Integrand size = 9, antiderivative size = 44 \[ \int (\csc (x)-\sin (x))^4 \, dx=\frac {35 x}{8}+\frac {35 \cot (x)}{8}-\frac {35 \cot ^3(x)}{24}+\frac {7}{8} \cos ^2(x) \cot ^3(x)+\frac {1}{4} \cos ^4(x) \cot ^3(x) \]
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int (\csc (x)-\sin (x))^4 \, dx=\frac {35 x}{8}+\frac {10 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x)+\frac {3}{4} \sin (2 x)+\frac {1}{32} \sin (4 x) \]
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 4889, 253, 253, 264, 264, 216}
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.
\(\displaystyle \int (\csc (x)-\sin (x))^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (\csc (x)-\sin (x))^4dx\) |
\(\Big \downarrow \) 4889 |
\(\displaystyle \int \frac {\cot ^4(x)}{\left (\tan ^2(x)+1\right )^3}d\tan (x)\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7}{4} \int \frac {\cot ^4(x)}{\left (\tan ^2(x)+1\right )^2}d\tan (x)+\frac {\cot ^3(x)}{4 \left (\tan ^2(x)+1\right )^2}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{2} \int \frac {\cot ^4(x)}{\tan ^2(x)+1}d\tan (x)+\frac {\cot ^3(x)}{2 \left (\tan ^2(x)+1\right )}\right )+\frac {\cot ^3(x)}{4 \left (\tan ^2(x)+1\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{2} \left (-\int \frac {\cot ^2(x)}{\tan ^2(x)+1}d\tan (x)-\frac {1}{3} \cot ^3(x)\right )+\frac {\cot ^3(x)}{2 \left (\tan ^2(x)+1\right )}\right )+\frac {\cot ^3(x)}{4 \left (\tan ^2(x)+1\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{2} \left (\int \frac {1}{\tan ^2(x)+1}d\tan (x)-\frac {1}{3} \cot ^3(x)+\cot (x)\right )+\frac {\cot ^3(x)}{2 \left (\tan ^2(x)+1\right )}\right )+\frac {\cot ^3(x)}{4 \left (\tan ^2(x)+1\right )^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{2} \left (\arctan (\tan (x))-\frac {1}{3} \cot ^3(x)+\cot (x)\right )+\frac {\cot ^3(x)}{2 \left (\tan ^2(x)+1\right )}\right )+\frac {\cot ^3(x)}{4 \left (\tan ^2(x)+1\right )^2}\) |
Cot[x]^3/(4*(1 + Tan[x]^2)^2) + (7*((5*(ArcTan[Tan[x]] + Cot[x] - Cot[x]^3 /3))/2 + Cot[x]^3/(2*(1 + Tan[x]^2))))/4
3.4.3.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
Time = 2.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {35 x}{8}+2 \cos \left (x \right ) \sin \left (x \right )+4 \cot \left (x \right )+\left (-\frac {2}{3}-\frac {\csc \left (x \right )^{2}}{3}\right ) \cot \left (x \right )\) | \(39\) |
parts | \(-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {35 x}{8}+2 \cos \left (x \right ) \sin \left (x \right )+4 \cot \left (x \right )+\left (-\frac {2}{3}-\frac {\csc \left (x \right )^{2}}{3}\right ) \cot \left (x \right )\) | \(39\) |
parallelrisch | \(\frac {\csc \left (x \right )^{3} \left (2520 x \sin \left (x \right )-840 x \sin \left (3 x \right )+525 \cos \left (x \right )+3 \cos \left (7 x \right )+63 \cos \left (5 x \right )-847 \cos \left (3 x \right )\right )}{768}\) | \(42\) |
