3.366 \(\int (d \cos (a+b x))^n (c \sin (a+b x))^{5/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0534167, 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} \[ -\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{n+1} \, _2F_1\left (-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sin ^2(a+b x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(5/2),x]

[Out]

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

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 (c \sin (a+b x))^{5/2} \, 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 ) (c \sin (a+b x))^{3/2}}{b d (1+n) \sin ^2(a+b x)^{3/4}}\\ \end{align*}

Mathematica [B]  time = 0.415062, size = 158, normalized size = 2.08 \[ \frac{\cot (a+b x) (c \sin (a+b x))^{5/2} (d \cos (a+b x))^n \left ((n+1) \cos ^2(a+b x) \, _2F_1\left (\frac{1}{4},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(a+b x)\right )-(n+3) \, _2F_1\left (-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )-(n+3) \, _2F_1\left (\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )\right )}{2 b (n+1) (n+3) \sin ^2(a+b x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(5/2),x]

[Out]

((d*Cos[a + b*x])^n*Cot[a + b*x]*(-((3 + n)*Hypergeometric2F1[-3/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]) - (
3 + n)*Hypergeometric2F1[1/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2] + (1 + n)*Cos[a + b*x]^2*Hypergeometric2F1
[1/4, (3 + n)/2, (5 + n)/2, Cos[a + b*x]^2])*(c*Sin[a + b*x])^(5/2))/(2*b*(1 + n)*(3 + n)*(Sin[a + b*x]^2)^(3/
4))

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( c\sin \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}\, 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))^(5/2),x)

[Out]

int((d*cos(b*x+a))^n*(c*sin(b*x+a))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}} \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))^(5/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(5/2)*(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 (-{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \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))^(5/2),x, algorithm="fricas")

[Out]

integral(-(c^2*cos(b*x + a)^2 - c^2)*sqrt(c*sin(b*x + a))*(d*cos(b*x + a))^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}} \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))^(5/2),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(5/2)*(d*cos(b*x + a))^n, x)