3.464 \(\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{3}{2}}(e+f x)} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 b}{f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac{2 \sqrt{\sin (e+f x)} E\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}} \]

[Out]

(-2*b)/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]]) - (2*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(f*
Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])

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Rubi [A]  time = 0.119976, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2584, 2585, 2572, 2639} \[ -\frac{2 b}{f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac{2 \sqrt{\sin (e+f x)} E\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b)/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]]) - (2*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(f*
Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*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 2585

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

Rule 2572

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(Sqrt[a*Sin[e +
f*x]]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[2*e + 2*f*x]], Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{3}{2}}(e+f x)} \, dx &=-\frac{2 b}{f (b \sec (e+f x))^{3/2} \sqrt{\sin (e+f x)}}-2 \int \frac{\sqrt{\sin (e+f x)}}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{2 b}{f (b \sec (e+f x))^{3/2} \sqrt{\sin (e+f x)}}-\frac{2 \int \sqrt{b \cos (e+f x)} \sqrt{\sin (e+f x)} \, dx}{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{2 b}{f (b \sec (e+f x))^{3/2} \sqrt{\sin (e+f x)}}-\frac{\left (2 \sqrt{\sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{\sqrt{b \sec (e+f x)} \sqrt{\sin (2 e+2 f x)}}\\ &=-\frac{2 b}{f (b \sec (e+f x))^{3/2} \sqrt{\sin (e+f x)}}-\frac{2 E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{\sin (e+f x)}}{f \sqrt{b \sec (e+f x)} \sqrt{\sin (2 e+2 f x)}}\\ \end{align*}

Mathematica [C]  time = 0.245375, size = 63, normalized size = 0.78 \[ \frac{2 b \left (\sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right )-1\right )}{f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*(-1 + Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(f*(b*Sec[e + f*x])^(3/
2)*Sqrt[Sin[e + f*x]])

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Maple [B]  time = 0.109, size = 484, normalized size = 6. \begin{align*}{\frac{\sqrt{2}}{f\cos \left ( fx+e \right ) } \left ( 2\,\cos \left ( fx+e \right ) \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) -\cos \left ( fx+e \right ) \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +2\,\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) -\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) -\sqrt{2}\cos \left ( fx+e \right ) \right ){\frac{1}{\sqrt{\sin \left ( fx+e \right ) }}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*2^(1/2)*(2*cos(f*x+e)*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))
^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))-
cos(f*x+e)*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos
(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))+2*((1-cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/
2)*EllipticE(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))-((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^
(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((1-cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))-2^(1/2)*cos(f*x+e))/cos(f*x+e)/sin(f*x+e)^(1/2)/(b/cos(f*x+e))^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \sec \left (f x + e\right )} \sqrt{\sin \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right )^{2} - b\right )} \sec \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(b*sec(f*x + e))*sqrt(sin(f*x + e))/((b*cos(f*x + e)^2 - b)*sec(f*x + e)), 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(1/sin(f*x+e)**(3/2)/(b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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