\(\int (d \cos (a+b x))^n \sqrt {c \sin (a+b x)} \, dx\) [368]

Optimal result

Integrand size = 23, antiderivative size = 76 \[ \int (d \cos (a+b x))^n \sqrt {c \sin (a+b x)} \, dx=-\frac {c (d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b d (1+n) \sqrt {c \sin (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2656} \[ \int (d \cos (a+b x))^n \sqrt {c \sin (a+b x)} \, dx=-\frac {c \sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(a+b x)\right )}{b d (n+1) \sqrt {c \sin (a+b x)}} \]

[In]

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

[Out]

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

Rule 2656

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^(2*IntPar
t[(n - 1)/2] + 1))*(b*Sin[e + f*x])^(2*FracPart[(n - 1)/2])*((a*Cos[e + f*x])^(m + 1)/(a*f*(m + 1)*(Sin[e + f*
x]^2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2], x] /; FreeQ[{a
, b, e, f, m, n}, x] && SimplerQ[n, m]

Rubi steps \begin{align*} \text {integral}& = -\frac {c (d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b d (1+n) \sqrt {c \sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int (d \cos (a+b x))^n \sqrt {c \sin (a+b x)} \, dx=-\frac {\cos (a+b x) (d \cos (a+b x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (a+b x) \sqrt {c \sin (a+b x)}}{b (1+n) \sin ^2(a+b x)^{3/4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c*sin(b*x + a))*(d*cos(b*x + a))^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a))*(d*cos(b*x + a))^n, x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a))*(d*cos(b*x + a))^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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