3.3.96 \(\int (b \csc (e+f x))^n (c \sec (e+f x))^{3/2} \, dx\) [296]

Optimal. Leaf size=81 \[ \frac {b \cos ^2(e+f x)^{5/4} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {5}{4},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right ) (c \sec (e+f x))^{5/2}}{c f (1-n)} \]

[Out]

b*(cos(f*x+e)^2)^(5/4)*(b*csc(f*x+e))^(-1+n)*hypergeom([5/4, 1/2-1/2*n],[3/2-1/2*n],sin(f*x+e)^2)*(c*sec(f*x+e
))^(5/2)/c/f/(1-n)

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Rubi [A]
time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2711, 2657} \begin {gather*} \frac {b \cos ^2(e+f x)^{5/4} (c \sec (e+f x))^{5/2} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {5}{4},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{c f (1-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Csc[e + f*x])^n*(c*Sec[e + f*x])^(3/2),x]

[Out]

(b*(Cos[e + f*x]^2)^(5/4)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[5/4, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^
2]*(c*Sec[e + f*x])^(5/2))/(c*f*(1 - n))

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 2711

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

Rubi steps

\begin {align*} \int (b \csc (e+f x))^n (c \sec (e+f x))^{3/2} \, dx &=\frac {\left (b^2 (c \cos (e+f x))^{5/2} (b \csc (e+f x))^{-1+n} (c \sec (e+f x))^{5/2} (b \sin (e+f x))^{-1+n}\right ) \int \frac {(b \sin (e+f x))^{-n}}{(c \cos (e+f x))^{3/2}} \, dx}{c^2}\\ &=\frac {b \cos ^2(e+f x)^{5/4} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {5}{4},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right ) (c \sec (e+f x))^{5/2}}{c f (1-n)}\\ \end {align*}

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Mathematica [A]
time = 11.93, size = 92, normalized size = 1.14 \begin {gather*} \frac {2 \cot (e+f x) (b \csc (e+f x))^n \, _2F_1\left (\frac {1+n}{2},\frac {1}{4} (3+2 n);\frac {1}{4} (7+2 n);\sec ^2(e+f x)\right ) (c \sec (e+f x))^{3/2} \left (-\tan ^2(e+f x)\right )^{\frac {1+n}{2}}}{f (3+2 n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Csc[e + f*x])^n*(c*Sec[e + f*x])^(3/2),x]

[Out]

(2*Cot[e + f*x]*(b*Csc[e + f*x])^n*Hypergeometric2F1[(1 + n)/2, (3 + 2*n)/4, (7 + 2*n)/4, Sec[e + f*x]^2]*(c*S
ec[e + f*x])^(3/2)*(-Tan[e + f*x]^2)^((1 + n)/2))/(f*(3 + 2*n))

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Maple [F]
time = 0.45, size = 0, normalized size = 0.00 \[\int \left (b \csc \left (f x +e \right )\right )^{n} \left (c \sec \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*csc(f*x+e))^n*(c*sec(f*x+e))^(3/2),x)

[Out]

int((b*csc(f*x+e))^n*(c*sec(f*x+e))^(3/2),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((b*csc(f*x+e))^n*(c*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((c*sec(f*x + e))^(3/2)*(b*csc(f*x + e))^n, 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((b*csc(f*x+e))^n*(c*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*sec(f*x + e))*(b*csc(f*x + e))^n*c*sec(f*x + e), 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((b*csc(f*x+e))**n*(c*sec(f*x+e))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3433 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((b*csc(f*x+e))^n*(c*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((c*sec(f*x + e))^(3/2)*(b*csc(f*x + e))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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