Integrand size = 17, antiderivative size = 61 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {\sin ^5(a+b x)}{5 b}-\frac {3 \sin ^7(a+b x)}{7 b}+\frac {\sin ^9(a+b x)}{3 b}-\frac {\sin ^{11}(a+b x)}{11 b} \] Output:
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)^1 1/b
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {(3042+3335 \cos (2 (a+b x))+910 \cos (4 (a+b x))+105 \cos (6 (a+b x))) \sin ^5(a+b x)}{36960 b} \] Input:
Integrate[Cos[a + b*x]^7*Sin[a + b*x]^4,x]
Output:
((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)
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3042, 3044, 244, 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.
\(\displaystyle \int \sin ^4(a+b x) \cos ^7(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^4 \cos (a+b x)^7dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {\int \sin ^4(a+b x) \left (1-\sin ^2(a+b x)\right )^3d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \left (-\sin ^{10}(a+b x)+3 \sin ^8(a+b x)-3 \sin ^6(a+b x)+\sin ^4(a+b x)\right )d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{11} \sin ^{11}(a+b x)+\frac {1}{3} \sin ^9(a+b x)-\frac {3}{7} \sin ^7(a+b x)+\frac {1}{5} \sin ^5(a+b x)}{b}\) |
Input:
Int[Cos[a + b*x]^7*Sin[a + b*x]^4,x]
Output:
(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
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[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] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Time = 54.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {\frac {\sin \left (b x +a \right )^{11}}{11}-\frac {\sin \left (b x +a \right )^{9}}{3}+\frac {3 \sin \left (b x +a \right )^{7}}{7}-\frac {\sin \left (b x +a \right )^{5}}{5}}{b}\) | \(47\) |
default | \(-\frac {\frac {\sin \left (b x +a \right )^{11}}{11}-\frac {\sin \left (b x +a \right )^{9}}{3}+\frac {3 \sin \left (b x +a \right )^{7}}{7}-\frac {\sin \left (b x +a \right )^{5}}{5}}{b}\) | \(47\) |
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 | \(\frac {\left (\sin \left (\frac {5 b x}{2}+\frac {5 a}{2}\right )-5 \sin \left (\frac {3 b x}{2}+\frac {3 a}{2}\right )+10 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (3335 \cos \left (2 b x +2 a \right )+105 \cos \left (6 b x +6 a \right )+910 \cos \left (4 b x +4 a \right )+3042\right ) \left (\cos \left (\frac {5 b x}{2}+\frac {5 a}{2}\right )+5 \cos \left (\frac {3 b x}{2}+\frac {3 a}{2}\right )+10 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{295680 b}\) | \(105\) |
orering | \(-\frac {14312974 \left (-7 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{5} b +4 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{3} b \right )}{12006225 b^{2}}-\frac {1997021 \left (-210 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{7} b^{3}+721 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{5} b^{3}-376 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{3} b^{3}+24 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right ) b^{3}\right )}{9823275 b^{4}}-\frac {112268 \left (-2520 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{9} b^{5}+35700 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{7} b^{5}-81067 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{5} b^{5}+38764 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{3} b^{5}-3000 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right ) b^{5}\right )}{9823275 b^{6}}-\frac {871 \left (-5040 b^{7} \sin \left (b x +a \right )^{11}+546840 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{9} b^{7}-5005350 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{7} b^{7}+9425941 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{5} b^{7}-4178416 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{3} b^{7}+325584 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right ) b^{7}\right )}{3274425 b^{8}}-\frac {26 \left (-86320080 b^{9} \sin \left (b x +a \right )^{9} \cos \left (b x +a \right )^{2}+1149120 b^{9} \sin \left (b x +a \right )^{11}+657509160 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{7} b^{9}-1113457807 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{5} b^{9}+464347924 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{3} b^{9}-35163600 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right ) b^{9}\right )}{9823275 b^{10}}-\frac {-83671893690 b^{11} \sin \left (b x +a \right )^{7} \cos \left (b x +a \right )^{4}+12073556880 b^{11} \sin \left (b x +a \right )^{9} \cos \left (b x +a \right )^{2}-185280480 b^{11} \sin \left (b x +a \right )^{11}+132674372761 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{5} b^{11}-52830407656 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{3} b^{11}+3876159144 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right ) b^{11}}{108056025 b^{12}}\) | \(609\) |
Input:
int(cos(b*x+a)^7*sin(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
-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)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {{\left (105 \, \cos \left (b x + a\right )^{10} - 140 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{1155 \, b} \] Input:
integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="fricas")
Output:
1/1155*(105*cos(b*x + a)^10 - 140*cos(b*x + a)^8 + 5*cos(b*x + a)^6 + 6*co s(b*x + a)^4 + 8*cos(b*x + a)^2 + 16)*sin(b*x + a)/b
Time = 1.76 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\begin {cases} \frac {16 \sin ^{11}{\left (a + b x \right )}}{1155 b} + \frac {8 \sin ^{9}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{105 b} + \frac {6 \sin ^{7}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{35 b} + \frac {\sin ^{5}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \cos ^{7}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**7*sin(b*x+a)**4,x)
Output:
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))
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=-\frac {105 \, \sin \left (b x + a\right )^{11} - 385 \, \sin \left (b x + a\right )^{9} + 495 \, \sin \left (b x + a\right )^{7} - 231 \, \sin \left (b x + a\right )^{5}}{1155 \, b} \] Input:
integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="maxima")
Output:
-1/1155*(105*sin(b*x + a)^11 - 385*sin(b*x + a)^9 + 495*sin(b*x + a)^7 - 2 31*sin(b*x + a)^5)/b
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=-\frac {105 \, \sin \left (b x + a\right )^{11} - 385 \, \sin \left (b x + a\right )^{9} + 495 \, \sin \left (b x + a\right )^{7} - 231 \, \sin \left (b x + a\right )^{5}}{1155 \, b} \] Input:
integrate(cos(b*x+a)^7*sin(b*x+a)^4,x, algorithm="giac")
Output:
-1/1155*(105*sin(b*x + a)^11 - 385*sin(b*x + a)^9 + 495*sin(b*x + a)^7 - 2 31*sin(b*x + a)^5)/b
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {-\frac {{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {{\sin \left (a+b\,x\right )}^9}{3}-\frac {3\,{\sin \left (a+b\,x\right )}^7}{7}+\frac {{\sin \left (a+b\,x\right )}^5}{5}}{b} \] Input:
int(cos(a + b*x)^7*sin(a + b*x)^4,x)
Output:
(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
Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx=\frac {\sin \left (b x +a \right )^{5} \left (-105 \sin \left (b x +a \right )^{6}+385 \sin \left (b x +a \right )^{4}-495 \sin \left (b x +a \right )^{2}+231\right )}{1155 b} \] Input:
int(cos(b*x+a)^7*sin(b*x+a)^4,x)
Output:
(sin(a + b*x)**5*( - 105*sin(a + b*x)**6 + 385*sin(a + b*x)**4 - 495*sin(a + b*x)**2 + 231))/(1155*b)