Integrand size = 38, antiderivative size = 239 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\frac {(A+B) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-4-p}}{f g (7+p)}+\frac {(3 A-B (4+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-3-p}}{c f g (5+p) (7+p)}+\frac {2 (3 A-B (4+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-2-p}}{c^2 f g (3+p) (5+p) (7+p)}+\frac {2 (3 A-B (4+p)) (g \cos (e+f x))^{1+p} (c-c \sin (e+f x))^{-1-p}}{c^3 f g (1+p) (3+p) (5+p) (7+p)} \] Output:
(A+B)*(g*cos(f*x+e))^(p+1)*(c-c*sin(f*x+e))^(-4-p)/f/g/(7+p)+(3*A-B*(4+p)) *(g*cos(f*x+e))^(p+1)*(c-c*sin(f*x+e))^(-3-p)/c/f/g/(5+p)/(7+p)+2*(3*A-B*( 4+p))*(g*cos(f*x+e))^(p+1)*(c-c*sin(f*x+e))^(-2-p)/c^2/f/g/(3+p)/(5+p)/(7+ p)+2*(3*A-B*(4+p))*(g*cos(f*x+e))^(p+1)*(c-c*sin(f*x+e))^(-1-p)/c^3/f/g/(p +1)/(3+p)/(5+p)/(7+p)
Time = 0.60 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.67 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\frac {\cos (e+f x) (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p} \left (-B \left (13+8 p+p^2\right )+A \left (36+41 p+12 p^2+p^3\right )+\left (13+8 p+p^2\right ) (-3 A+B (4+p)) \sin (e+f x)-2 (4+p) (-3 A+B (4+p)) \sin ^2(e+f x)+(-6 A+2 B (4+p)) \sin ^3(e+f x)\right )}{c^4 f (1+p) (3+p) (5+p) (7+p) (-1+\sin (e+f x))^4} \] Input:
Integrate[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - p),x]
Output:
(Cos[e + f*x]*(g*Cos[e + f*x])^p*(-(B*(13 + 8*p + p^2)) + A*(36 + 41*p + 1 2*p^2 + p^3) + (13 + 8*p + p^2)*(-3*A + B*(4 + p))*Sin[e + f*x] - 2*(4 + p )*(-3*A + B*(4 + p))*Sin[e + f*x]^2 + (-6*A + 2*B*(4 + p))*Sin[e + f*x]^3) )/(c^4*f*(1 + p)*(3 + p)*(5 + p)*(7 + p)*(-1 + Sin[e + f*x])^4*(c - c*Sin[ e + f*x])^p)
Time = 0.92 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.87, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3338, 3042, 3151, 3042, 3151, 3042, 3150}
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.
| \(\displaystyle \int (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^p \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^pdx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle \frac {(3 A-B (p+4)) \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p-3}dx}{c (p+7)}+\frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(3 A-B (p+4)) \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p-3}dx}{c (p+7)}+\frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {(3 A-B (p+4)) \left (\frac {2 \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p-2}dx}{c (p+5)}+\frac {(c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}\right )}{c (p+7)}+\frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(3 A-B (p+4)) \left (\frac {2 \int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p-2}dx}{c (p+5)}+\frac {(c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}\right )}{c (p+7)}+\frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {(3 A-B (p+4)) \left (\frac {2 \left (\frac {\int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p-1}dx}{c (p+3)}+\frac {(c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}\right )}{c (p+5)}+\frac {(c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}\right )}{c (p+7)}+\frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(3 A-B (p+4)) \left (\frac {2 \left (\frac {\int (g \cos (e+f x))^p (c-c \sin (e+f x))^{-p-1}dx}{c (p+3)}+\frac {(c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}\right )}{c (p+5)}+\frac {(c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}\right )}{c (p+7)}+\frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle \frac {(A+B) (c-c \sin (e+f x))^{-p-4} (g \cos (e+f x))^{p+1}}{f g (p+7)}+\frac {(3 A-B (p+4)) \left (\frac {(c-c \sin (e+f x))^{-p-3} (g \cos (e+f x))^{p+1}}{f g (p+5)}+\frac {2 \left (\frac {(c-c \sin (e+f x))^{-p-2} (g \cos (e+f x))^{p+1}}{f g (p+3)}+\frac {(c-c \sin (e+f x))^{-p-1} (g \cos (e+f x))^{p+1}}{c f g (p+1) (p+3)}\right )}{c (p+5)}\right )}{c (p+7)}\) |
