\(\int \cos ^6(a+b x) \sin ^4(a+b x) \, dx\) [89]

Optimal result

Integrand size = 17, antiderivative size = 111 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {3 x}{256}+\frac {3 \cos (a+b x) \sin (a+b x)}{256 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{128 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{160 b}-\frac {3 \cos ^7(a+b x) \sin (a+b x)}{80 b}-\frac {\cos ^7(a+b x) \sin ^3(a+b x)}{10 b} \]

[Out]

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+1/160*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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2648, 2715, 8} \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}-\frac {3 \sin (a+b x) \cos ^7(a+b x)}{80 b}+\frac {\sin (a+b x) \cos ^5(a+b x)}{160 b}+\frac {\sin (a+b x) \cos ^3(a+b x)}{128 b}+\frac {3 \sin (a+b x) \cos (a+b x)}{256 b}+\frac {3 x}{256} \]

[In]

Int[Cos[a + b*x]^6*Sin[a + b*x]^4,x]

[Out]

(3*x)/256 + (3*Cos[a + b*x]*Sin[a + b*x])/(256*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^5*Si
n[a + b*x])/(160*b) - (3*Cos[a + b*x]^7*Sin[a + b*x])/(80*b) - (Cos[a + b*x]^7*Sin[a + b*x]^3)/(10*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2648

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] + Dist[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]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^7(a+b x) \sin ^3(a+b x)}{10 b}+\frac {3}{10} \int \cos ^6(a+b x) \sin ^2(a+b x) \, dx \\ & = -\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}+\frac {3}{80} \int \cos ^6(a+b x) \, dx \\ & = \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}+\frac {1}{32} \int \cos ^4(a+b x) \, dx \\ & = \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}+\frac {3}{128} \int \cos ^2(a+b x) \, dx \\ & = \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}+\frac {3 \int 1 \, dx}{256} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.56 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {120 b x+20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))}{10240 b} \]

[In]

Integrate[Cos[a + b*x]^6*Sin[a + b*x]^4,x]

[Out]

(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)

Maple [A] (verified)

Time = 0.33 (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	\(\frac {-\frac {\left (\sin ^{3}\left (b x +a \right )\right ) \left (\cos ^{7}\left (b x +a \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{80}+\frac {\left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{160}+\frac {3 b x}{256}+\frac {3 a}{256}}{b}\)	\(82\)
default	\(\frac {-\frac {\left (\sin ^{3}\left (b x +a \right )\right ) \left (\cos ^{7}\left (b x +a \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{80}+\frac {\left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{160}+\frac {3 b x}{256}+\frac {3 a}{256}}{b}\)	\(82\)

[In]

int(cos(b*x+a)^6*sin(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.59 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {15 \, b x + {\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{1280 \, b} \]

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (100) = 200\).

Time = 1.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.08 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\begin {cases} \frac {3 x \sin ^{10}{\left (a + b x \right )}}{256} + \frac {15 x \sin ^{8}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{256} + \frac {15 x \sin ^{6}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{128} + \frac {15 x \sin ^{4}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{128} + \frac {15 x \sin ^{2}{\left (a + b x \right )} \cos ^{8}{\left (a + b x \right )}}{256} + \frac {3 x \cos ^{10}{\left (a + b x \right )}}{256} + \frac {3 \sin ^{9}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{256 b} + \frac {7 \sin ^{7}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} + \frac {\sin ^{5}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{10 b} - \frac {7 \sin ^{3}{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} - \frac {3 \sin {\left (a + b x \right )} \cos ^{9}{\left (a + b x \right )}}{256 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(b*x+a)**6*sin(b*x+a)**4,x)

[Out]

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/(2
56*b), Ne(b, 0)), (x*sin(a)**4*cos(a)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.43 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {32 \, \sin \left (2 \, b x + 2 \, a\right )^{5} + 120 \, b x + 120 \, a + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right )}{10240 \, b} \]

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {3}{256} \, x + \frac {\sin \left (10 \, b x + 10 \, a\right )}{5120 \, b} + \frac {\sin \left (8 \, b x + 8 \, a\right )}{2048 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{1024 \, b} - \frac {\sin \left (4 \, b x + 4 \, a\right )}{256 \, b} + \frac {\sin \left (2 \, b x + 2 \, a\right )}{512 \, b} \]

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.76 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98 \[ \int \cos ^6(a+b x) \sin ^4(a+b x) \, dx=\frac {3\,x}{256}+\frac {\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^9}{256}+\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^5}{10}-\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^3}{128}-\frac {3\,\mathrm {tan}\left (a+b\,x\right )}{256}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^{10}+5\,{\mathrm {tan}\left (a+b\,x\right )}^8+10\,{\mathrm {tan}\left (a+b\,x\right )}^6+10\,{\mathrm {tan}\left (a+b\,x\right )}^4+5\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]

[In]

int(cos(a + b*x)^6*sin(a + b*x)^4,x)

[Out]

(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*t
an(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))