Integrand size = 17, antiderivative size = 111 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {3 x}{256}+\frac {3 \cos (a+b x) \sin (a+b x)}{256 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac {3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac {\cos ^7(a+b x) \sin ^3(a+b x)}{10 b} \] Output:
3/256*x+3/256*cos(b*x+a)*sin(b*x+a)/b+1/128*cos(b*x+a)^3*sin(b*x+a)/b+1/16 0*cos(b*x+a)^5*sin(b*x+a)/b-3/80*cos(b*x+a)^7*sin(b*x+a)/b-1/10*cos(b*x+a) ^7*sin(b*x+a)^3/b
Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.56 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {120 b x+20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))}{10240 b} \] Input:
Integrate[Cos[a + b*x]^6*Sin[a + b*x]^4,x]
Output:
(120*b*x + 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] - 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] + 2*Sin[10*(a + b*x)])/(10240*b)
Time = 0.93 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {3042, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
\(\displaystyle \int \sin ^4(a+b x) \cos ^6(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^4 \cos (a+b x)^6dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{10} \int \cos ^6(a+b x) \sin ^2(a+b x)dx-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \int \cos (a+b x)^6 \sin (a+b x)^2dx-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \int \cos ^6(a+b x)dx-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \int \sin \left (a+b x+\frac {\pi }{2}\right )^6dx-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(a+b x)dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\) |
Input:
Int[Cos[a + b*x]^6*Sin[a + b*x]^4,x]
Output:
-1/10*(Cos[a + b*x]^7*Sin[a + b*x]^3)/b + (3*(-1/8*(Cos[a + b*x]^7*Sin[a + b*x])/b + ((Cos[a + b*x]^5*Sin[a + b*x])/(6*b) + (5*((Cos[a + b*x]^3*Sin[ a + b*x])/(4*b) + (3*(x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b)))/4))/6)/8)) /10
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Time = 21.42 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.59
method | result | size |
parallelrisch | \(\frac {120 b x +2 \sin \left (10 b x +10 a \right )+5 \sin \left (8 b x +8 a \right )-10 \sin \left (6 b x +6 a \right )-40 \sin \left (4 b x +4 a \right )+20 \sin \left (2 b x +2 a \right )}{10240 b}\) | \(66\) |
risch | \(\frac {3 x}{256}+\frac {\sin \left (10 b x +10 a \right )}{5120 b}+\frac {\sin \left (8 b x +8 a \right )}{2048 b}-\frac {\sin \left (6 b x +6 a \right )}{1024 b}-\frac {\sin \left (4 b x +4 a \right )}{256 b}+\frac {\sin \left (2 b x +2 a \right )}{512 b}\) | \(75\) |
derivativedivides | \(\frac {-\frac {\sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{7}}{10}-\frac {3 \sin \left (b x +a \right ) \cos \left (b x +a \right )^{7}}{80}+\frac {\left (\cos \left (b x +a \right )^{5}+\frac {5 \cos \left (b x +a \right )^{3}}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{160}+\frac {3 b x}{256}+\frac {3 a}{256}}{b}\) | \(82\) |
default | \(\frac {-\frac {\sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{7}}{10}-\frac {3 \sin \left (b x +a \right ) \cos \left (b x +a \right )^{7}}{80}+\frac {\left (\cos \left (b x +a \right )^{5}+\frac {5 \cos \left (b x +a \right )^{3}}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{160}+\frac {3 b x}{256}+\frac {3 a}{256}}{b}\) | \(82\) |
orering | \(\text {Expression too large to display}\) | \(998\) |
Input:
int(cos(b*x+a)^6*sin(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/10240*(120*b*x+2*sin(10*b*x+10*a)+5*sin(8*b*x+8*a)-10*sin(6*b*x+6*a)-40* sin(4*b*x+4*a)+20*sin(2*b*x+2*a))/b
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.59 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {15 \, b x + {\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{1280 \, b} \] Input:
integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="fricas")
Output:
1/1280*(15*b*x + (128*cos(b*x + a)^9 - 176*cos(b*x + a)^7 + 8*cos(b*x + a) ^5 + 10*cos(b*x + a)^3 + 15*cos(b*x + a))*sin(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (100) = 200\).
