3.647 \(\int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx\)

Optimal. Leaf size=598 \[ -\frac{a 2^{-\frac{m}{2}-\frac{1}{2}} \left (a^2 (m+2)+2 a b-b^2\right ) (1-\sin (c+d x))^2 (e \cos (c+d x))^{-m-3} \left (\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (\frac{1-m}{2},\frac{m+3}{2};\frac{3-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (1-m) (m+3) (a-b) (a+b)^3}+\frac{a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) \left (a^2-b^2\right )}-\frac{a b 2^{\frac{3}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left (\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e^3 (m+1) (m+3) (a-b)^2 (a+b)}+\frac{2 b (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+1) (m+3) (a-b)^2}+\frac{a (a (m+2)+3 b) (1-\sin (c+d x)) (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (m+3) (a-b) (a+b)^2}-\frac{(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.01811, antiderivative size = 598, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2700, 2699, 2920, 132, 129, 155, 12} \[ -\frac{a 2^{-\frac{m}{2}-\frac{1}{2}} \left (a^2 (m+2)+2 a b-b^2\right ) (1-\sin (c+d x))^2 (e \cos (c+d x))^{-m-3} \left (\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (\frac{1-m}{2},\frac{m+3}{2};\frac{3-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e (1-m) (m+3) (a-b) (a+b)^3}+\frac{a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) \left (a^2-b^2\right )}-\frac{a b 2^{\frac{3}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left (\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left (\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right )}{d e^3 (m+1) (m+3) (a-b)^2 (a+b)}+\frac{2 b (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+1) (m+3) (a-b)^2}+\frac{a (a (m+2)+3 b) (1-\sin (c+d x)) (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (m+3) (a-b) (a+b)^2}-\frac{(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(-4 - m)*(a + b*Sin[c + d*x])^m,x]

[Out]

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

Rule 2700

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

Rule 2699

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

Rule 2920

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

Rule 132

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

Rule 129

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx &=-\frac{(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac{a \int \frac{(e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^m}{1-\sin (c+d x)} \, dx}{(a-b) e^2}-\frac{(2 b) \int (e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^m \, dx}{(a-b) e^2 (3+m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac{2 b (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 d e^3 (1+m) (3+m)}-\frac{(2 a b) \int \frac{(e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m}{1-\sin (c+d x)} \, dx}{(a-b)^2 e^4 (3+m)}+\frac{\left (a (e \cos (c+d x))^{-3-m} (1-\sin (c+d x))^{\frac{3+m}{2}} (1+\sin (c+d x))^{\frac{3+m}{2}}\right ) \operatorname{Subst}\left (\int (1-x)^{-1+\frac{1}{2} (-3-m)} (1+x)^{\frac{1}{2} (-3-m)} (a+b x)^m \, dx,x,\sin (c+d x)\right )}{(a-b) d e}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac{2 b (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 d e^3 (1+m) (3+m)}+\frac{a (e \cos (c+d x))^{-3-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (3+m)}-\frac{\left (2 a b (e \cos (c+d x))^{-1-m} (1-\sin (c+d x))^{\frac{1+m}{2}} (1+\sin (c+d x))^{\frac{1+m}{2}}\right ) \operatorname{Subst}\left (\int (1-x)^{-1+\frac{1}{2} (-1-m)} (1+x)^{\frac{1}{2} (-1-m)} (a+b x)^m \, dx,x,\sin (c+d x)\right )}{(a-b)^2 d e^3 (3+m)}-\frac{\left (a (e \cos (c+d x))^{-3-m} (1-\sin (c+d x))^{\frac{3+m}{2}} (1+\sin (c+d x))^{\frac{3+m}{2}}\right ) \operatorname{Subst}\left (\int (1-x)^{\frac{1}{2} (-3-m)} (1+x)^{\frac{1}{2} (-3-m)} (-2 b-a (2+m)-b x) (a+b x)^m \, dx,x,\sin (c+d x)\right )}{(a-b) (a+b) d e (3+m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac{2 b (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 d e^3 (1+m) (3+m)}+\frac{a (e \cos (c+d x))^{-3-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (3+m)}+\frac{a (3 b+a (2+m)) (e \cos (c+d x))^{-3-m} (1-\sin (c+d x)) (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^2 d e (1+m) (3+m)}-\frac{2^{\frac{3}{2}-\frac{m}{2}} a b (e \cos (c+d x))^{-1-m} \, _2F_1\left (\frac{1}{2} (-1-m),\frac{1+m}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) \left (\frac{(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac{1+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 (a+b) d e^3 (1+m) (3+m)}+\frac{\left (a (e \cos (c+d x))^{-3-m} (1-\sin (c+d x))^{\frac{3+m}{2}} (1+\sin (c+d x))^{\frac{3+m}{2}}\right ) \operatorname{Subst}\left (\int (1+m) \left (2 a b-b^2+a^2 (2+m)\right ) (1-x)^{1+\frac{1}{2} (-3-m)} (1+x)^{\frac{1}{2} (-3-m)} (a+b x)^m \, dx,x,\sin (c+d x)\right )}{2 (-a-b) (a-b) (a+b) d e \left (1+\frac{1}{2} (-3-m)\right ) (3+m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac{2 b (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 d e^3 (1+m) (3+m)}+\frac{a (e \cos (c+d x))^{-3-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (3+m)}+\frac{a (3 b+a (2+m)) (e \cos (c+d x))^{-3-m} (1-\sin (c+d x)) (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^2 d e (1+m) (3+m)}-\frac{2^{\frac{3}{2}-\frac{m}{2}} a b (e \cos (c+d x))^{-1-m} \, _2F_1\left (\frac{1}{2} (-1-m),\frac{1+m}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) \left (\frac{(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac{1+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 (a+b) d e^3 (1+m) (3+m)}+\frac{\left (a (1+m) \left (2 a b-b^2+a^2 (2+m)\right ) (e \cos (c+d x))^{-3-m} (1-\sin (c+d x))^{\frac{3+m}{2}} (1+\sin (c+d x))^{\frac{3+m}{2}}\right ) \operatorname{Subst}\left (\int (1-x)^{1+\frac{1}{2} (-3-m)} (1+x)^{\frac{1}{2} (-3-m)} (a+b x)^m \, dx,x,\sin (c+d x)\right )}{2 (-a-b) (a-b) (a+b) d e \left (1+\frac{1}{2} (-3-m)\right ) (3+m)}\\ &=-\frac{(e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d e (3+m)}+\frac{2 b (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 d e^3 (1+m) (3+m)}+\frac{a (e \cos (c+d x))^{-3-m} (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{\left (a^2-b^2\right ) d e (3+m)}+\frac{a (3 b+a (2+m)) (e \cos (c+d x))^{-3-m} (1-\sin (c+d x)) (1+\sin (c+d x)) (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^2 d e (1+m) (3+m)}-\frac{2^{\frac{3}{2}-\frac{m}{2}} a b (e \cos (c+d x))^{-1-m} \, _2F_1\left (\frac{1}{2} (-1-m),\frac{1+m}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) \left (\frac{(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac{1+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b)^2 (a+b) d e^3 (1+m) (3+m)}-\frac{2^{-\frac{1}{2}-\frac{m}{2}} a \left (2 a b-b^2+a^2 (2+m)\right ) (e \cos (c+d x))^{-3-m} \, _2F_1\left (\frac{1-m}{2},\frac{3+m}{2};\frac{3-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^2 \left (\frac{(a+b) (1+\sin (c+d x))}{a+b \sin (c+d x)}\right )^{\frac{3+m}{2}} (a+b \sin (c+d x))^{1+m}}{(a-b) (a+b)^3 d e (1-m) (3+m)}\\ \end{align*}

