3.7.47 \(\int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx\) [647]

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

[Out]

-(e*cos(d*x+c))^(-3-m)*(a+b*sin(d*x+c))^(1+m)/(a-b)/d/e/(3+m)+2*b*(e*cos(d*x+c))^(-1-m)*(a+b*sin(d*x+c))^(1+m)
/(a-b)^2/d/e^3/(1+m)/(3+m)+a*(e*cos(d*x+c))^(-3-m)*(1+sin(d*x+c))*(a+b*sin(d*x+c))^(1+m)/(a^2-b^2)/d/e/(3+m)+a
*(3*b+a*(2+m))*(e*cos(d*x+c))^(-3-m)*(1-sin(d*x+c))*(1+sin(d*x+c))*(a+b*sin(d*x+c))^(1+m)/(a-b)/(a+b)^2/d/e/(1
+m)/(3+m)-2^(3/2-1/2*m)*a*b*(e*cos(d*x+c))^(-1-m)*hypergeom([-1/2-1/2*m, 1/2+1/2*m],[1/2-1/2*m],1/2*(a-b)*(1-s
in(d*x+c))/(a+b*sin(d*x+c)))*((a+b)*(1+sin(d*x+c))/(a+b*sin(d*x+c)))^(1/2+1/2*m)*(a+b*sin(d*x+c))^(1+m)/(a-b)^
2/(a+b)/d/e^3/(m^2+4*m+3)-2^(-1/2-1/2*m)*a*(2*a*b-b^2+a^2*(2+m))*(e*cos(d*x+c))^(-3-m)*hypergeom([1/2-1/2*m, 3
/2+1/2*m],[3/2-1/2*m],1/2*(a-b)*(1-sin(d*x+c))/(a+b*sin(d*x+c)))*(1-sin(d*x+c))^2*((a+b)*(1+sin(d*x+c))/(a+b*s
in(d*x+c)))^(3/2+1/2*m)*(a+b*sin(d*x+c))^(1+m)/(a-b)/(a+b)^3/d/e/(1-m)/(3+m)

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 598, normalized size of antiderivative = 1.00, 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 = {2779, 2778, 2999, 134, 136, 160, 12} \begin {gather*} -\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)} \end {gather*}

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 12

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

Rule 134

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)/((b*e - a*f)*(m + 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)*((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 136

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] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rule 160

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 2778

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*Cos
[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*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] && N
eQ[a^2 - b^2, 0] && EqQ[m + p + 2, 0]

Rule 2779

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*Cos
[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 2999

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 - S
in[e + f*x])^((p - 1)/2))), Subst[Int[(1 + (b/a)*x)^(m + (p - 1)/2)*(1 - (b/a)*x)^((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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 6.07, size = 826, normalized size = 1.38 \begin {gather*} \frac {\cos (c+d x) (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^{1+m}}{(a-b) d (-3-m)}+\frac {2 b \cos ^{4+m}(c+d x) (e \cos (c+d x))^{-4-m} \left (\frac {\cos ^{-1-m}(c+d x) (a+b \sin (c+d x))^{1+m}}{(a-b) d (-1-m)}+\frac {2^{1+\frac {1}{2} (-1-m)} a \cos ^{-1-m}(c+d x) \, _2F_1\left (\frac {1}{2} (-1-m),\frac {1+m}{2};1+\frac {1}{2} (-1-m);-\frac {(-a+b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^{\frac {1}{2} (-1-m)+\frac {1+m}{2}} (1+\sin (c+d x))^{\frac {1}{2} (-1-m)+\frac {1+m}{2}} \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) (a-b) d (-1-m)}\right )}{(a-b) (-3-m)}+\frac {a \cos (c+d x) (e \cos (c+d x))^{-4-m} (1-\sin (c+d x))^{\frac {3+m}{2}} (1+\sin (c+d x))^{\frac {3+m}{2}} \left (\frac {(1-\sin (c+d x))^{\frac {1}{2} (-3-m)} (1+\sin (c+d x))^{1+\frac {1}{2} (-3-m)} (a+b \sin (c+d x))^{1+m}}{(-a-b) (-3-m)}-\frac {-\frac {(3 b+a (2+m)) (1-\sin (c+d x))^{1+\frac {1}{2} (-3-m)} (1+\sin (c+d x))^{1+\frac {1}{2} (-3-m)} (a+b \sin (c+d x))^{1+m}}{2 (-a-b) \left (1+\frac {1}{2} (-3-m)\right )}-\frac {2^{-1+\frac {1}{2} (-3-m)} (1+m) \left (2 a b-b^2+a^2 (2+m)\right ) \, _2F_1\left (2+\frac {1}{2} (-3-m),\frac {3+m}{2};3+\frac {1}{2} (-3-m);-\frac {(-a+b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right ) (1-\sin (c+d x))^{2+\frac {1}{2} (-3-m)} (1+\sin (c+d x))^{\frac {1}{2} (-3-m)} \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)^2 \left (1+\frac {1}{2} (-3-m)\right ) \left (2+\frac {1}{2} (-3-m)\right )}}{(-a-b) (-3-m)}\right )}{(a-b) d} \end {gather*}

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, -1/2*((
-a + b)*(1 - Sin[c + d*x]))/(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)) - (-1/2*((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))/((-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, -1/2*((-a + b)*(1 - Sin[c + d*x]))/(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 = 0.17, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{-4-m} \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 5007 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{m+4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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