3.4.51 \(\int \sqrt {d \cos (a+b x)} (c \sin (a+b x))^m \, dx\) [351]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {2657} \begin {gather*} \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)}} \end {gather*}

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 2657

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

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Mathematica [A]
time = 0.05, size = 75, normalized size = 1.00 \begin {gather*} \frac {\sqrt {d \cos (a+b x)} \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))^m \tan (a+b x)}{b (1+m)} \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \sqrt {d \cos \left (b x +a \right )}\, \left (c \sin \left (b x +a \right )\right )^{m}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \sin {\left (a + b x \right )}\right )^{m} \sqrt {d \cos {\left (a + b x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {d\,\cos \left (a+b\,x\right )}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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