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

Optimal. Leaf size=43 \[ \frac {2 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]

[Out]

2/7*b^3/f/(b*sec(f*x+e))^(7/2)-2/3*b/f/(b*sec(f*x+e))^(3/2)

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Rubi [A]  time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2622, 14} \[ \frac {2 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \frac {\sin ^3(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {-1+\frac {x^2}{b^2}}{x^{9/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {b^3 \operatorname {Subst}\left (\int \left (-\frac {1}{x^{9/2}}+\frac {1}{b^2 x^{5/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {2 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 32, normalized size = 0.74 \[ \frac {b (3 \cos (2 (e+f x))-11)}{21 f (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(-11 + 3*Cos[2*(e + f*x)]))/(21*f*(b*Sec[e + f*x])^(3/2))

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fricas [A]  time = 0.61, size = 41, normalized size = 0.95 \[ \frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{4} - 7 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{21 \, b f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*cos(f*x + e)^4 - 7*cos(f*x + e)^2)*sqrt(b/cos(f*x + e))/(b*f)

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giac [A]  time = 1.69, size = 50, normalized size = 1.16 \[ \frac {2 \, {\left (3 \, b^{4} - \frac {7 \, b^{4}}{\cos \left (f x + e\right )^{2}}\right )} \cos \left (f x + e\right )^{3}}{21 \, b^{4} f \sqrt {\frac {b}{\cos \left (f x + e\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*b^4 - 7*b^4/cos(f*x + e)^2)*cos(f*x + e)^3/(b^4*f*sqrt(b/cos(f*x + e)))

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maple [A]  time = 0.16, size = 36, normalized size = 0.84 \[ \frac {2 \left (3 \left (\cos ^{2}\left (f x +e \right )\right )-7\right ) \cos \left (f x +e \right )}{21 f \sqrt {\frac {b}{\cos \left (f x +e \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 37, normalized size = 0.86 \[ \frac {2 \, {\left (3 \, b^{2} - \frac {7 \, b^{2}}{\cos \left (f x + e\right )^{2}}\right )} b}{21 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (e+f\,x\right )}^3}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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