3.4.79 \(\int \sqrt {b \sec (e+f x)} \sin ^4(e+f x) \, dx\) [379]

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

[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 = 0.07, 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 = {2707, 3856, 2720} \begin {gather*} -\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} \end {gather*}

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 2707

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 2720

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

Rule 3856

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 = 0.08, size = 61, normalized size = 0.64 \begin {gather*} \frac {\sqrt {b \sec (e+f x)} \left (32 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )-10 \sin (2 (e+f x))+\sin (4 (e+f x))\right )}{28 f} \end {gather*}

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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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 143, normalized size = 1.51

method	result	size
default	\(-\frac {2 \left (-1+\cos \left (f x +e \right )\right ) \left (4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-\left (\cos ^{4}\left (f x +e \right )\right )+\cos ^{3}\left (f x +e \right )+3 \left (\cos ^{2}\left (f x +e \right )\right )-3 \cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{7 f \sin \left (f x +e \right )^{3}}\)	\(143\)

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,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 102, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left ({\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 2 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{7 \, f} \end {gather*}

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]

2/7*((cos(f*x + e)^3 - 3*cos(f*x + e))*sqrt(b/cos(f*x + e))*sin(f*x + e) - 2*I*sqrt(2)*sqrt(b)*weierstrassPInv
erse(-4, 0, cos(f*x + e) + I*sin(f*x + e)) + 2*I*sqrt(2)*sqrt(b)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*s
in(f*x + e)))/f

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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