Mathematica [A]  time = 6.09413, size = 826, normalized size = 1.38 \[ \frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} (e \cos (c+d x))^{-m-4}}{(a-b) d (-m-3)}+\frac{2 b \cos ^{m+4}(c+d x) \left (\frac{2^{\frac{1}{2} (-m-1)+1} a \, _2F_1\left (\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1}{2} (-m-1)+1;\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^{\frac{1}{2} (-m-1)+\frac{m+1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-m-1)+\frac{m+1}{2}} \left (-\frac{(-a-b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \cos ^{-m-1}(c+d x)}{(-a-b) (a-b) d (-m-1)}+\frac{(a+b \sin (c+d x))^{m+1} \cos ^{-m-1}(c+d x)}{(a-b) d (-m-1)}\right ) (e \cos (c+d x))^{-m-4}}{(a-b) (-m-3)}+\frac{a \cos (c+d x) (1-\sin (c+d x))^{\frac{m+3}{2}} (\sin (c+d x)+1)^{\frac{m+3}{2}} \left (\frac{(1-\sin (c+d x))^{\frac{1}{2} (-m-3)} (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)+1} (a+b \sin (c+d x))^{m+1}}{(-a-b) (-m-3)}-\frac{-\frac{(3 b+a (m+2)) (1-\sin (c+d x))^{\frac{1}{2} (-m-3)+1} (a+b \sin (c+d x))^{m+1} (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)+1}}{2 (-a-b) \left (\frac{1}{2} (-m-3)+1\right )}-\frac{2^{\frac{1}{2} (-m-3)-1} (m+1) \left ((m+2) a^2+2 b a-b^2\right ) \, _2F_1\left (\frac{1}{2} (-m-3)+2,\frac{m+3}{2};\frac{1}{2} (-m-3)+3;\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^{\frac{1}{2} (-m-3)+2} \left (-\frac{(-a-b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right )^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)}}{(-a-b)^2 \left (\frac{1}{2} (-m-3)+1\right ) \left (\frac{1}{2} (-m-3)+2\right )}}{(-a-b) (-m-3)}\right ) (e \cos (c+d x))^{-m-4}}{(a-b) d} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(-4 - m)*(a + b*Sin[c + d*x])^m,x]

[Out]

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

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Maple [F]  time = 0.202, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{-4-m} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(-4-m)*(a+b*sin(d*x+c))^m,x)

[Out]

int((e*cos(d*x+c))^(-4-m)*(a+b*sin(d*x+c))^m,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{-m - 4}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(-4-m)*(a+b*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(-m - 4)*(b*sin(d*x + c) + a)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e \cos \left (d x + c\right )\right )^{-m - 4}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(-4-m)*(a+b*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

integral((e*cos(d*x + c))^(-m - 4)*(b*sin(d*x + c) + a)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**(-4-m)*(a+b*sin(d*x+c))**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \cos \left (d x + c\right )\right )^{-m - 4}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(-4-m)*(a+b*sin(d*x+c))^m,x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(-m - 4)*(b*sin(d*x + c) + a)^m, x)