Time = 1.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.08 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\begin {cases} \frac {3 x \sin ^{10}{\left (a + b x \right )}}{256} + \frac {15 x \sin ^{8}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{256} + \frac {15 x \sin ^{6}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{128} + \frac {15 x \sin ^{4}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{128} + \frac {15 x \sin ^{2}{\left (a + b x \right )} \cos ^{8}{\left (a + b x \right )}}{256} + \frac {3 x \cos ^{10}{\left (a + b x \right )}}{256} + \frac {3 \sin ^{9}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{256 b} + \frac {7 \sin ^{7}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} + \frac {\sin ^{5}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{10 b} - \frac {7 \sin ^{3}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} - \frac {3 \sin {\left (a + b x \right )} \cos ^{9}{\left (a + b x \right )}}{256 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**6*sin(b*x+a)**4,x)
Output:
Piecewise((3*x*sin(a + b*x)**10/256 + 15*x*sin(a + b*x)**8*cos(a + b*x)**2 /256 + 15*x*sin(a + b*x)**6*cos(a + b*x)**4/128 + 15*x*sin(a + b*x)**4*cos (a + b*x)**6/128 + 15*x*sin(a + b*x)**2*cos(a + b*x)**8/256 + 3*x*cos(a + b*x)**10/256 + 3*sin(a + b*x)**9*cos(a + b*x)/(256*b) + 7*sin(a + b*x)**7* cos(a + b*x)**3/(128*b) + sin(a + b*x)**5*cos(a + b*x)**5/(10*b) - 7*sin(a + b*x)**3*cos(a + b*x)**7/(128*b) - 3*sin(a + b*x)*cos(a + b*x)**9/(256*b ), Ne(b, 0)), (x*sin(a)**4*cos(a)**6, True))
Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.43 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {32 \, \sin \left (2 \, b x + 2 \, a\right )^{5} + 120 \, b x + 120 \, a + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right )}{10240 \, b} \] Input:
integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="maxima")
Output:
1/10240*(32*sin(2*b*x + 2*a)^5 + 120*b*x + 120*a + 5*sin(8*b*x + 8*a) - 40 *sin(4*b*x + 4*a))/b
Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {3}{256} \, x + \frac {\sin \left (10 \, b x + 10 \, a\right )}{5120 \, b} + \frac {\sin \left (8 \, b x + 8 \, a\right )}{2048 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{1024 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{256 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{512 \, b} \] Input:
integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="giac")
Output:
3/256*x + 1/5120*sin(10*b*x + 10*a)/b + 1/2048*sin(8*b*x + 8*a)/b - 1/1024 *sin(6*b*x + 6*a)/b - 1/256*sin(4*b*x + 4*a)/b + 1/512*sin(2*b*x + 2*a)/b
Time = 26.86 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {3\,x}{256}+\frac {\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^9}{256}+\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^5}{10}-\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^3}{128}-\frac {3\,\mathrm {tan}\left (a+b\,x\right )}{256}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^{10}+5\,{\mathrm {tan}\left (a+b\,x\right )}^8+10\,{\mathrm {tan}\left (a+b\,x\right )}^6+10\,{\mathrm {tan}\left (a+b\,x\right )}^4+5\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \] Input:
int(cos(a + b*x)^6*sin(a + b*x)^4,x)
Output:
(3*x)/256 + (tan(a + b*x)^5/10 - (7*tan(a + b*x)^3)/128 - (3*tan(a + b*x)) /256 + (7*tan(a + b*x)^7)/128 + (3*tan(a + b*x)^9)/256)/(b*(5*tan(a + b*x) ^2 + 10*tan(a + b*x)^4 + 10*tan(a + b*x)^6 + 5*tan(a + b*x)^8 + tan(a + b* x)^10 + 1))
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {128 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{9}-336 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{7}+248 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{5}-10 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}-15 \cos \left (b x +a \right ) \sin \left (b x +a \right )+15 b x}{1280 b} \] Input:
int(cos(b*x+a)^6*sin(b*x+a)^4,x)
Output:
(128*cos(a + b*x)*sin(a + b*x)**9 - 336*cos(a + b*x)*sin(a + b*x)**7 + 248 *cos(a + b*x)*sin(a + b*x)**5 - 10*cos(a + b*x)*sin(a + b*x)**3 - 15*cos(a + b*x)*sin(a + b*x) + 15*b*x)/(1280*b)