3.229 \(\int (d \sec (a+b x))^{9/2} \sin ^3(a+b x) \, dx\)

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

[Out]

(-2*d^3*(d*Sec[a + b*x])^(3/2))/(3*b) + (2*d*(d*Sec[a + b*x])^(7/2))/(7*b)

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Rubi [A]  time = 0.0485292, antiderivative size = 43, normalized size of antiderivative = 1., 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} \[ \frac{2 d (d \sec (a+b x))^{7/2}}{7 b}-\frac{2 d^3 (d \sec (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[a + b*x])^(9/2)*Sin[a + b*x]^3,x]

[Out]

(-2*d^3*(d*Sec[a + b*x])^(3/2))/(3*b) + (2*d*(d*Sec[a + b*x])^(7/2))/(7*b)

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])

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]]

Rubi steps

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

Mathematica [A]  time = 0.102751, size = 42, normalized size = 0.98 \[ -\frac{d^4 (7 \cos (2 (a+b x))+1) \sec ^3(a+b x) \sqrt{d \sec (a+b x)}}{21 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[a + b*x])^(9/2)*Sin[a + b*x]^3,x]

[Out]

-(d^4*(1 + 7*Cos[2*(a + b*x)])*Sec[a + b*x]^3*Sqrt[d*Sec[a + b*x]])/(21*b)

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Maple [A]  time = 0.149, size = 36, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 14\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-6 \right ) \cos \left ( bx+a \right ) }{21\,b} \left ({\frac{d}{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(b*x+a))^(9/2)*sin(b*x+a)^3,x)

[Out]

-2/21/b*(7*cos(b*x+a)^2-3)*(d/cos(b*x+a))^(9/2)*cos(b*x+a)

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Maxima [A]  time = 1.2052, size = 51, normalized size = 1.19 \begin{align*} -\frac{2 \,{\left (7 \, d^{2} \left (\frac{d}{\cos \left (b x + a\right )}\right )^{\frac{3}{2}} - 3 \, \left (\frac{d}{\cos \left (b x + a\right )}\right )^{\frac{7}{2}}\right )} d}{21 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(9/2)*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-2/21*(7*d^2*(d/cos(b*x + a))^(3/2) - 3*(d/cos(b*x + a))^(7/2))*d/b

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Fricas [A]  time = 1.69515, size = 105, normalized size = 2.44 \begin{align*} -\frac{2 \,{\left (7 \, d^{4} \cos \left (b x + a\right )^{2} - 3 \, d^{4}\right )} \sqrt{\frac{d}{\cos \left (b x + a\right )}}}{21 \, b \cos \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(9/2)*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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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((d*sec(b*x+a))**(9/2)*sin(b*x+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.3429, size = 66, normalized size = 1.53 \begin{align*} -\frac{2 \,{\left (7 \, d^{5} \cos \left (b x + a\right )^{2} - 3 \, d^{5}\right )} \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{21 \, \sqrt{d \cos \left (b x + a\right )} b \cos \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(9/2)*sin(b*x+a)^3,x, algorithm="giac")

[Out]

-2/21*(7*d^5*cos(b*x + a)^2 - 3*d^5)*sgn(cos(b*x + a))/(sqrt(d*cos(b*x + a))*b*cos(b*x + a)^3)