\(\int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1027]

Optimal result

Integrand size = 31, antiderivative size = 129 \[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {2^{\frac {7}{2}+m} a^3 (B m+A (7+m)) \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^{-3+m}}{7 f (7+m)}-\frac {B \cos ^7(e+f x) (a+a \sin (e+f x))^m}{f (7+m)} \]

[Out]

-1/7*2^(7/2+m)*a^3*(B*m+A*(7+m))*cos(f*x+e)^7*hypergeom([7/2, -5/2-m],[9/2],1/2-1/2*sin(f*x+e))*(1+sin(f*x+e))
^(-1/2-m)*(a+a*sin(f*x+e))^(-3+m)/f/(7+m)-B*cos(f*x+e)^7*(a+a*sin(f*x+e))^m/f/(7+m)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2939, 2768, 72, 71} \[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {a^3 2^{m+\frac {7}{2}} (A (m+7)+B m) \cos ^7(e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^{m-3} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-m-\frac {5}{2},\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{7 f (m+7)}-\frac {B \cos ^7(e+f x) (a \sin (e+f x)+a)^m}{f (m+7)} \]

[In]

Int[Cos[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

-1/7*(2^(7/2 + m)*a^3*(B*m + A*(7 + m))*Cos[e + f*x]^7*Hypergeometric2F1[7/2, -5/2 - m, 9/2, (1 - Sin[e + f*x]
)/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(f*(7 + m)) - (B*Cos[e + f*x]^7*(a + a*Sin[e
+ f*x])^m)/(f*(7 + m))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 2768

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[a^2*(
(g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))), Subst[Int[(
a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&
EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]

Rule 2939

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

Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos ^7(e+f x) (a+a \sin (e+f x))^m}{f (7+m)}+\left (A+\frac {B m}{7+m}\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^m \, dx \\ & = -\frac {B \cos ^7(e+f x) (a+a \sin (e+f x))^m}{f (7+m)}+\frac {\left (a^2 \left (A+\frac {B m}{7+m}\right ) \cos ^7(e+f x)\right ) \text {Subst}\left (\int (a-a x)^{5/2} (a+a x)^{\frac {5}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{7/2} (a+a \sin (e+f x))^{7/2}} \\ & = -\frac {B \cos ^7(e+f x) (a+a \sin (e+f x))^m}{f (7+m)}+\frac {\left (2^{\frac {5}{2}+m} a^4 \left (A+\frac {B m}{7+m}\right ) \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{-\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {5}{2}+m} (a-a x)^{5/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{7/2}} \\ & = -\frac {2^{\frac {7}{2}+m} a^3 \left (A+\frac {B m}{7+m}\right ) \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^{-3+m}}{7 f}-\frac {B \cos ^7(e+f x) (a+a \sin (e+f x))^m}{f (7+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {\cos ^7(e+f x) (1+\sin (e+f x))^{-\frac {7}{2}-m} (a (1+\sin (e+f x)))^m \left (2^{\frac {7}{2}+m} (B m+A (7+m)) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-\frac {5}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right )+7 B (1+\sin (e+f x))^{\frac {7}{2}+m}\right )}{7 f (7+m)} \]

[In]

Integrate[Cos[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

-1/7*(Cos[e + f*x]^7*(1 + Sin[e + f*x])^(-7/2 - m)*(a*(1 + Sin[e + f*x]))^m*(2^(7/2 + m)*(B*m + A*(7 + m))*Hyp
ergeometric2F1[7/2, -5/2 - m, 9/2, (1 - Sin[e + f*x])/2] + 7*B*(1 + Sin[e + f*x])^(7/2 + m)))/(f*(7 + m))

Maple [F]

\[\int \left (\cos ^{6}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )d x\]

[In]

int(cos(f*x+e)^6*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

[Out]

int(cos(f*x+e)^6*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

Fricas [F]

\[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{6} \,d x } \]

[In]

integrate(cos(f*x+e)^6*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*cos(f*x + e)^6*sin(f*x + e) + A*cos(f*x + e)^6)*(a*sin(f*x + e) + a)^m, x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**6*(a+a*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)^6*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{6} \,d x } \]

[In]

integrate(cos(f*x+e)^6*(a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m*cos(f*x + e)^6, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int {\cos \left (e+f\,x\right )}^6\,\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]

[In]

int(cos(e + f*x)^6*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m,x)

[Out]

int(cos(e + f*x)^6*(A + B*sin(e + f*x))*(a + a*sin(e + f*x))^m, x)