3.454 \(\int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{8 b}{5 a^3 f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b}{5 a f (a \sin (e+f x))^{5/2} \sqrt{b \sec (e+f x)}} \]

[Out]

(-2*b)/(5*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(5/2)) - (8*b)/(5*a^3*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e
+ f*x]])

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Rubi [A]  time = 0.108842, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2584, 2578} \[ -\frac{8 b}{5 a^3 f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b}{5 a f (a \sin (e+f x))^{5/2} \sqrt{b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b)/(5*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(5/2)) - (8*b)/(5*a^3*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e
+ f*x]])

Rule 2584

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] + Dist[(m - n + 2)/(a^2*(m + 1)), Int[(a*Sin[e + f*
x])^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2578

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{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{7/2}} \, dx &=-\frac{2 b}{5 a f \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{5/2}}+\frac{4 \int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{3/2}} \, dx}{5 a^2}\\ &=-\frac{2 b}{5 a f \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{5/2}}-\frac{8 b}{5 a^3 f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.197654, size = 52, normalized size = 0.73 \[ \frac{2 (2 \cos (2 (e+f x))-3) \cot (e+f x) \sqrt{b \sec (e+f x)}}{5 a^2 f (a \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.122, size = 52, normalized size = 0.7 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-10 \right ) \cos \left ( fx+e \right ) \sin \left ( fx+e \right ) }{5\,f}\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}} \left ( a\sin \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5/f*(4*cos(f*x+e)^2-5)*cos(f*x+e)*(b/cos(f*x+e))^(1/2)*sin(f*x+e)/(a*sin(f*x+e))^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.61584, size = 176, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left (4 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{5 \,{\left (a^{4} f \cos \left (f x + e\right )^{2} - a^{4} f\right )} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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