3.370 \(\int \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=78 \[ -\frac {\sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac {5}{4},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b c d (n+1) \sqrt {c \sin (a+b x)}} \]

[Out]

-(d*cos(b*x+a))^(1+n)*hypergeom([5/4, 1/2+1/2*n],[3/2+1/2*n],cos(b*x+a)^2)*(sin(b*x+a)^2)^(1/4)/b/c/d/(1+n)/(c
*sin(b*x+a))^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 78, 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 = {2576} \[ -\frac {\sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac {5}{4},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b c d (n+1) \sqrt {c \sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[5/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(1/4
))/(b*c*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 \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx &=-\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (\frac {5}{4},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b c d (1+n) \sqrt {c \sin (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 79, normalized size = 1.01 \[ -\frac {\sqrt [4]{\sin ^2(a+b x)} \cot (a+b x) \sqrt {c \sin (a+b x)} (d \cos (a+b x))^n \, _2F_1\left (\frac {5}{4},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b c^2 (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c \sin \left (b x + a\right )} \left (d \cos \left (b x + a\right )\right )^{n}}{c^{2} \cos \left (b x + a\right )^{2} - c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n/(c*sin(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \cos \left (b x + a\right )\right )^{n}}{\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n/(c*sin(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^n/(c*sin(b*x + a))^(3/2), x)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \cos \left (b x +a \right )\right )^{n}}{\left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^n/(c*sin(b*x+a))^(3/2),x)

[Out]

int((d*cos(b*x+a))^n/(c*sin(b*x+a))^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \cos \left (b x + a\right )\right )^{n}}{\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n/(c*sin(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^n/(c*sin(b*x + a))^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^n}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(a + b*x))^n/(c*sin(a + b*x))^(3/2),x)

[Out]

int((d*cos(a + b*x))^n/(c*sin(a + b*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \cos {\left (a + b x \right )}\right )^{n}}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))**n/(c*sin(b*x+a))**(3/2),x)

[Out]

Integral((d*cos(a + b*x))**n/(c*sin(a + b*x))**(3/2), x)

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