3.333 \(\int \frac{1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{8 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \sec (e+f x)}}{3 b^2 d^2 f \sqrt{b \tan (e+f x)}}-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{2}{3 b f (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{3/2}} \]

[Out]

-2/(3*b*f*(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2)) - (8*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[d*Sec[e +
f*x]]*Sqrt[Sin[e + f*x]])/(3*b^2*d^2*f*Sqrt[b*Tan[e + f*x]]) - (4*Sqrt[b*Tan[e + f*x]])/(3*b^3*f*(d*Sec[e + f*
x])^(3/2))

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Rubi [A]  time = 0.184395, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2609, 2612, 2616, 2642, 2641} \[ -\frac{8 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \sec (e+f x)}}{3 b^2 d^2 f \sqrt{b \tan (e+f x)}}-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{2}{3 b f (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(5/2)),x]

[Out]

-2/(3*b*f*(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2)) - (8*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[d*Sec[e +
f*x]]*Sqrt[Sin[e + f*x]])/(3*b^2*d^2*f*Sqrt[b*Tan[e + f*x]]) - (4*Sqrt[b*Tan[e + f*x]])/(3*b^3*f*(d*Sec[e + f*
x])^(3/2))

Rule 2609

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*Sec[e +
f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*(n + 1)), x] - Dist[(m + n + 1)/(b^2*(n + 1)), Int[(a*Sec[e + f*x])^m*(
b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && IntegersQ[2*m, 2*n]

Rule 2612

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((a*Sec[e +
f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*m), x] + Dist[(m + n + 1)/(a^2*m), Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[
e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (LtQ[m, -1] || (EqQ[m, -1] && EqQ[n, -2^(-1)])) && Integer
sQ[2*m, 2*n]

Rule 2616

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

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

Rubi steps

\begin{align*} \int \frac{1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2}} \, dx &=-\frac{2}{3 b f (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}-\frac{2 \int \frac{1}{(d \sec (e+f x))^{3/2} \sqrt{b \tan (e+f x)}} \, dx}{b^2}\\ &=-\frac{2}{3 b f (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{4 \int \frac{\sqrt{d \sec (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx}{3 b^2 d^2}\\ &=-\frac{2}{3 b f (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{\left (4 \sqrt{d \sec (e+f x)} \sqrt{b \sin (e+f x)}\right ) \int \frac{1}{\sqrt{b \sin (e+f x)}} \, dx}{3 b^2 d^2 \sqrt{b \tan (e+f x)}}\\ &=-\frac{2}{3 b f (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac{\left (4 \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\sin (e+f x)}} \, dx}{3 b^2 d^2 \sqrt{b \tan (e+f x)}}\\ &=-\frac{2}{3 b f (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}-\frac{8 F\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}{3 b^2 d^2 f \sqrt{b \tan (e+f x)}}-\frac{4 \sqrt{b \tan (e+f x)}}{3 b^3 f (d \sec (e+f x))^{3/2}}\\ \end{align*}

Mathematica [C]  time = 2.37566, size = 112, normalized size = 0.85 \[ \frac{\left (-\tan ^2(e+f x)\right )^{3/4} \csc ^2(e+f x) \sqrt{b \tan (e+f x)} \left (\sqrt [4]{-\tan ^2(e+f x)} \left (\cos (2 (e+f x))+2 \csc ^2(e+f x)-1\right )-8 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\sec ^2(e+f x)\right )\right )}{3 b^3 f (d \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(5/2)),x]

[Out]

(Csc[e + f*x]^2*Sqrt[b*Tan[e + f*x]]*(-Tan[e + f*x]^2)^(3/4)*(-8*Hypergeometric2F1[1/4, 3/4, 5/4, Sec[e + f*x]
^2] + (-1 + Cos[2*(e + f*x)] + 2*Csc[e + f*x]^2)*(-Tan[e + f*x]^2)^(1/4)))/(3*b^3*f*(d*Sec[e + f*x])^(3/2))

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Maple [C]  time = 0.209, size = 335, normalized size = 2.5 \begin{align*}{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{4}} \left ( -4\,i\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -i-\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{-i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }}}-4\,i\sin \left ( fx+e \right ) \sqrt{{\frac{-i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -i-\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) -i+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sqrt{2}-2\,\cos \left ( fx+e \right ) \sqrt{2} \right ) \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{3}{2}}} \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*sec(f*x+e))^(3/2)/(b*tan(f*x+e))^(5/2),x)

[Out]

1/3/f*2^(1/2)*(-4*I*cos(f*x+e)*sin(f*x+e)*((I*cos(f*x+e)-I+sin(f*x+e))/sin(f*x+e))^(1/2)*(-(I*cos(f*x+e)-I-sin
(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)-I+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-I*(cos(f*x+
e)-1)/sin(f*x+e))^(1/2)-4*I*sin(f*x+e)*(-I*(cos(f*x+e)-1)/sin(f*x+e))^(1/2)*((I*cos(f*x+e)-I+sin(f*x+e))/sin(f
*x+e))^(1/2)*(-(I*cos(f*x+e)-I-sin(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)-I+sin(f*x+e))/sin(f*x+e)
)^(1/2),1/2*2^(1/2))+cos(f*x+e)^3*2^(1/2)-2*cos(f*x+e)*2^(1/2))*sin(f*x+e)/(d/cos(f*x+e))^(3/2)/(b*sin(f*x+e)/
cos(f*x+e))^(5/2)/cos(f*x+e)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))^(3/2)/(b*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((d*sec(f*x + e))^(3/2)*(b*tan(f*x + e))^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))^(3/2)/(b*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*sec(f*x + e))*sqrt(b*tan(f*x + e))/(b^3*d^2*sec(f*x + e)^2*tan(f*x + e)^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(1/(d*sec(f*x+e))**(3/2)/(b*tan(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))^(3/2)/(b*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((d*sec(f*x + e))^(3/2)*(b*tan(f*x + e))^(5/2)), x)