\(\int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\) [461]

Optimal result

Integrand size = 23, antiderivative size = 115 \[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {7 b \sin ^{\frac {3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7 E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{20 f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \]

[Out]

-7/30*b*sin(f*x+e)^(3/2)/f/(b*sec(f*x+e))^(3/2)-1/5*b*sin(f*x+e)^(7/2)/f/(b*sec(f*x+e))^(3/2)-7/20*(sin(e+1/4*
Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*sin(f*x+e)^(1/2)/f/(b*sec(f*x+e))^(1/2
)/sin(2*f*x+2*e)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2663, 2665, 2652, 2719} \[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac {7 b \sin ^{\frac {3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}+\frac {7 \sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{20 f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}} \]

[In]

Int[Sin[e + f*x]^(9/2)/Sqrt[b*Sec[e + f*x]],x]

[Out]

(-7*b*Sin[e + f*x]^(3/2))/(30*f*(b*Sec[e + f*x])^(3/2)) - (b*Sin[e + f*x]^(7/2))/(5*f*(b*Sec[e + f*x])^(3/2))
+ (7*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(20*f*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2663

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

Rule 2665

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(b*Cos[e + f*
x])^n*(b*Sec[e + f*x])^n, Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&
IntegerQ[m - 1/2] && IntegerQ[n - 1/2]

Rule 2719

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

Rubi steps \begin{align*} \text {integral}& = -\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7}{10} \int \frac {\sin ^{\frac {5}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx \\ & = -\frac {7 b \sin ^{\frac {3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7}{20} \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx \\ & = -\frac {7 b \sin ^{\frac {3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7 \int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)} \, dx}{20 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}} \\ & = -\frac {7 b \sin ^{\frac {3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {\left (7 \sqrt {\sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{20 \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {7 b \sin ^{\frac {3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac {b \sin ^{\frac {7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac {7 E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{20 f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.67 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {b \left (23-26 \cos (2 (e+f x))+3 \cos (4 (e+f x))+42 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {1}{2},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}\right )}{120 f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}} \]

[In]

Integrate[Sin[e + f*x]^(9/2)/Sqrt[b*Sec[e + f*x]],x]

[Out]

-1/120*(b*(23 - 26*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] + 42*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2
]*(-Tan[e + f*x]^2)^(1/4)))/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs. \(2(120)=240\).

Time = 0.98 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.46

method	result	size
default	\(-\frac {\sqrt {2}\, \left (12 \sqrt {2}\, \left (\cos ^{5}\left (f x +e \right )\right )-38 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-21 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )+42 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, E\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right )+47 \sqrt {2}\, \cos \left (f x +e \right )-21 \sqrt {2}\right )}{120 f \sqrt {b \sec \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right )}}\)	\(398\)

[In]

int(sin(f*x+e)^(9/2)/(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/120/f*2^(1/2)/(b*sec(f*x+e))^(1/2)/sin(f*x+e)^(1/2)*(12*2^(1/2)*cos(f*x+e)^5-38*2^(1/2)*cos(f*x+e)^3-21*(-c
ot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot(f*
x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))+42*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot
(f*x+e)-csc(f*x+e))^(1/2)*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))-21*(-cot(f*x+e)+csc(f*x+e)+1
)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/
2),1/2*2^(1/2))*sec(f*x+e)+42*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc
(f*x+e))^(1/2)*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*sec(f*x+e)+47*2^(1/2)*cos(f*x+e)-21*2^(
1/2))

Fricas [F]

\[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{\frac {9}{2}}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]

[In]

integrate(sin(f*x+e)^(9/2)/(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))*sqrt(sin(f*x + e))/(b*sec(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\text {Timed out} \]

[In]

integrate(sin(f*x+e)**(9/2)/(b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{\frac {9}{2}}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]

[In]

integrate(sin(f*x+e)^(9/2)/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sin(f*x + e)^(9/2)/sqrt(b*sec(f*x + e)), x)

Giac [F]

\[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{\frac {9}{2}}}{\sqrt {b \sec \left (f x + e\right )}} \,d x } \]

[In]

integrate(sin(f*x+e)^(9/2)/(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sin(f*x + e)^(9/2)/sqrt(b*sec(f*x + e)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^{\frac {9}{2}}(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^{9/2}}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int(sin(e + f*x)^(9/2)/(b/cos(e + f*x))^(1/2),x)

[Out]

int(sin(e + f*x)^(9/2)/(b/cos(e + f*x))^(1/2), x)