220}
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.
\(\displaystyle \int \sin ^3(x) \csc (4 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)^3}{\sin (4 x)}dx\) |
\(\Big \downarrow \) 4878 |
\(\displaystyle \int \frac {\sin ^2(x)}{4 \left (2 \sin ^4(x)-3 \sin ^2(x)+1\right )}d\sin (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {\sin ^2(x)}{2 \sin ^4(x)-3 \sin ^2(x)+1}d\sin (x)\) |
\(\Big \downarrow \) 1450 |
\(\displaystyle \frac {1}{4} \left (2 \int \frac {1}{2 \sin ^2(x)-2}d\sin (x)-\int \frac {1}{2 \sin ^2(x)-1}d\sin (x)\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{4} \left (\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}}-\text {arctanh}(\sin (x))\right )\) |
3.4.89.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1) Int[(d*x)^(m - 2)/(b/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Sin[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeF actors[Sin[v], x], u/Cos[v], x]]
Time = 0.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{8}+\frac {\ln \left (\sin \left (x \right )-1\right )}{8}-\frac {\ln \left (\sin \left (x \right )+1\right )}{8}\) | \(28\) |
risch | \(-\frac {\ln \left (i+{\mathrm e}^{i x}\right )}{4}+\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{4}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{16}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{16}\) | \(72\) |
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \csc (4 x) \sin ^3(x) \, dx=\frac {1}{16} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
1/16*sqrt(2)*log(-(2*cos(x)^2 - 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1)) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (22) = 44\).
Time = 7.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 11.31 \[ \int \csc (4 x) \sin ^3(x) \, dx=\frac {4093147632754948 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {2894292447518688 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {4093147632754948 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {2894292447518688 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} \]
4093147632754948*log(tan(x/2) - 1)/(16372590531019792 + 11577169790074752* sqrt(2)) + 2894292447518688*sqrt(2)*log(tan(x/2) - 1)/(16372590531019792 + 11577169790074752*sqrt(2)) - 4093147632754948*log(tan(x/2) + 1)/(16372590 531019792 + 11577169790074752*sqrt(2)) - 2894292447518688*sqrt(2)*log(tan( x/2) + 1)/(16372590531019792 + 11577169790074752*sqrt(2)) + 14471462237593 44*log(tan(x/2) - 1 + sqrt(2))/(16372590531019792 + 11577169790074752*sqrt (2)) + 1023286908188737*sqrt(2)*log(tan(x/2) - 1 + sqrt(2))/(1637259053101 9792 + 11577169790074752*sqrt(2)) + 1447146223759344*log(tan(x/2) + 1 + sq rt(2))/(16372590531019792 + 11577169790074752*sqrt(2)) + 1023286908188737* sqrt(2)*log(tan(x/2) + 1 + sqrt(2))/(16372590531019792 + 11577169790074752 *sqrt(2)) - 1447146223759344*log(tan(x/2) - sqrt(2) - 1)/(1637259053101979 2 + 11577169790074752*sqrt(2)) - 1023286908188737*sqrt(2)*log(tan(x/2) - s qrt(2) - 1)/(16372590531019792 + 11577169790074752*sqrt(2)) - 144714622375 9344*log(tan(x/2) - sqrt(2) + 1)/(16372590531019792 + 11577169790074752*sq rt(2)) - 1023286908188737*sqrt(2)*log(tan(x/2) - sqrt(2) + 1)/(16372590531 019792 + 11577169790074752*sqrt(2))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 6.58 \[ \int \csc (4 x) \sin ^3(x) \, dx=\frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) + \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \]
1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 2*sqrt(2)*si n(x) + 2) - 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) - 2*sqrt(2)*sin(x) + 2) + 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqrt( 2)*cos(x) + 2*sqrt(2)*sin(x) + 2) - 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x) ^2 - 2*sqrt(2)*cos(x) - 2*sqrt(2)*sin(x) + 2) - 1/8*log(cos(x)^2 + sin(x)^ 2 + 2*sin(x) + 1) + 1/8*log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \csc (4 x) \sin ^3(x) \, dx=-\frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
-1/16*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x))) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x) + 1)
Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \csc (4 x) \sin ^3(x) \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )}{8}-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2} \]
\[ \int \csc (4 x) \sin ^3(x) \, dx=\int \frac {\sin \left (x \right )^{3}}{\sin \left (4 x \right )}d x \]