Integrand size = 17, antiderivative size = 80 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {35 x}{8}+\frac {35 \cot (a+b x)}{8 b}-\frac {35 \cot ^3(a+b x)}{24 b}+\frac {7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac {\cos ^4(a+b x) \cot ^3(a+b x)}{4 b} \]
35/8*x+35/8*cot(b*x+a)/b-35/24*cot(b*x+a)^3/b+7/8*cos(b*x+a)^2*cot(b*x+a)^ 3/b+1/4*cos(b*x+a)^4*cot(b*x+a)^3/b
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.66 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {420 (a+b x)-32 \cot (a+b x) \left (-10+\csc ^2(a+b x)\right )+72 \sin (2 (a+b x))+3 \sin (4 (a+b x))}{96 b} \]
(420*(a + b*x) - 32*Cot[a + b*x]*(-10 + Csc[a + b*x]^2) + 72*Sin[2*(a + b* x)] + 3*Sin[4*(a + b*x)])/(96*b)
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 3071, 252, 252, 254, 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.
\(\displaystyle \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (a+b x+\frac {\pi }{2}\right )^4 \tan \left (a+b x+\frac {\pi }{2}\right )^4dx\) |
\(\Big \downarrow \) 3071 |
\(\displaystyle -\frac {\int \frac {\cot ^8(a+b x)}{\left (\cot ^2(a+b x)+1\right )^3}d\cot (a+b x)}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -\frac {\frac {7}{4} \int \frac {\cot ^6(a+b x)}{\left (\cot ^2(a+b x)+1\right )^2}d\cot (a+b x)-\frac {\cot ^7(a+b x)}{4 \left (\cot ^2(a+b x)+1\right )^2}}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -\frac {\frac {7}{4} \left (\frac {5}{2} \int \frac {\cot ^4(a+b x)}{\cot ^2(a+b x)+1}d\cot (a+b x)-\frac {\cot ^5(a+b x)}{2 \left (\cot ^2(a+b x)+1\right )}\right )-\frac {\cot ^7(a+b x)}{4 \left (\cot ^2(a+b x)+1\right )^2}}{b}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle -\frac {\frac {7}{4} \left (\frac {5}{2} \int \left (\cot ^2(a+b x)+\frac {1}{\cot ^2(a+b x)+1}-1\right )d\cot (a+b x)-\frac {\cot ^5(a+b x)}{2 \left (\cot ^2(a+b x)+1\right )}\right )-\frac {\cot ^7(a+b x)}{4 \left (\cot ^2(a+b x)+1\right )^2}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {7}{4} \left (\frac {5}{2} \left (\arctan (\cot (a+b x))+\frac {1}{3} \cot ^3(a+b x)-\cot (a+b x)\right )-\frac {\cot ^5(a+b x)}{2 \left (\cot ^2(a+b x)+1\right )}\right )-\frac {\cot ^7(a+b x)}{4 \left (\cot ^2(a+b x)+1\right )^2}}{b}\) |
-((-1/4*Cot[a + b*x]^7/(1 + Cot[a + b*x]^2)^2 + (7*(-1/2*Cot[a + b*x]^5/(1 + Cot[a + b*x]^2) + (5*(ArcTan[Cot[a + b*x]] - Cot[a + b*x] + Cot[a + b*x ]^3/3))/2))/4)/b)
3.2.58.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(\frac {\left (2520 b x \sin \left (b x +a \right )-840 b x \sin \left (3 b x +3 a \right )+525 \cos \left (b x +a \right )+63 \cos \left (5 b x +5 a \right )-847 \cos \left (3 b x +3 a \right )+3 \cos \left (7 b x +7 a \right )\right ) \left (\sec ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\csc ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{6144 b}\) | \(93\) |
derivativedivides | \(\frac {-\frac {\cos ^{9}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {2 \left (\cos ^{9}\left (b x +a \right )\right )}{\sin \left (b x +a \right )}+2 \left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )+\frac {35 b x}{8}+\frac {35 a}{8}}{b}\) | \(94\) |
default | \(\frac {-\frac {\cos ^{9}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {2 \left (\cos ^{9}\left (b x +a \right )\right )}{\sin \left (b x +a \right )}+2 \left (\cos ^{7}\left (b x +a \right )+\frac {7 \left (\cos ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \cos \left (b x +a \right )}{16}\right ) \sin \left (b x +a \right )+\frac {35 b x}{8}+\frac {35 a}{8}}{b}\) | \(94\) |
risch | \(\frac {35 x}{8}-\frac {i {\mathrm e}^{4 i \left (b x +a \right )}}{64 b}-\frac {3 i {\mathrm e}^{2 i \left (b x +a \right )}}{8 b}+\frac {3 i {\mathrm e}^{-2 i \left (b x +a \right )}}{8 b}+\frac {i {\mathrm e}^{-4 i \left (b x +a \right )}}{64 b}+\frac {4 i \left (6 \,{\mathrm e}^{4 i \left (b x +a \right )}-9 \,{\mathrm e}^{2 i \left (b x +a \right )}+5\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\) | \(108\) |
