3.385 \(\int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx\)

Optimal. Leaf size=83 \[ \frac{2 b^7}{11 f (b \sec (e+f x))^{11/2}}-\frac{6 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac{2 b^3}{f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)}}{f} \]

[Out]

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

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Rubi [A]  time = 0.0627426, antiderivative size = 83, 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} \[ \frac{2 b^7}{11 f (b \sec (e+f x))^{11/2}}-\frac{6 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac{2 b^3}{f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^7,x]

[Out]

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

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

Mathematica [A]  time = 0.142235, size = 52, normalized size = 0.63 \[ \frac{b (809 \cos (2 (e+f x))-90 \cos (4 (e+f x))+7 \cos (6 (e+f x))+3370) \sqrt{b \sec (e+f x)}}{1232 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^7,x]

[Out]

(b*(3370 + 809*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 7*Cos[6*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(1232*f)

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Maple [B]  time = 0.237, size = 969, normalized size = 11.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sec(f*x+e))^(3/2)*sin(f*x+e)^7,x)

[Out]

1/154/f*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^2*(28*cos(f*x+e)^7-77*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/
2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+
e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+77*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-c
os(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+
e)^2)-231*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)
^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+231*cos(f*x+e)^2*(-co
s(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos
(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-132*cos(f*x+e)^5-231*cos(f*x+e)*(-cos(f*x+e)/(
cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-
cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+231*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*
(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2
)^(1/2)-1)/sin(f*x+e)^2)-77*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)
-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)+77*ln(-2*(2*cos(
f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)
-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)+308*cos(f*x+e)^3+308*cos(f*x+e))*(b/cos(f*x+e))^(3/2)/s
in(f*x+e)^4

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Maxima [A]  time = 0.995003, size = 97, normalized size = 1.17 \begin{align*} \frac{2 \, b{\left (\frac{7 \, b^{6}}{\left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{11}{2}}} - \frac{33 \, b^{4}}{\left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{7}{2}}} + \frac{77 \, b^{2}}{\left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{3}{2}}} + 77 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}}\right )}}{77 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^7,x, algorithm="maxima")

[Out]

2/77*b*(7*b^6/(b/cos(f*x + e))^(11/2) - 33*b^4/(b/cos(f*x + e))^(7/2) + 77*b^2/(b/cos(f*x + e))^(3/2) + 77*sqr
t(b/cos(f*x + e)))/f

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Fricas [A]  time = 2.23216, size = 136, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (7 \, b \cos \left (f x + e\right )^{6} - 33 \, b \cos \left (f x + e\right )^{4} + 77 \, b \cos \left (f x + e\right )^{2} + 77 \, b\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{77 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^7,x, algorithm="fricas")

[Out]

2/77*(7*b*cos(f*x + e)^6 - 33*b*cos(f*x + e)^4 + 77*b*cos(f*x + e)^2 + 77*b)*sqrt(b/cos(f*x + e))/f

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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((b*sec(f*x+e))**(3/2)*sin(f*x+e)**7,x)

[Out]

Timed out

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Giac [A]  time = 1.14018, size = 143, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (7 \, \sqrt{b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{5} - 33 \, \sqrt{b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{3} + 77 \, \sqrt{b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right ) + \frac{77 \, b^{6}}{\sqrt{b \cos \left (f x + e\right )}}\right )} \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{77 \, b^{4} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^7,x, algorithm="giac")

[Out]

2/77*(7*sqrt(b*cos(f*x + e))*b^5*cos(f*x + e)^5 - 33*sqrt(b*cos(f*x + e))*b^5*cos(f*x + e)^3 + 77*sqrt(b*cos(f
*x + e))*b^5*cos(f*x + e) + 77*b^6/sqrt(b*cos(f*x + e)))*sgn(cos(f*x + e))/(b^4*f)