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

Optimal. Leaf size=76 \[ -\frac{c \sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sqrt{c \sin (a+b x)}} \]

[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]]))

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Rubi [A]  time = 0.0454629, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2576} \[ -\frac{c \sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sqrt{c \sin (a+b x)}} \]

Antiderivative was successfully verified.

[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 2576

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)*Hypergeometric2F1[(1 + m)/
2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a,
b, e, f, m, n}, x] && SimplerQ[n, m]

Rubi steps

\begin{align*} \int (d \cos (a+b x))^n \sqrt{c \sin (a+b x)} \, dx &=-\frac{c (d \cos (a+b x))^{1+n} \, _2F_1\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]  time = 0.106635, size = 82, normalized size = 1.08 \[ -\frac{\sin (a+b x) \cos (a+b x) \sqrt{c \sin (a+b x)} (d \cos (a+b x))^n \, _2F_1\left (\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b (n+1) \sin ^2(a+b x)^{3/4}} \]

Antiderivative was successfully verified.

[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)))

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n}\sqrt{c\sin \left ( bx+a \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \sin{\left (a + b x \right )}} \left (d \cos{\left (a + b x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)