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

Optimal. Leaf size=68 \[ -\frac{8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac{2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}} \]

[Out]

(-8*a^2*b*(a*Sin[e + f*x])^(3/2))/(21*f*(b*Tan[e + f*x])^(3/2)) - (2*b*(a*Sin[e + f*x])^(7/2))/(7*f*(b*Tan[e +
f*x])^(3/2))

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Rubi [A]  time = 0.101757, antiderivative size = 68, 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 = {2598, 2589} \[ -\frac{8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac{2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^2*b*(a*Sin[e + f*x])^(3/2))/(21*f*(b*Tan[e + f*x])^(3/2)) - (2*b*(a*Sin[e + f*x])^(7/2))/(7*f*(b*Tan[e +
f*x])^(3/2))

Rule 2598

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

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))^{7/2}}{\sqrt{b \tan (e+f x)}} \, dx &=-\frac{2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}}+\frac{1}{7} \left (4 a^2\right ) \int \frac{(a \sin (e+f x))^{3/2}}{\sqrt{b \tan (e+f x)}} \, dx\\ &=-\frac{8 a^2 b (a \sin (e+f x))^{3/2}}{21 f (b \tan (e+f x))^{3/2}}-\frac{2 b (a \sin (e+f x))^{7/2}}{7 f (b \tan (e+f x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.163352, size = 52, normalized size = 0.76 \[ \frac{a^3 \cos (e+f x) (3 \cos (2 (e+f x))-11) \sqrt{a \sin (e+f x)}}{21 f \sqrt{b \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Cos[e + f*x]*(-11 + 3*Cos[2*(e + f*x)])*Sqrt[a*Sin[e + f*x]])/(21*f*Sqrt[b*Tan[e + f*x]])

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Maple [A]  time = 0.148, size = 60, normalized size = 0.9 \begin{align*}{\frac{ \left ( 6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-14 \right ) \cos \left ( fx+e \right ) }{21\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21/f*(3*cos(f*x+e)^2-7)*(a*sin(f*x+e))^(7/2)*cos(f*x+e)/(b*sin(f*x+e)/cos(f*x+e))^(1/2)/sin(f*x+e)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65726, size = 170, normalized size = 2.5 \begin{align*} \frac{2 \,{\left (3 \, a^{3} \cos \left (f x + e\right )^{4} - 7 \, a^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{21 \, b f \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*a^3*cos(f*x + e)^4 - 7*a^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b*sin(f*x + e)/cos(f*x + e))/(b*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((a*sin(f*x+e))**(7/2)/(b*tan(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{\left (a \sin \left (f x + e\right )\right )^{\frac{7}{2}}}{\sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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