3.228 \(\int (d \sec (a+b x))^{5/2} \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=41 \[ \frac{2 d^3}{b \sqrt{d \sec (a+b x)}}+\frac{2 d (d \sec (a+b x))^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.048539, antiderivative size = 41, 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} \[ \frac{2 d^3}{b \sqrt{d \sec (a+b x)}}+\frac{2 d (d \sec (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[a + b*x])^(5/2)*Sin[a + b*x]^3,x]

[Out]

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

Mathematica [A]  time = 0.219513, size = 32, normalized size = 0.78 \[ \frac{d (3 \cos (2 (a+b x))+5) (d \sec (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[a + b*x])^(5/2)*Sin[a + b*x]^3,x]

[Out]

(d*(5 + 3*Cos[2*(a + b*x)])*(d*Sec[a + b*x])^(3/2))/(3*b)

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Maple [B]  time = 0.199, size = 357, normalized size = 8.7 \begin{align*} -{\frac{ \left ( -1+\cos \left ( bx+a \right ) \right ) \cos \left ( bx+a \right ) }{6\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}} \left ( 12\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{3}+12\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{2}-3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\ln \left ( -{\frac{1}{ \left ( \sin \left ( bx+a \right ) \right ) ^{2}} \left ( 2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{2}- \left ( \cos \left ( bx+a \right ) \right ) ^{2}-2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}+2\,\cos \left ( bx+a \right ) -1 \right ) } \right ) +3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\ln \left ( -2\,{\frac{1}{ \left ( \sin \left ( bx+a \right ) \right ) ^{2}} \left ( 2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( bx+a \right ) \right ) ^{2}- \left ( \cos \left ( bx+a \right ) \right ) ^{2}-2\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}+2\,\cos \left ( bx+a \right ) -1 \right ) } \right ) +4\,\cos \left ( bx+a \right ) \sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}+4\,\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}} \right ) \left ({\frac{d}{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{-{\frac{\cos \left ( bx+a \right ) }{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(b*x+a))^(5/2)*sin(b*x+a)^3,x)

[Out]

-1/6/b*(-1+cos(b*x+a))*(12*(-cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*cos(b*x+a)^3+12*(-cos(b*x+a)/(cos(b*x+a)+1)^2)
^(1/2)*cos(b*x+a)^2-3*cos(b*x+a)^2*ln(-(2*(-cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*cos(b*x+a)^2-cos(b*x+a)^2-2*(-c
os(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+2*cos(b*x+a)-1)/sin(b*x+a)^2)+3*cos(b*x+a)^2*ln(-2*(2*(-cos(b*x+a)/(cos(b*x+
a)+1)^2)^(1/2)*cos(b*x+a)^2-cos(b*x+a)^2-2*(-cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+2*cos(b*x+a)-1)/sin(b*x+a)^2)+
4*cos(b*x+a)*(-cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+4*(-cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2))*cos(b*x+a)*(d/cos(b*
x+a))^(5/2)/(-cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/sin(b*x+a)^2

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Maxima [A]  time = 1.09725, size = 49, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (\frac{3 \, d^{2}}{\sqrt{\frac{d}{\cos \left (b x + a\right )}}} + \left (\frac{d}{\cos \left (b x + a\right )}\right )^{\frac{3}{2}}\right )} d}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(5/2)*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

2/3*(3*d^2/sqrt(d/cos(b*x + a)) + (d/cos(b*x + a))^(3/2))*d/b

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Fricas [A]  time = 1.65811, size = 97, normalized size = 2.37 \begin{align*} \frac{2 \,{\left (3 \, d^{2} \cos \left (b x + a\right )^{2} + d^{2}\right )} \sqrt{\frac{d}{\cos \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(5/2)*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

2/3*(3*d^2*cos(b*x + a)^2 + d^2)*sqrt(d/cos(b*x + a))/(b*cos(b*x + a))

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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))**(5/2)*sin(b*x+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.18904, size = 66, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{d \cos \left (b x + a\right )} d + \frac{d^{2}}{\sqrt{d \cos \left (b x + a\right )} \cos \left (b x + a\right )}\right )} d \mathrm{sgn}\left (\cos \left (b x + a\right )\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(5/2)*sin(b*x+a)^3,x, algorithm="giac")

[Out]

2/3*(3*sqrt(d*cos(b*x + a))*d + d^2/(sqrt(d*cos(b*x + a))*cos(b*x + a)))*d*sgn(cos(b*x + a))/b