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

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

[Out]

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

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Rubi [A]  time = 0.0484558, 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} \[ -\frac{c \sin ^2(a+b x)^{3/4} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) (c \sin (a+b x))^{3/2}} \]

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

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 \frac{(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{3}{4},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right ) \sin ^2(a+b x)^{3/4}}{b d (1+n) (c \sin (a+b x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.118294, size = 82, normalized size = 1.08 \[ -\frac{\sin (a+b x) \cos (a+b x) (d \cos (a+b x))^n \, _2F_1\left (\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b (n+1) \sqrt [4]{\sin ^2(a+b x)} \sqrt{c \sin (a+b x)}} \]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\cos \left ( bx+a \right ) \right ) ^{n}{\frac{1}{\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 \frac{\left (d \cos \left (b x + a\right )\right )^{n}}{\sqrt{c \sin \left (b x + a\right )}}\,{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((d*cos(b*x + a))^n/sqrt(c*sin(b*x + a)), x)

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

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

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