\(\int \frac {(a+b \sin (c+d x))^m}{\sqrt {e \cos (c+d x)}} \, dx\) [644]

Optimal result

Integrand size = 25, antiderivative size = 134 \[ \int \frac {(a+b \sin (c+d x))^m}{\sqrt {e \cos (c+d x)}} \, dx=\frac {e \operatorname {AppellF1}\left (1+m,\frac {3}{4},\frac {3}{4},2+m,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{3/4}}{b d (1+m) (e \cos (c+d x))^{3/2}} \]

[Out]

e*AppellF1(1+m,3/4,3/4,2+m,(a+b*sin(d*x+c))/(a-b),(a+b*sin(d*x+c))/(a+b))*(a+b*sin(d*x+c))^(1+m)*(1+(-a-b*sin(
d*x+c))/(a-b))^(3/4)*(1+(-a-b*sin(d*x+c))/(a+b))^(3/4)/b/d/(1+m)/(e*cos(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2783, 143} \[ \int \frac {(a+b \sin (c+d x))^m}{\sqrt {e \cos (c+d x)}} \, dx=\frac {e \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{3/4} (a+b \sin (c+d x))^{m+1} \operatorname {AppellF1}\left (m+1,\frac {3}{4},\frac {3}{4},m+2,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) (e \cos (c+d x))^{3/2}} \]

[In]

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

[Out]

(e*AppellF1[1 + m, 3/4, 3/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c +
d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(b*d*(1 + m)
*(e*Cos[c + d*x])^(3/2))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 2783

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (e \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{\left (-\frac {b}{a-b}-\frac {b x}{a-b}\right )^{3/4} \left (\frac {b}{a+b}-\frac {b x}{a+b}\right )^{3/4}} \, dx,x,\sin (c+d x)\right )}{d (e \cos (c+d x))^{3/2}} \\ & = \frac {e \operatorname {AppellF1}\left (1+m,\frac {3}{4},\frac {3}{4},2+m,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{3/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{3/4}}{b d (1+m) (e \cos (c+d x))^{3/2}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(a+b \sin (c+d x))^m}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {(a+b \sin (c+d x))^m}{\sqrt {e \cos (c+d x)}} \, dx \]

[In]

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

[Out]

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

Maple [F]

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

[In]

int((a+b*sin(d*x+c))^m/(e*cos(d*x+c))^(1/2),x)

[Out]

int((a+b*sin(d*x+c))^m/(e*cos(d*x+c))^(1/2),x)

Fricas [F]

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

[In]

integrate((a+b*sin(d*x+c))^m/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((a+b*sin(d*x+c))**m/(e*cos(d*x+c))**(1/2),x)

[Out]

Integral((a + b*sin(c + d*x))**m/sqrt(e*cos(c + d*x)), x)

Maxima [F]

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

[In]

integrate((a+b*sin(d*x+c))^m/(e*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*sin(d*x+c))^m/(e*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(1/2),x)

[Out]

int((a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(1/2), x)