3.379 \(\int \sqrt {b \sec (e+f x)} \sin ^4(e+f x) \, dx\)

Optimal. Leaf size=95 \[ -\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{7 f} \]

[Out]

-4/7*b*sin(f*x+e)/f/(b*sec(f*x+e))^(1/2)-2/7*b*sin(f*x+e)^3/f/(b*sec(f*x+e))^(1/2)+8/7*(cos(1/2*f*x+1/2*e)^2)^
(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(b*sec(f*x+e))^(1/2)/f

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Rubi [A]  time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2627, 3771, 2641} \[ -\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{7 f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^4,x]

[Out]

(8*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(7*f) - (4*b*Sin[e + f*x])/(7*f*Sqrt[b*S
ec[e + f*x]]) - (2*b*Sin[e + f*x]^3)/(7*f*Sqrt[b*Sec[e + f*x]])

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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(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 \sqrt {b \sec (e+f x)} \sin ^4(e+f x) \, dx &=-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {6}{7} \int \sqrt {b \sec (e+f x)} \sin ^2(e+f x) \, dx\\ &=-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {4}{7} \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {1}{7} \left (4 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{7 f}-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 61, normalized size = 0.64 \[ \frac {\sqrt {b \sec (e+f x)} \left (-10 \sin (2 (e+f x))+\sin (4 (e+f x))+32 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{28 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^4,x]

[Out]

(Sqrt[b*Sec[e + f*x]]*(32*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - 10*Sin[2*(e + f*x)] + Sin[4*(e + f*x)
]))/(28*f)

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b \sec \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(b*sec(f*x + e)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4*(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(f*x + e))*sin(f*x + e)^4, x)

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maple [C]  time = 0.23, size = 143, normalized size = 1.51 \[ \frac {2 \left (-1+\cos \left (f x +e \right )\right ) \left (-4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+\cos ^{4}\left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right )-3 \left (\cos ^{2}\left (f x +e \right )\right )+3 \cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{7 f \sin \left (f x +e \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^4*(b*sec(f*x+e))^(1/2),x)

[Out]

2/7/f*(-1+cos(f*x+e))*(-4*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)^4-cos(f*x+e)^3-3*cos(f*x+e)^2+3*cos(f*x+e))*(cos(f*x+e)+1)^2*(b/cos(f
*x+e))^(1/2)/sin(f*x+e)^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sec(f*x + e))*sin(f*x + e)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (e+f\,x\right )}^4\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^4*(b/cos(e + f*x))^(1/2),x)

[Out]

int(sin(e + f*x)^4*(b/cos(e + f*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)**4*(b*sec(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(b*sec(e + f*x))*sin(e + f*x)**4, x)

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