\(\int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx\) [385]

Optimal result

Integrand size = 21, antiderivative size = 83 \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\frac {2 b^7}{11 f (b \sec (e+f x))^{11/2}}-\frac {6 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac {2 b^3}{f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f} \]

[Out]

2/11*b^7/f/(b*sec(f*x+e))^(11/2)-6/7*b^5/f/(b*sec(f*x+e))^(7/2)+2*b^3/f/(b*sec(f*x+e))^(3/2)+2*b*(b*sec(f*x+e)
)^(1/2)/f

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2702, 276} \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\frac {2 b^7}{11 f (b \sec (e+f x))^{11/2}}-\frac {6 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac {2 b^3}{f (b \sec (e+f x))^{3/2}}+\frac {2 b \sqrt {b \sec (e+f x)}}{f} \]

[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 276

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]

Rule 2702

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])

Rubi steps \begin{align*} \text {integral}& = \frac {b^7 \text {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 \text {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] (verified)

Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.63 \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\frac {b (3370+809 \cos (2 (e+f x))-90 \cos (4 (e+f x))+7 \cos (6 (e+f x))) \sqrt {b \sec (e+f x)}}{1232 f} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(71)=142\).

Time = 0.80 (sec) , antiderivative size = 256, normalized size of antiderivative = 3.08

method	result	size
default	\(-\frac {b \left (\left (77 \cos \left (f x +e \right )+77\right ) \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+\left (-77 \cos \left (f x +e \right )-77\right ) \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-28 \left (\cos ^{6}\left (f x +e \right )\right )+132 \left (\cos ^{4}\left (f x +e \right )\right )-308 \left (\cos ^{2}\left (f x +e \right )\right )-308\right ) \sqrt {b \sec \left (f x +e \right )}}{154 f}\)	\(256\)

[In]

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

[Out]

-1/154/f*b*((77*cos(f*x+e)+77)*ln(2*(2*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+2*(-cos(f*x+e)/(cos(f*x
+e)+1)^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e)+1))*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+(-77*cos(f*x+e)-77)*ln((2*c
os(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e
)+1))*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-28*cos(f*x+e)^6+132*cos(f*x+e)^4-308*cos(f*x+e)^2-308)*(b*sec(f*x+e
))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65 \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (7 \, b \cos \left (f x + e\right )^{6} - 33 \, b \cos \left (f x + e\right )^{4} + 77 \, b \cos \left (f x + e\right )^{2} + 77 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{77 \, f} \]

[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

Sympy [F(-1)]

Timed out. \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\frac {2 \, b {\left (\frac {7 \, b^{6}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {11}{2}}} - \frac {33 \, b^{4}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {7}{2}}} + \frac {77 \, b^{2}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}}} + 77 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{77 \, f} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.18 \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (7 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{5} - 33 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{3} + 77 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right ) + \frac {77 \, b^{6}}{\sqrt {b \cos \left (f x + e\right )}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{77 \, b^{4} f} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^7\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]

[In]

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

[Out]

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