3.136 \(\int \frac {(a \sin (e+f x))^{5/2}}{(b \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac {2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 32, 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 = {2589} \[ -\frac {2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sin[e + f*x])^(5/2)/(b*Tan[e + f*x])^(3/2),x]

[Out]

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

Rule 2589

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 45, normalized size = 1.41 \[ -\frac {2 a^2 \cos ^2(e+f x) \sqrt {a \sin (e+f x)}}{5 b f \sqrt {b \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[e + f*x])^(5/2)/(b*Tan[e + f*x])^(3/2),x]

[Out]

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

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fricas [B]  time = 0.64, size = 55, normalized size = 1.72 \[ -\frac {2 \, \sqrt {a \sin \left (f x + e\right )} a^{2} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{3}}{5 \, b^{2} f \sin \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(f*x + e))^(5/2)/(b*tan(f*x + e))^(3/2), x)

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maple [A]  time = 0.45, size = 48, normalized size = 1.50 \[ -\frac {2 \left (a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \cos \left (f x +e \right )}{5 f \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(f*x+e))^(5/2)/(b*tan(f*x+e))^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(f*x + e))^(5/2)/(b*tan(f*x + e))^(3/2), x)

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mupad [B]  time = 4.01, size = 81, normalized size = 2.53 \[ \frac {a^2\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\right )\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{10\,b^2\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(e + f*x))^(5/2)/(b*tan(e + f*x))^(3/2),x)

[Out]

(a^2*(a*sin(e + f*x))^(1/2)*(2*sin(2*e + 2*f*x) + sin(4*e + 4*f*x))*((b*sin(2*e + 2*f*x))/(cos(2*e + 2*f*x) +
1))^(1/2))/(10*b^2*f*(cos(2*e + 2*f*x) - 1))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(f*x+e))**(5/2)/(b*tan(f*x+e))**(3/2),x)

[Out]

Timed out

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