∫ 1___________ (d sec(e+fx))3∕2(b tan(e+fx))3∕2 dx [326]

Optimal. Leaf size=72 \[ \frac {2}{3 b f (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}-\frac {8 \sqrt {d \sec (e+f x)}}{3 b d^2 f \sqrt {b \tan (e+f x)}} \]

2/3/b/f/(d*sec(f*x+e))^(3/2)/(b*tan(f*x+e))^(1/2)-8/3*(d*sec(f*x+e))^(1/2)/b/d^2/f/(b*tan(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 67, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2689, 2685} \begin {gather*} -\frac {8 (b \tan (e+f x))^{3/2}}{3 b^3 f (d \sec (e+f x))^{3/2}}-\frac {2}{b f \sqrt {b \tan (e+f x)} (d \sec (e+f x))^{3/2}} \end {gather*}

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

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] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 1, 0]

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]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.21, size = 52, normalized size = 0.72 \begin {gather*} \frac {(-7+\cos (2 (e+f x))) \sec ^2(e+f x)}{3 b f (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}} \end {gather*}

((-7 + Cos[2*(e + f*x)])*Sec[e + f*x]^2)/(3*b*f*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])

________________________________________________________________________________________


method	result	size

default	\(\frac {2 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )-4\right )}{3 f \cos \left (f x +e \right )^{3} \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}\)	\(60\)

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

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 72, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (\cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{3 \, b^{2} d^{2} f \sin \left (f x + e\right )} \end {gather*}

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

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

________________________________________________________________________________________

Sympy [A]
time = 37.90, size = 90, normalized size = 1.25 \begin {gather*} \begin {cases} - \frac {8 \tan ^{3}{\left (e + f x \right )}}{3 f \left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}} - \frac {2 \tan {\left (e + f x \right )}}{f \left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}} & \text {for}\: f \neq 0 \\\frac {x}{\left (b \tan {\left (e \right )}\right )^{\frac {3}{2}} \left (d \sec {\left (e \right )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Piecewise((-8*tan(e + f*x)**3/(3*f*(b*tan(e + f*x))**(3/2)*(d*sec(e + f*x))**(3/2)) - 2*tan(e + f*x)/(f*(b*tan

(e + f*x))**(3/2)*(d*sec(e + f*x))**(3/2)), Ne(f, 0)), (x/((b*tan(e))**(3/2)*(d*sec(e))**(3/2)), True))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 3.20, size = 60, normalized size = 0.83 \begin {gather*} \frac {\left (\cos \left (2\,e+2\,f\,x\right )-7\right )\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}}{3\,b\,d^2\,f\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}

((cos(2*e + 2*f*x) - 7)*(d/cos(e + f*x))^(1/2))/(3*b*d^2*f*((b*sin(2*e + 2*f*x))/(cos(2*e + 2*f*x) + 1))^(1/2)

________________________________________________________________________________________

3.4.26 \(\int \frac {1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}} \, dx\) [326]