3.283 \(\int \frac{\csc ^p(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{\cos ^2(a+b x)^{3/4} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{7}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) (d \cos (a+b x))^{3/2}} \]

[Out]

((Cos[a + b*x]^2)^(3/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[7/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b
*d*(1 - p)*(d*Cos[a + b*x])^(3/2))

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Rubi [A]  time = 0.110108, antiderivative size = 78, normalized size of antiderivative = 1., 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 = {2587, 2577} \[ \frac{\cos ^2(a+b x)^{3/4} \csc ^{p-1}(a+b x) \, _2F_1\left (\frac{7}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^p/(d*Cos[a + b*x])^(5/2),x]

[Out]

((Cos[a + b*x]^2)^(3/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[7/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b
*d*(1 - p)*(d*Cos[a + b*x])^(3/2))

Rule 2587

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]

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 \frac{\csc ^p(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx &=\left (\csc ^p(a+b x) \sin ^p(a+b x)\right ) \int \frac{\sin ^{-p}(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx\\ &=\frac{\cos ^2(a+b x)^{3/4} \csc ^{-1+p}(a+b x) \, _2F_1\left (\frac{7}{4},\frac{1-p}{2};\frac{3-p}{2};\sin ^2(a+b x)\right )}{b d (1-p) (d \cos (a+b x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.286083, size = 70, normalized size = 0.9 \[ \frac{2 \sin ^2(a+b x)^{\frac{p-1}{2}} \csc ^{p-1}(a+b x) \, _2F_1\left (-\frac{3}{4},\frac{p+1}{2};\frac{1}{4};\cos ^2(a+b x)\right )}{3 b d (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^p/(d*Cos[a + b*x])^(5/2),x]

[Out]

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

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Maple [F]  time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \csc \left ( bx+a \right ) \right ) ^{p} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^p/(d*cos(b*x+a))^(5/2),x)

[Out]

int(csc(b*x+a)^p/(d*cos(b*x+a))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{p}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^p/(d*cos(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^p/(d*cos(b*x + a))^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{p}}{d^{3} \cos \left (b x + a\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^p/(d*cos(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*csc(b*x + a)^p/(d^3*cos(b*x + a)^3), 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(csc(b*x+a)**p/(d*cos(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 \frac{\csc \left (b x + a\right )^{p}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^p/(d*cos(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^p/(d*cos(b*x + a))^(5/2), x)