\(\int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx\) [618]

Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=-\frac {b (e \cos (c+d x))^{1+p}}{d e (1+p)}-\frac {a (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d e (1+p) \sqrt {\sin ^2(c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2748, 2722} \[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=-\frac {a \sin (c+d x) (e \cos (c+d x))^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {p+1}{2},\frac {p+3}{2},\cos ^2(c+d x)\right )}{d e (p+1) \sqrt {\sin ^2(c+d x)}}-\frac {b (e \cos (c+d x))^{p+1}}{d e (p+1)} \]

[In]

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

[Out]

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

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 2748

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

Rubi steps \begin{align*} \text {integral}& = -\frac {b (e \cos (c+d x))^{1+p}}{d e (1+p)}+a \int (e \cos (c+d x))^p \, dx \\ & = -\frac {b (e \cos (c+d x))^{1+p}}{d e (1+p)}-\frac {a (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d e (1+p) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=-\frac {(e \cos (c+d x))^p \left (b \cos (c+d x)+a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{d (1+p)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (e \cos \left (d x +c \right )\right )^{p} \left (a +b \sin \left (d x +c \right )\right )d x\]

[In]

int((e*cos(d*x+c))^p*(a+b*sin(d*x+c)),x)

[Out]

int((e*cos(d*x+c))^p*(a+b*sin(d*x+c)),x)

Fricas [F]

\[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]

[In]

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

[Out]

integral((b*sin(d*x + c) + a)*(e*cos(d*x + c))^p, x)

Sympy [F]

\[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=\int \left (e \cos {\left (c + d x \right )}\right )^{p} \left (a + b \sin {\left (c + d x \right )}\right )\, dx \]

[In]

integrate((e*cos(d*x+c))**p*(a+b*sin(d*x+c)),x)

[Out]

Integral((e*cos(c + d*x))**p*(a + b*sin(c + d*x)), x)

Maxima [F]

\[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]

[In]

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

[Out]

integrate((b*sin(d*x + c) + a)*(e*cos(d*x + c))^p, x)

Giac [F]

\[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]

[In]

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

[Out]

integrate((b*sin(d*x + c) + a)*(e*cos(d*x + c))^p, x)

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^p\,\left (a+b\,\sin \left (c+d\,x\right )\right ) \,d x \]

[In]

int((e*cos(c + d*x))^p*(a + b*sin(c + d*x)),x)

[Out]

int((e*cos(c + d*x))^p*(a + b*sin(c + d*x)), x)