Input:
Int[(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - p), x]
Output:
((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-4 - p))/(f*g*(7 + p)) + ((3*A - B*(4 + p))*(((g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^ (-3 - p))/(f*g*(5 + p)) + (2*(((g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x ])^(-2 - p))/(f*g*(3 + p)) + ((g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x] )^(-1 - p))/(c*f*g*(1 + p)*(3 + p))))/(c*(5 + p))))/(c*(7 + p))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{-4-p}d x\]
Input:
int((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-p),x)
Output:
int((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-p),x)
Time = 0.10 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.82 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\frac {{\left (2 \, {\left (B p^{2} - {\left (3 \, A - 8 \, B\right )} p - 12 \, A + 16 \, B\right )} \cos \left (f x + e\right )^{3} + {\left (A p^{3} + 3 \, {\left (4 \, A - B\right )} p^{2} + {\left (47 \, A - 24 \, B\right )} p + 60 \, A - 45 \, B\right )} \cos \left (f x + e\right ) - {\left (2 \, {\left (B p - 3 \, A + 4 \, B\right )} \cos \left (f x + e\right )^{3} - {\left (B p^{3} - 3 \, {\left (A - 4 \, B\right )} p^{2} - {\left (24 \, A - 47 \, B\right )} p - 45 \, A + 60 \, B\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 4}}{f p^{4} + 16 \, f p^{3} + 86 \, f p^{2} + 176 \, f p + 105 \, f} \] Input:
integrate((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-p),x, alg orithm="fricas")
Output:
(2*(B*p^2 - (3*A - 8*B)*p - 12*A + 16*B)*cos(f*x + e)^3 + (A*p^3 + 3*(4*A - B)*p^2 + (47*A - 24*B)*p + 60*A - 45*B)*cos(f*x + e) - (2*(B*p - 3*A + 4 *B)*cos(f*x + e)^3 - (B*p^3 - 3*(A - 4*B)*p^2 - (24*A - 47*B)*p - 45*A + 6 0*B)*cos(f*x + e))*sin(f*x + e))*(g*cos(f*x + e))^p*(-c*sin(f*x + e) + c)^ (-p - 4)/(f*p^4 + 16*f*p^3 + 86*f*p^2 + 176*f*p + 105*f)
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\int \left (g \cos {\left (e + f x \right )}\right )^{p} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{- p - 4} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \] Input:
integrate((g*cos(f*x+e))**p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(-4-p),x)
Output:
Integral((g*cos(e + f*x))**p*(-c*(sin(e + f*x) - 1))**(-p - 4)*(A + B*sin( e + f*x)), x)
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 4} \,d x } \] Input:
integrate((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-p),x, alg orithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(-c*sin(f*x + e) + c)^(- p - 4), x)
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-p - 4} \,d x } \] Input:
integrate((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-p),x, alg orithm="giac")
Output:
integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^p*(-c*sin(f*x + e) + c)^(- p - 4), x)
