3.3.79 \(\int (d \cos (a+b x))^{3/2} \csc ^p(a+b x) \, dx\) [279]

Optimal. Leaf size=76 \[ \frac {d \sqrt {d \cos (a+b x)} \csc ^{-1+p}(a+b x) \text {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1-p}{2},\frac {3-p}{2},\sin ^2(a+b x)\right )}{b (1-p) \sqrt [4]{\cos ^2(a+b x)}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 2657} \begin {gather*} \frac {d \sqrt {d \cos (a+b x)} \csc ^{p-1}(a+b x) \, _2F_1\left (-\frac {1}{4},\frac {1-p}{2};\frac {3-p}{2};\sin ^2(a+b x)\right )}{b (1-p) \sqrt [4]{\cos ^2(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^p,x]

[Out]

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

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]

Rule 2667

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[b^2*(b*Cos[e
+ f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1), Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e,
f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rubi steps

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

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Mathematica [A]
time = 41.82, size = 105, normalized size = 1.38 \begin {gather*} -\frac {2 (d \cos (a+b x))^{5/2} \csc ^{-1+p}(a+b x) \left (9 \text {Hypergeometric2F1}\left (\frac {5}{4},\frac {1}{2} (-1+p),\frac {9}{4},\cos ^2(a+b x)\right )+5 \cos ^2(a+b x) \text {Hypergeometric2F1}\left (\frac {9}{4},\frac {1+p}{2},\frac {13}{4},\cos ^2(a+b x)\right )\right ) \sin ^2(a+b x)^{\frac {1}{2} (-1+p)}}{45 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^p,x]

[Out]

(-2*(d*Cos[a + b*x])^(5/2)*Csc[a + b*x]^(-1 + p)*(9*Hypergeometric2F1[5/4, (-1 + p)/2, 9/4, Cos[a + b*x]^2] +
5*Cos[a + b*x]^2*Hypergeometric2F1[9/4, (1 + p)/2, 13/4, Cos[a + b*x]^2])*(Sin[a + b*x]^2)^((-1 + p)/2))/(45*b
*d)

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \left (\csc ^{p}\left (b x +a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(3/2)*csc(b*x+a)^p,x)

[Out]

int((d*cos(b*x+a))^(3/2)*csc(b*x+a)^p,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))^(3/2)*csc(b*x+a)^p,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(3/2)*csc(b*x + a)^p, 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))^(3/2)*csc(b*x+a)^p,x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*d*csc(b*x + a)^p*cos(b*x + a), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))**(3/2)*csc(b*x+a)**p,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6436 deep

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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))^(3/2)*csc(b*x+a)^p,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(3/2)*csc(b*x + a)^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}\,{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(a + b*x))^(3/2)*(1/sin(a + b*x))^p,x)

[Out]

int((d*cos(a + b*x))^(3/2)*(1/sin(a + b*x))^p, x)

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