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]

8/3*b^3*sin(f*x+e)/f/(b*sec(f*x+e))^(3/2)+20/9*b^3*sin(f*x+e)^3/f/(b*sec(f*x+e))^(3/2)-16/3*b^2*(cos(1/2*f*x+1
/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(b*sec(f*x+e))^(1/2
)+2*b*sin(f*x+e)^5*(b*sec(f*x+e))^(1/2)/f

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Rubi [A]  time = 0.15, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 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]

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]

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*}

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Mathematica [A]  time = 0.15, 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]

-1/72*(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)]))/f

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b \cos \left (f x + e\right )^{6} - 3 \, b \cos \left (f x + e\right )^{4} + 3 \, b \cos \left (f x + e\right )^{2} - b\right )} \sqrt {b \sec \left (f x + e\right )} \sec \left (f x + e\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{6}\,{d x} \]

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)

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maple [C]  time = 0.29, size = 330, normalized size = 2.58 \[ \frac {2 \left (\cos ^{6}\left (f x +e \right )-24 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )+24 i \cos \left (f x +e \right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-24 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+24 i \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-5 \left (\cos ^{4}\left (f x +e \right )\right )+19 \left (\cos ^{2}\left (f x +e \right )\right )-24 \cos \left (f x +e \right )+9\right ) \cos \left (f x +e \right ) \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{9 f \sin \left (f x +e \right )} \]

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*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e)
)/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)+24*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)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-24*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+
1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)+24*I*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f
*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{6}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (e+f\,x\right )}^6\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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