risch | \(\frac {35 x}{8}-\frac {i {\mathrm e}^{4 i x}}{64}-\frac {3 i {\mathrm e}^{2 i x}}{8}+\frac {3 i {\mathrm e}^{-2 i x}}{8}+\frac {i {\mathrm e}^{-4 i x}}{64}+\frac {4 i \left (6 \,{\mathrm e}^{4 i x}-9 \,{\mathrm e}^{2 i x}+5\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\) | \(65\) |
norman | \(\frac {-\frac {1}{24}+\frac {35 \tan \left (\frac {x}{2}\right )^{2}}{24}+\frac {63 \tan \left (\frac {x}{2}\right )^{4}}{8}+\frac {35 \tan \left (\frac {x}{2}\right )^{6}}{8}-\frac {35 \tan \left (\frac {x}{2}\right )^{8}}{8}-\frac {63 \tan \left (\frac {x}{2}\right )^{10}}{8}-\frac {35 \tan \left (\frac {x}{2}\right )^{12}}{24}+\frac {\tan \left (\frac {x}{2}\right )^{14}}{24}+\frac {35 x \tan \left (\frac {x}{2}\right )^{3}}{8}+\frac {35 x \tan \left (\frac {x}{2}\right )^{5}}{2}+\frac {105 x \tan \left (\frac {x}{2}\right )^{7}}{4}+\frac {35 x \tan \left (\frac {x}{2}\right )^{9}}{2}+\frac {35 x \tan \left (\frac {x}{2}\right )^{11}}{8}}{\tan \left (\frac {x}{2}\right )^{3} \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}\) | \(121\) |
-1/4*(sin(x)^3+3/2*sin(x))*cos(x)+35/8*x+2*cos(x)*sin(x)+4*cot(x)+(-2/3-1/ 3*csc(x)^2)*cot(x)
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int (\csc (x)-\sin (x))^4 \, dx=-\frac {6 \, \cos \left (x\right )^{7} + 21 \, \cos \left (x\right )^{5} - 140 \, \cos \left (x\right )^{3} - 105 \, {\left (x \cos \left (x\right )^{2} - x\right )} \sin \left (x\right ) + 105 \, \cos \left (x\right )}{24 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
-1/24*(6*cos(x)^7 + 21*cos(x)^5 - 140*cos(x)^3 - 105*(x*cos(x)^2 - x)*sin( x) + 105*cos(x))/((cos(x)^2 - 1)*sin(x))
Time = 3.45 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int (\csc (x)-\sin (x))^4 \, dx=\frac {35 x}{8} + 2 \sin {\left (x \right )} \cos {\left (x \right )} - \frac {\sin {\left (2 x \right )}}{4} + \frac {\sin {\left (4 x \right )}}{32} - \frac {\cot ^{3}{\left (x \right )}}{3} - \cot {\left (x \right )} + \frac {4 \cos {\left (x \right )}}{\sin {\left (x \right )}} \]
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int (\csc (x)-\sin (x))^4 \, dx=\frac {35}{8} \, x + \frac {4}{\tan \left (x\right )} - \frac {3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} + \frac {1}{32} \, \sin \left (4 \, x\right ) + \frac {3}{4} \, \sin \left (2 \, x\right ) \]
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int (\csc (x)-\sin (x))^4 \, dx=\frac {35}{8} \, x + \frac {11 \, \tan \left (x\right )^{3} + 13 \, \tan \left (x\right )}{8 \, {\left (\tan \left (x\right )^{2} + 1\right )}^{2}} + \frac {9 \, \tan \left (x\right )^{2} - 1}{3 \, \tan \left (x\right )^{3}} \]
Time = 29.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int (\csc (x)-\sin (x))^4 \, dx=\frac {\frac {{\cos \left (x\right )}^7}{4}+\frac {7\,{\cos \left (x\right )}^5}{8}-\frac {35\,{\cos \left (x\right )}^3}{6}+\frac {35\,\cos \left (x\right )}{8}}{\sin \left (x\right )-{\cos \left (x\right )}^2\,\sin \left (x\right )}-\frac {\frac {35\,x}{8}-\frac {35\,x\,{\cos \left (x\right )}^2}{8}}{{\cos \left (x\right )}^2-1} \]