\(\int \cos ^7(a+b x) \sin ^4(a+b x) \, dx\) [80]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {\sin ^5(a+b x)}{5 b}-\frac {3 \sin ^7(a+b x)}{7 b}+\frac {\sin ^9(a+b x)}{3 b}-\frac {\sin ^{11}(a+b x)}{11 b} \]

[Out]

1/5*sin(b*x+a)^5/b-3/7*sin(b*x+a)^7/b+1/3*sin(b*x+a)^9/b-1/11*sin(b*x+a)^11/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2644, 276} \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=-\frac {\sin ^{11}(a+b x)}{11 b}+\frac {\sin ^9(a+b x)}{3 b}-\frac {3 \sin ^7(a+b x)}{7 b}+\frac {\sin ^5(a+b x)}{5 b} \]

[In]

Int[Cos[a + b*x]^7*Sin[a + b*x]^4,x]

[Out]

Sin[a + b*x]^5/(5*b) - (3*Sin[a + b*x]^7)/(7*b) + Sin[a + b*x]^9/(3*b) - Sin[a + b*x]^11/(11*b)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\sin ^5(a+b x)}{5 b}-\frac {3 \sin ^7(a+b x)}{7 b}+\frac {\sin ^9(a+b x)}{3 b}-\frac {\sin ^{11}(a+b x)}{11 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {(3042+3335 \cos (2 (a+b x))+910 \cos (4 (a+b x))+105 \cos (6 (a+b x))) \sin ^5(a+b x)}{36960 b} \]

[In]

Integrate[Cos[a + b*x]^7*Sin[a + b*x]^4,x]

[Out]

((3042 + 3335*Cos[2*(a + b*x)] + 910*Cos[4*(a + b*x)] + 105*Cos[6*(a + b*x)])*Sin[a + b*x]^5)/(36960*b)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(-\frac {\frac {\left (\sin ^{11}\left (b x +a \right )\right )}{11}-\frac {\left (\sin ^{9}\left (b x +a \right )\right )}{3}+\frac {3 \left (\sin ^{7}\left (b x +a \right )\right )}{7}-\frac {\left (\sin ^{5}\left (b x +a \right )\right )}{5}}{b}\)	\(47\)
default	\(-\frac {\frac {\left (\sin ^{11}\left (b x +a \right )\right )}{11}-\frac {\left (\sin ^{9}\left (b x +a \right )\right )}{3}+\frac {3 \left (\sin ^{7}\left (b x +a \right )\right )}{7}-\frac {\left (\sin ^{5}\left (b x +a \right )\right )}{5}}{b}\)	\(47\)
risch	\(\frac {7 \sin \left (b x +a \right )}{512 b}+\frac {\sin \left (11 b x +11 a \right )}{11264 b}+\frac {\sin \left (9 b x +9 a \right )}{3072 b}-\frac {\sin \left (7 b x +7 a \right )}{7168 b}-\frac {11 \sin \left (5 b x +5 a \right )}{5120 b}-\frac {\sin \left (3 b x +3 a \right )}{512 b}\)	\(83\)
parallelrisch	\(\frac {\left (\sin \left (\frac {5 b x}{2}+\frac {5 a}{2}\right )-5 \sin \left (\frac {3 b x}{2}+\frac {3 a}{2}\right )+10 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (105 \cos \left (6 b x +6 a \right )+3335 \cos \left (2 b x +2 a \right )+910 \cos \left (4 b x +4 a \right )+3042\right ) \left (\cos \left (\frac {5 b x}{2}+\frac {5 a}{2}\right )+5 \cos \left (\frac {3 b x}{2}+\frac {3 a}{2}\right )+10 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{295680 b}\)	\(105\)

[In]

int(cos(b*x+a)^7*sin(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

-1/b*(1/11*sin(b*x+a)^11-1/3*sin(b*x+a)^9+3/7*sin(b*x+a)^7-1/5*sin(b*x+a)^5)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {{\left (105 \, \cos \left (b x + a\right )^{10} - 140 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{1155 \, b} \]

[In]

integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

1/1155*(105*cos(b*x + a)^10 - 140*cos(b*x + a)^8 + 5*cos(b*x + a)^6 + 6*cos(b*x + a)^4 + 8*cos(b*x + a)^2 + 16
)*sin(b*x + a)/b

Sympy [A] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\begin {cases} \frac {16 \sin ^{11}{\left (a + b x \right )}}{1155 b} + \frac {8 \sin ^{9}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{105 b} + \frac {6 \sin ^{7}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{35 b} + \frac {\sin ^{5}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{7}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(b*x+a)**7*sin(b*x+a)**4,x)

[Out]

Piecewise((16*sin(a + b*x)**11/(1155*b) + 8*sin(a + b*x)**9*cos(a + b*x)**2/(105*b) + 6*sin(a + b*x)**7*cos(a
+ b*x)**4/(35*b) + sin(a + b*x)**5*cos(a + b*x)**6/(5*b), Ne(b, 0)), (x*sin(a)**4*cos(a)**7, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=-\frac {105 \, \sin \left (b x + a\right )^{11} - 385 \, \sin \left (b x + a\right )^{9} + 495 \, \sin \left (b x + a\right )^{7} - 231 \, \sin \left (b x + a\right )^{5}}{1155 \, b} \]

[In]

integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/1155*(105*sin(b*x + a)^11 - 385*sin(b*x + a)^9 + 495*sin(b*x + a)^7 - 231*sin(b*x + a)^5)/b

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.34 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {\sin \left (11 \, b x + 11 \, a\right )}{11264 \, b} + \frac {\sin \left (9 \, b x + 9 \, a\right )}{3072 \, b} - \frac {\sin \left (7 \, b x + 7 \, a\right )}{7168 \, b} - \frac {11 \, \sin \left (5 \, b x + 5 \, a\right )}{5120 \, b} - \frac {\sin \left (3 \, b x + 3 \, a\right )}{512 \, b} + \frac {7 \, \sin \left (b x + a\right )}{512 \, b} \]

[In]

integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="giac")

[Out]

1/11264*sin(11*b*x + 11*a)/b + 1/3072*sin(9*b*x + 9*a)/b - 1/7168*sin(7*b*x + 7*a)/b - 11/5120*sin(5*b*x + 5*a
)/b - 1/512*sin(3*b*x + 3*a)/b + 7/512*sin(b*x + a)/b

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {-\frac {{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {{\sin \left (a+b\,x\right )}^9}{3}-\frac {3\,{\sin \left (a+b\,x\right )}^7}{7}+\frac {{\sin \left (a+b\,x\right )}^5}{5}}{b} \]

[In]

int(cos(a + b*x)^7*sin(a + b*x)^4,x)

[Out]

(sin(a + b*x)^5/5 - (3*sin(a + b*x)^7)/7 + sin(a + b*x)^9/3 - sin(a + b*x)^11/11)/b