Time = 44.79 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.85 \[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (g\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}\right )\right )}^p\,\left (A\,168{}\mathrm {i}-B\,84{}\mathrm {i}+A\,p\,170{}\mathrm {i}-B\,p\,48{}\mathrm {i}+A\,p^2\,48{}\mathrm {i}+A\,p^3\,4{}\mathrm {i}-B\,p^2\,6{}\mathrm {i}\right )}{4\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{p+4}\,\left (p^4\,1{}\mathrm {i}+p^3\,16{}\mathrm {i}+p^2\,86{}\mathrm {i}+p\,176{}\mathrm {i}+105{}\mathrm {i}\right )}-\frac {\sin \left (4\,e+4\,f\,x\right )\,{\left (g\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}\right )\right )}^p\,\left (4\,B-3\,A+B\,p\right )\,1{}\mathrm {i}}{4\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{p+4}\,\left (p^4\,1{}\mathrm {i}+p^3\,16{}\mathrm {i}+p^2\,86{}\mathrm {i}+p\,176{}\mathrm {i}+105{}\mathrm {i}\right )}+\frac {\cos \left (3\,e+3\,f\,x\right )\,{\left (g\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}\right )\right )}^p\,\left (p+4\right )\,\left (-A\,3{}\mathrm {i}+B\,4{}\mathrm {i}+B\,p\,1{}\mathrm {i}\right )}{2\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{p+4}\,\left (p^4\,1{}\mathrm {i}+p^3\,16{}\mathrm {i}+p^2\,86{}\mathrm {i}+p\,176{}\mathrm {i}+105{}\mathrm {i}\right )}+\frac {\sin \left (2\,e+2\,f\,x\right )\,{\left (g\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}\right )\right )}^p\,\left (4\,B-3\,A+B\,p\right )\,\left (p^2+8\,p+14\right )\,1{}\mathrm {i}}{2\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{p+4}\,\left (p^4\,1{}\mathrm {i}+p^3\,16{}\mathrm {i}+p^2\,86{}\mathrm {i}+p\,176{}\mathrm {i}+105{}\mathrm {i}\right )} \] Input:
int(((g*cos(e + f*x))^p*(A + B*sin(e + f*x)))/(c - c*sin(e + f*x))^(p + 4) ,x)
Output:
(cos(e + f*x)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^p*(A*168 i - B*84i + A*p*170i - B*p*48i + A*p^2*48i + A*p^3*4i - B*p^2*6i))/(4*f*(c - c*sin(e + f*x))^(p + 4)*(p*176i + p^2*86i + p^3*16i + p^4*1i + 105i)) - (sin(4*e + 4*f*x)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^p*( 4*B - 3*A + B*p)*1i)/(4*f*(c - c*sin(e + f*x))^(p + 4)*(p*176i + p^2*86i + p^3*16i + p^4*1i + 105i)) + (cos(3*e + 3*f*x)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^p*(p + 4)*(B*4i - A*3i + B*p*1i))/(2*f*(c - c*sin (e + f*x))^(p + 4)*(p*176i + p^2*86i + p^3*16i + p^4*1i + 105i)) + (sin(2* e + 2*f*x)*(g*(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^p*(4*B - 3* A + B*p)*(8*p + p^2 + 14)*1i)/(2*f*(c - c*sin(e + f*x))^(p + 4)*(p*176i + p^2*86i + p^3*16i + p^4*1i + 105i))
\[ \int (g \cos (e+f x))^p (A+B \sin (e+f x)) (c-c \sin (e+f x))^{-4-p} \, dx=\frac {g^{p} \left (\left (\int \frac {\cos \left (f x +e \right )^{p}}{\left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )^{4}-4 \left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )^{3}+6 \left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )^{2}-4 \left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )+\left (-\sin \left (f x +e \right ) c +c \right )^{p}}d x \right ) a +\left (\int \frac {\cos \left (f x +e \right )^{p} \sin \left (f x +e \right )}{\left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )^{4}-4 \left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )^{3}+6 \left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )^{2}-4 \left (-\sin \left (f x +e \right ) c +c \right )^{p} \sin \left (f x +e \right )+\left (-\sin \left (f x +e \right ) c +c \right )^{p}}d x \right ) b \right )}{c^{4}} \] Input:
int((g*cos(f*x+e))^p*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(-4-p),x)
Output:
(g**p*(int(cos(e + f*x)**p/(( - sin(e + f*x)*c + c)**p*sin(e + f*x)**4 - 4 *( - sin(e + f*x)*c + c)**p*sin(e + f*x)**3 + 6*( - sin(e + f*x)*c + c)**p *sin(e + f*x)**2 - 4*( - sin(e + f*x)*c + c)**p*sin(e + f*x) + ( - sin(e + f*x)*c + c)**p),x)*a + int((cos(e + f*x)**p*sin(e + f*x))/(( - sin(e + f* x)*c + c)**p*sin(e + f*x)**4 - 4*( - sin(e + f*x)*c + c)**p*sin(e + f*x)** 3 + 6*( - sin(e + f*x)*c + c)**p*sin(e + f*x)**2 - 4*( - sin(e + f*x)*c + c)**p*sin(e + f*x) + ( - sin(e + f*x)*c + c)**p),x)*b))/c**4