3.410 \(\int \frac{\sin ^7(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\)

Optimal. Leaf size=87 \[ \frac{2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac{6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac{6 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}} \]

[Out]

(2*b^7)/(15*f*(b*Sec[e + f*x])^(15/2)) - (6*b^5)/(11*f*(b*Sec[e + f*x])^(11/2)) + (6*b^3)/(7*f*(b*Sec[e + f*x]
)^(7/2)) - (2*b)/(3*f*(b*Sec[e + f*x])^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0570967, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2622, 270} \[ \frac{2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac{6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac{6 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^7/Sqrt[b*Sec[e + f*x]],x]

[Out]

(2*b^7)/(15*f*(b*Sec[e + f*x])^(15/2)) - (6*b^5)/(11*f*(b*Sec[e + f*x])^(11/2)) + (6*b^3)/(7*f*(b*Sec[e + f*x]
)^(7/2)) - (2*b)/(3*f*(b*Sec[e + f*x])^(3/2))

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\sin ^7(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=\frac{b^7 \operatorname{Subst}\left (\int \frac{\left (-1+\frac{x^2}{b^2}\right )^3}{x^{17/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac{b^7 \operatorname{Subst}\left (\int \left (-\frac{1}{x^{17/2}}+\frac{3}{b^2 x^{13/2}}-\frac{3}{b^4 x^{9/2}}+\frac{1}{b^6 x^{5/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac{2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac{6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac{6 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.218208, size = 52, normalized size = 0.6 \[ \frac{b (4035 \cos (2 (e+f x))-798 \cos (4 (e+f x))+77 \cos (6 (e+f x))-7410)}{18480 f (b \sec (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^7/Sqrt[b*Sec[e + f*x]],x]

[Out]

(b*(-7410 + 4035*Cos[2*(e + f*x)] - 798*Cos[4*(e + f*x)] + 77*Cos[6*(e + f*x)]))/(18480*f*(b*Sec[e + f*x])^(3/
2))

________________________________________________________________________________________

Maple [A]  time = 0.155, size = 56, normalized size = 0.6 \begin{align*}{\frac{ \left ( 154\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}-630\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+990\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-770 \right ) \cos \left ( fx+e \right ) }{1155\,f}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^7/(b*sec(f*x+e))^(1/2),x)

[Out]

2/1155/f*(77*cos(f*x+e)^6-315*cos(f*x+e)^4+495*cos(f*x+e)^2-385)*cos(f*x+e)/(b/cos(f*x+e))^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.00202, size = 85, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (77 \, b^{6} - \frac{315 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac{495 \, b^{6}}{\cos \left (f x + e\right )^{4}} - \frac{385 \, b^{6}}{\cos \left (f x + e\right )^{6}}\right )} b}{1155 \, f \left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{15}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^7/(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

2/1155*(77*b^6 - 315*b^6/cos(f*x + e)^2 + 495*b^6/cos(f*x + e)^4 - 385*b^6/cos(f*x + e)^6)*b/(f*(b/cos(f*x + e
))^(15/2))

________________________________________________________________________________________

Fricas [A]  time = 2.71872, size = 159, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (77 \, \cos \left (f x + e\right )^{8} - 315 \, \cos \left (f x + e\right )^{6} + 495 \, \cos \left (f x + e\right )^{4} - 385 \, \cos \left (f x + e\right )^{2}\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{1155 \, b f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^7/(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/1155*(77*cos(f*x + e)^8 - 315*cos(f*x + e)^6 + 495*cos(f*x + e)^4 - 385*cos(f*x + e)^2)*sqrt(b/cos(f*x + e))
/(b*f)

________________________________________________________________________________________

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)**7/(b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{7}}{\sqrt{b \sec \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^7/(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sin(f*x + e)^7/sqrt(b*sec(f*x + e)), x)