\(\int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx\) [371]

Optimal result

Integrand size = 27, antiderivative size = 70 \[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\frac {(e \cos (c+d x))^{-2 (1+m)} \operatorname {Hypergeometric2F1}\left (2,-1-m,-m,\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{1+m}}{4 a d e (1+m)} \]

[Out]

1/4*hypergeom([2, -1-m],[-m],1/2-1/2*sin(d*x+c))*(a+a*sin(d*x+c))^(1+m)/a/d/e/(1+m)/((e*cos(d*x+c))^(2+2*m))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2768, 7, 70} \[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\frac {(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \operatorname {Hypergeometric2F1}\left (2,-m-1,-m,\frac {1}{2} (1-\sin (c+d x))\right )}{4 a d e (m+1)} \]

[In]

Int[(e*Cos[c + d*x])^(-3 - 2*m)*(a + a*Sin[c + d*x])^m,x]

[Out]

(Hypergeometric2F1[2, -1 - m, -m, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1 + m))/(4*a*d*e*(1 + m)*(e*Cos[
c + d*x])^(2*(1 + m)))

Rule 7

Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[u*Px^Simplify[p], x] /; PolyQ[Px, x] &&  !RationalQ[p] && FreeQ[p, x] &
& RationalQ[Simplify[p]]

Rule 70

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 (e \cos (c+d x))^{-2-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (2+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (2+2 m)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (-4-2 m)} (a+a x)^{\frac {1}{2} (-4-2 m)+m} \, dx,x,\sin (c+d x)\right )}{d e} \\ & = \frac {\left (a^2 (e \cos (c+d x))^{-2-2 m} (a-a \sin (c+d x))^{\frac {1}{2} (2+2 m)} (a+a \sin (c+d x))^{\frac {1}{2} (2+2 m)}\right ) \text {Subst}\left (\int \frac {(a-a x)^{\frac {1}{2} (-4-2 m)}}{(a+a x)^2} \, dx,x,\sin (c+d x)\right )}{d e} \\ & = \frac {(e \cos (c+d x))^{-2 (1+m)} \operatorname {Hypergeometric2F1}\left (2,-1-m,-m,\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{1+m}}{4 a d e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.09 \[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\frac {(e \cos (c+d x))^{-2 m} \operatorname {Hypergeometric2F1}\left (2,-1-m,-m,\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^2(c+d x) (a (1+\sin (c+d x)))^{1+m}}{4 a d e^3 (1+m)} \]

[In]

Integrate[(e*Cos[c + d*x])^(-3 - 2*m)*(a + a*Sin[c + d*x])^m,x]

[Out]

(Hypergeometric2F1[2, -1 - m, -m, (1 - Sin[c + d*x])/2]*Sec[c + d*x]^2*(a*(1 + Sin[c + d*x]))^(1 + m))/(4*a*d*
e^3*(1 + m)*(e*Cos[c + d*x])^(2*m))

Maple [F]

\[\int \left (e \cos \left (d x +c \right )\right )^{-3-2 m} \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]

[In]

int((e*cos(d*x+c))^(-3-2*m)*(a+a*sin(d*x+c))^m,x)

[Out]

int((e*cos(d*x+c))^(-3-2*m)*(a+a*sin(d*x+c))^m,x)

Fricas [F]

\[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-2 \, m - 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(-3-2*m)*(a+a*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

integral((e*cos(d*x + c))^(-2*m - 3)*(a*sin(d*x + c) + a)^m, x)

Sympy [F]

\[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \left (e \cos {\left (c + d x \right )}\right )^{- 2 m - 3}\, dx \]

[In]

integrate((e*cos(d*x+c))**(-3-2*m)*(a+a*sin(d*x+c))**m,x)

[Out]

Integral((a*(sin(c + d*x) + 1))**m*(e*cos(c + d*x))**(-2*m - 3), x)

Maxima [F]

\[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-2 \, m - 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(-3-2*m)*(a+a*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(-2*m - 3)*(a*sin(d*x + c) + a)^m, x)

Giac [F]

\[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{-2 \, m - 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(-3-2*m)*(a+a*sin(d*x+c))^m,x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(-2*m - 3)*(a*sin(d*x + c) + a)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{2\,m+3}} \,d x \]

[In]

int((a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(2*m + 3),x)

[Out]

int((a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(2*m + 3), x)