3.5.76 \(\int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{7/2}} \, dx\) [476]

Optimal. Leaf size=35 \[ -\frac {2 b}{5 a f (b \sec (e+f x))^{5/2} (a \sin (e+f x))^{5/2}} \]

[Out]

-2/5*b/a/f/(b*sec(f*x+e))^(5/2)/(a*sin(f*x+e))^(5/2)

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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2658} \begin {gather*} -\frac {2 b}{5 a f (a \sin (e+f x))^{5/2} (b \sec (e+f x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(7/2)),x]

[Out]

(-2*b)/(5*a*f*(b*Sec[e + f*x])^(5/2)*(a*Sin[e + f*x])^(5/2))

Rule 2658

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

Rubi steps

\begin {align*} \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{7/2}} \, dx &=-\frac {2 b}{5 a f (b \sec (e+f x))^{5/2} (a \sin (e+f x))^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 45, normalized size = 1.29 \begin {gather*} -\frac {2 \cot ^3(e+f x) \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}}{5 a^4 b^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(7/2)),x]

[Out]

(-2*Cot[e + f*x]^3*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])/(5*a^4*b^2*f)

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Maple [A]
time = 0.18, size = 40, normalized size = 1.14

method	result	size
default	\(-\frac {2 \cos \left (f x +e \right ) \sin \left (f x +e \right )}{5 f \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/f*cos(f*x+e)*sin(f*x+e)/(b/cos(f*x+e))^(3/2)/(a*sin(f*x+e))^(7/2)

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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(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(7/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (31) = 62\).
time = 0.42, size = 73, normalized size = 2.09 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{3}}{5 \, {\left (a^{4} b^{2} f \cos \left (f x + e\right )^{2} - a^{4} b^{2} f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/5*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))*cos(f*x + e)^3/((a^4*b^2*f*cos(f*x + e)^2 - a^4*b^2*f)*sin(f*x +
e))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*sec(f*x+e))**(3/2)/(a*sin(f*x+e))**(7/2),x)

[Out]

Timed out

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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(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate(1/((b*sec(f*x + e))^(3/2)*(a*sin(f*x + e))^(7/2)), x)

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Mupad [B]
time = 1.86, size = 84, normalized size = 2.40 \begin {gather*} \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}\,\left (\cos \left (3\,e+3\,f\,x\right )-2\,\cos \left (e+f\,x\right )+\cos \left (5\,e+5\,f\,x\right )\right )}{5\,a^3\,b^2\,f\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\cos \left (4\,e+4\,f\,x\right )-4\,\cos \left (2\,e+2\,f\,x\right )+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*sin(e + f*x))^(7/2)*(b/cos(e + f*x))^(3/2)),x)

[Out]

((b/cos(e + f*x))^(1/2)*(cos(3*e + 3*f*x) - 2*cos(e + f*x) + cos(5*e + 5*f*x)))/(5*a^3*b^2*f*(a*sin(e + f*x))^
(1/2)*(cos(4*e + 4*f*x) - 4*cos(2*e + 2*f*x) + 3))

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