3.444 \(\int \frac{\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}} \]

[Out]

(8*EllipticE[(e + f*x)/2, 2])/(65*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (4*b*Sin[e + f*x])/(39*f*(b
*Sec[e + f*x])^(7/2)) + (8*Sin[e + f*x])/(195*b*f*(b*Sec[e + f*x])^(3/2)) - (2*b*Sin[e + f*x]^3)/(13*f*(b*Sec[
e + f*x])^(7/2))

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Rubi [A]  time = 0.13199, antiderivative size = 126, 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 = {2627, 3769, 3771, 2639} \[ \frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^4/(b*Sec[e + f*x])^(5/2),x]

[Out]

(8*EllipticE[(e + f*x)/2, 2])/(65*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (4*b*Sin[e + f*x])/(39*f*(b
*Sec[e + f*x])^(7/2)) + (8*Sin[e + f*x])/(195*b*f*(b*Sec[e + f*x])^(3/2)) - (2*b*Sin[e + f*x]^3)/(13*f*(b*Sec[
e + f*x])^(7/2))

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 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 \frac{\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{6}{13} \int \frac{\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{4}{39} \int \frac{1}{(b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{4 \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx}{65 b^2}\\ &=-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{4 \int \sqrt{\cos (e+f x)} \, dx}{65 b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=\frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.4978, size = 83, normalized size = 0.66 \[ \frac{192 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )+(-6 \sin (e+f x)-55 \sin (3 (e+f x))+15 \sin (5 (e+f x))) \cos ^{\frac{3}{2}}(e+f x)}{1560 f \cos ^{\frac{5}{2}}(e+f x) (b \sec (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^4/(b*Sec[e + f*x])^(5/2),x]

[Out]

(192*EllipticE[(e + f*x)/2, 2] + Cos[e + f*x]^(3/2)*(-6*Sin[e + f*x] - 55*Sin[3*(e + f*x)] + 15*Sin[5*(e + f*x
)]))/(1560*f*Cos[e + f*x]^(5/2)*(b*Sec[e + f*x])^(5/2))

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Maple [C]  time = 0.158, size = 343, normalized size = 2.7 \begin{align*}{\frac{2}{195\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sin \left ( fx+e \right ) } \left ( -15\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+12\,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 ) -12\,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}}}+40\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+12\,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 ) -12\,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}}}-29\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+12\,\cos \left ( fx+e \right ) \right ) \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^4/(b*sec(f*x+e))^(5/2),x)

[Out]

2/195/f*(-15*cos(f*x+e)^8+12*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)-12*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)+40*cos(f*x+e)^6+12*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)-12*I*sin(f*x+e)*EllipticE
(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)-29*cos(f*x+e)^4-8*
cos(f*x+e)^2+12*cos(f*x+e))/cos(f*x+e)^3/sin(f*x+e)/(b/cos(f*x+e))^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(b*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(sin(f*x + e)^4/(b*sec(f*x + e))^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(b*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(b*sec(f*x + e))/(b^3*sec(f*x + e)^3), 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(sin(f*x+e)**4/(b*sec(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(b*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate(sin(f*x + e)^4/(b*sec(f*x + e))^(5/2), x)