\(\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx\) [81]

Optimal result

Integrand size = 38, antiderivative size = 182 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m}}{a c^2 f \left (35+24 m+4 m^2\right )}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-2-m}}{a c^3 f (7+2 m) \left (15+16 m+4 m^2\right )} \]

[Out]

cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)*(c-c*sin(f*x+e))^(-4-m)/a/c/f/(7+2*m)+2*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)*(c
-c*sin(f*x+e))^(-3-m)/a/c^2/f/(4*m^2+24*m+35)+2*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)*(c-c*sin(f*x+e))^(-2-m)/a/c^
3/f/(8*m^3+60*m^2+142*m+105)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2822, 2821} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-2}}{a c^3 f (2 m+7) \left (4 m^2+16 m+15\right )}+\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-3}}{a c^2 f \left (4 m^2+24 m+35\right )}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{m+1} (c-c \sin (e+f x))^{-m-4}}{a c f (2 m+7)} \]

[In]

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-5 - m),x]

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-4 - m))/(a*c*f*(7 + 2*m)) + (2*Cos[e + f*x]*
(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-3 - m))/(a*c^2*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a
+ a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c^3*f*(7 + 2*m)*(15 + 16*m + 4*m^2))

Rule 2821

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] /; FreeQ[{a, b, c, d, e, f
, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -2^(-1)]

Rule 2822

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] + Dist[(m + n + 1)/(a*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m
, 1] ||  !SumSimplerQ[n, 1])

Rule 2920

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m} \, dx}{a c^2 (7+2 m)} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m}}{a c^2 f (5+2 m) (7+2 m)}+\frac {2 \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-2-m} \, dx}{a c^3 (5+2 m) (7+2 m)} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-4-m}}{a c f (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-3-m}}{a c^2 f (5+2 m) (7+2 m)}+\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{-2-m}}{a c^3 f (3+2 m) (5+2 m) (7+2 m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.61 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.58 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\frac {\cos ^3(e+f x) (a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{-m} \left (-4 \left (6+5 m+m^2\right )+\cos (2 (e+f x))+2 (5+2 m) \sin (e+f x)\right )}{c^5 f (3+2 m) (5+2 m) (7+2 m) (-1+\sin (e+f x))^5} \]

[In]

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-5 - m),x]

[Out]

(Cos[e + f*x]^3*(a*(1 + Sin[e + f*x]))^m*(-4*(6 + 5*m + m^2) + Cos[2*(e + f*x)] + 2*(5 + 2*m)*Sin[e + f*x]))/(
c^5*f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(-1 + Sin[e + f*x])^5*(c - c*Sin[e + f*x])^m)

Maple [F]

\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-5-m}d x\]

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-5-m),x)

[Out]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-5-m),x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.58 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=-\frac {{\left (2 \, \cos \left (f x + e\right )^{5} + 2 \, {\left (2 \, m + 5\right )} \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - {\left (4 \, m^{2} + 20 \, m + 25\right )} \cos \left (f x + e\right )^{3}\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 5}}{8 \, f m^{3} + 60 \, f m^{2} + 142 \, f m + 105 \, f} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-5-m),x, algorithm="fricas")

[Out]

-(2*cos(f*x + e)^5 + 2*(2*m + 5)*cos(f*x + e)^3*sin(f*x + e) - (4*m^2 + 20*m + 25)*cos(f*x + e)^3)*(a*sin(f*x
+ e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 5)/(8*f*m^3 + 60*f*m^2 + 142*f*m + 105*f)

Sympy [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{- m - 5} \cos ^{2}{\left (e + f x \right )}\, dx \]

[In]

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(-5-m),x)

[Out]

Integral((a*(sin(e + f*x) + 1))**m*(-c*(sin(e + f*x) - 1))**(-m - 5)*cos(e + f*x)**2, x)

Maxima [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 5} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-5-m),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 5)*cos(f*x + e)^2, x)

Giac [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 5} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-5-m),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 5)*cos(f*x + e)^2, x)

Mupad [B] (verification not implemented)

Time = 17.22 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.84 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-5-m} \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (24\,m^2+120\,m+140\right )}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}-\frac {\cos \left (5\,e+5\,f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}+\frac {\cos \left (3\,e+3\,f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^2+40\,m+45\right )}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}+\frac {\sin \left (4\,e+4\,f\,x\right )\,\left (m\,4{}\mathrm {i}+10{}\mathrm {i}\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,1{}\mathrm {i}}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )}+\frac {\sin \left (2\,e+2\,f\,x\right )\,\left (m\,8{}\mathrm {i}+20{}\mathrm {i}\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,1{}\mathrm {i}}{8\,f\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{m+5}\,\left (8\,m^3+60\,m^2+142\,m+105\right )} \]

[In]

int((cos(e + f*x)^2*(a + a*sin(e + f*x))^m)/(c - c*sin(e + f*x))^(m + 5),x)

[Out]

(cos(e + f*x)*(a + a*sin(e + f*x))^m*(120*m + 24*m^2 + 140))/(8*f*(c - c*sin(e + f*x))^(m + 5)*(142*m + 60*m^2
+ 8*m^3 + 105)) - (cos(5*e + 5*f*x)*(a + a*sin(e + f*x))^m)/(8*f*(c - c*sin(e + f*x))^(m + 5)*(142*m + 60*m^2
+ 8*m^3 + 105)) + (sin(4*e + 4*f*x)*(m*4i + 10i)*(a + a*sin(e + f*x))^m*1i)/(8*f*(c - c*sin(e + f*x))^(m + 5)
*(142*m + 60*m^2 + 8*m^3 + 105)) + (sin(2*e + 2*f*x)*(m*8i + 20i)*(a + a*sin(e + f*x))^m*1i)/(8*f*(c - c*sin(e
+ f*x))^(m + 5)*(142*m + 60*m^2 + 8*m^3 + 105)) + (cos(3*e + 3*f*x)*(a + a*sin(e + f*x))^m*(40*m + 8*m^2 + 45
))/(8*f*(c - c*sin(e + f*x))^(m + 5)*(142*m + 60*m^2 + 8*m^3 + 105))