3.391 \(\int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx\)

Optimal. Leaf size=128 \[ \frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{16 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 b \sin ^5(e+f x) \sqrt{b \sec (e+f x)}}{f} \]

[Out]

(-16*b^2*EllipticE[(e + f*x)/2, 2])/(3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (8*b^3*Sin[e + f*x])/(3*f*
(b*Sec[e + f*x])^(3/2)) + (20*b^3*Sin[e + f*x]^3)/(9*f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin
[e + f*x]^5)/f

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Rubi [A]  time = 0.148598, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2624, 2627, 3771, 2639} \[ \frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{16 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 b \sin ^5(e+f x) \sqrt{b \sec (e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^6,x]

[Out]

(-16*b^2*EllipticE[(e + f*x)/2, 2])/(3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (8*b^3*Sin[e + f*x])/(3*f*
(b*Sec[e + f*x])^(3/2)) + (20*b^3*Sin[e + f*x]^3)/(9*f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin
[e + f*x]^5)/f

Rule 2624

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Csc[e +
f*x])^(m + 1)*(b*Sec[e + f*x])^(n - 1))/(f*a*(n - 1)), x] + Dist[(b^2*(m + 1))/(a^2*(n - 1)), Int[(a*Csc[e +
f*x])^(m + 2)*(b*Sec[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && LtQ[m, -1] && Integer
sQ[2*m, 2*n]

Rule 2627

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx &=\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\left (10 b^2\right ) \int \frac{\sin ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\frac{1}{3} \left (20 b^2\right ) \int \frac{\sin ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\frac{1}{3} \left (8 b^2\right ) \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\frac{\left (8 b^2\right ) \int \sqrt{\cos (e+f x)} \, dx}{3 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{16 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.150728, size = 70, normalized size = 0.55 \[ -\frac{b \sqrt{b \sec (e+f x)} \left (-158 \sin (e+f x)-13 \sin (3 (e+f x))+\sin (5 (e+f x))+384 \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{72 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^6,x]

[Out]

-(b*Sqrt[b*Sec[e + f*x]]*(384*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] - 158*Sin[e + f*x] - 13*Sin[3*(e +
f*x)] + Sin[5*(e + f*x)]))/(72*f)

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Maple [C]  time = 0.233, size = 330, normalized size = 2.6 \begin{align*}{\frac{2\,\cos \left ( fx+e \right ) }{9\,f\sin \left ( fx+e \right ) } \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{6}-24\,i\cos \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) +24\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-24\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) +24\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-5\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+19\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-24\,\cos \left ( fx+e \right ) +9 \right ) \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x)

[Out]

2/9/f*(cos(f*x+e)^6-24*I*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*
x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+24*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)*sin(f*x+e)*(1/
(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-24*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos
(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+24*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*
sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-5*cos(f*x+e)^4+19*cos(f*x+e)^2-24*cos(f*
x+e)+9)*cos(f*x+e)*(b/cos(f*x+e))^(3/2)/sin(f*x+e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e))^(3/2)*sin(f*x + e)^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (b \cos \left (f x + e\right )^{6} - 3 \, b \cos \left (f x + e\right )^{4} + 3 \, b \cos \left (f x + e\right )^{2} - b\right )} \sqrt{b \sec \left (f x + e\right )} \sec \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x, algorithm="fricas")

[Out]

integral(-(b*cos(f*x + e)^6 - 3*b*cos(f*x + e)^4 + 3*b*cos(f*x + e)^2 - b)*sqrt(b*sec(f*x + e))*sec(f*x + e),
x)

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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((b*sec(f*x+e))**(3/2)*sin(f*x+e)**6,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{6}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^6,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e))^(3/2)*sin(f*x + e)^6, x)