norman | \(\frac {-\frac {1}{24 b}+\frac {35 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {63 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}+\frac {35 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {35 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {63 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}-\frac {35 \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{24 b}+\frac {\tan ^{14}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}+\frac {35 x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {35 x \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+\frac {105 x \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {35 x \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}+\frac {35 x \left (\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}\) | \(216\) |
1/6144*(2520*b*x*sin(b*x+a)-840*b*x*sin(3*b*x+3*a)+525*cos(b*x+a)+63*cos(5 *b*x+5*a)-847*cos(3*b*x+3*a)+3*cos(7*b*x+7*a))*sec(1/2*b*x+1/2*a)^3*csc(1/ 2*b*x+1/2*a)^3/b
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=-\frac {6 \, \cos \left (b x + a\right )^{7} + 21 \, \cos \left (b x + a\right )^{5} - 140 \, \cos \left (b x + a\right )^{3} - 105 \, {\left (b x \cos \left (b x + a\right )^{2} - b x\right )} \sin \left (b x + a\right ) + 105 \, \cos \left (b x + a\right )}{24 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
-1/24*(6*cos(b*x + a)^7 + 21*cos(b*x + a)^5 - 140*cos(b*x + a)^3 - 105*(b* x*cos(b*x + a)^2 - b*x)*sin(b*x + a) + 105*cos(b*x + a))/((b*cos(b*x + a)^ 2 - b)*sin(b*x + a))
Time = 1.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.76 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\begin {cases} \frac {35 x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {35 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {35 x \cos ^{4}{\left (a + b x \right )}}{8} + \frac {35 \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} + \frac {175 \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{24 b} + \frac {7 \cos ^{5}{\left (a + b x \right )}}{3 b \sin {\left (a + b x \right )}} - \frac {\cos ^{7}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{8}{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text {otherwise} \end {cases} \]
Piecewise((35*x*sin(a + b*x)**4/8 + 35*x*sin(a + b*x)**2*cos(a + b*x)**2/4 + 35*x*cos(a + b*x)**4/8 + 35*sin(a + b*x)**3*cos(a + b*x)/(8*b) + 175*si n(a + b*x)*cos(a + b*x)**3/(24*b) + 7*cos(a + b*x)**5/(3*b*sin(a + b*x)) - cos(a + b*x)**7/(3*b*sin(a + b*x)**3), Ne(b, 0)), (x*cos(a)**8/sin(a)**4, True))
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {105 \, b x + 105 \, a + \frac {105 \, \tan \left (b x + a\right )^{6} + 175 \, \tan \left (b x + a\right )^{4} + 56 \, \tan \left (b x + a\right )^{2} - 8}{\tan \left (b x + a\right )^{7} + 2 \, \tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}}}{24 \, b} \]
1/24*(105*b*x + 105*a + (105*tan(b*x + a)^6 + 175*tan(b*x + a)^4 + 56*tan( b*x + a)^2 - 8)/(tan(b*x + a)^7 + 2*tan(b*x + a)^5 + tan(b*x + a)^3))/b
Time = 0.50 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {105 \, b x + 105 \, a + \frac {3 \, {\left (11 \, \tan \left (b x + a\right )^{3} + 13 \, \tan \left (b x + a\right )\right )}}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{2}} + \frac {8 \, {\left (9 \, \tan \left (b x + a\right )^{2} - 1\right )}}{\tan \left (b x + a\right )^{3}}}{24 \, b} \]
1/24*(105*b*x + 105*a + 3*(11*tan(b*x + a)^3 + 13*tan(b*x + a))/(tan(b*x + a)^2 + 1)^2 + 8*(9*tan(b*x + a)^2 - 1)/tan(b*x + a)^3)/b
Time = 1.46 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70 \[ \int \cos ^4(a+b x) \cot ^4(a+b x) \, dx=\frac {35\,x}{8}+\frac {{\cos \left (a+b\,x\right )}^4\,\left (\frac {35\,{\mathrm {tan}\left (a+b\,x\right )}^6}{8}+\frac {175\,{\mathrm {tan}\left (a+b\,x\right )}^4}{24}+\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^2}{3}-\frac {1}{3}\right )}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]