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

Optimal. Leaf size=75 \[ \frac{d \sqrt [4]{\cos ^2(a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1) \sqrt{d \cos (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0475906, antiderivative size = 75, 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 = {2577} \[ \frac{d \sqrt [4]{\cos ^2(a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1) \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2577

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

Rubi steps

\begin{align*} \int \sqrt{d \cos (a+b x)} (c \sin (a+b x))^m \, dx &=\frac{d \sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac{1}{4},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.06853, size = 75, normalized size = 1. \[ \frac{\sqrt [4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{d \cos (a+b x)} (c \sin (a+b x))^m \, _2F_1\left (\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*S
in[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int \sqrt{d\cos \left ( bx+a \right ) } \left ( c\sin \left ( bx+a \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin{\left (a + b x \right )}\right )^{m} \sqrt{d \cos{\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))**(1/2)*(c*sin(b*x+a))**m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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