\(\int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx\) [371]

Optimal result

Integrand size = 21, antiderivative size = 85 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac {2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac {6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]

[Out]

2/13*b^7/f/(b*sec(f*x+e))^(13/2)-2/3*b^5/f/(b*sec(f*x+e))^(9/2)+6/5*b^3/f/(b*sec(f*x+e))^(5/2)-2*b/f/(b*sec(f*
x+e))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, 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} \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac {2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac {6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]

[In]

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

[Out]

(2*b^7)/(13*f*(b*Sec[e + f*x])^(13/2)) - (2*b^5)/(3*f*(b*Sec[e + f*x])^(9/2)) + (6*b^3)/(5*f*(b*Sec[e + f*x])^
(5/2)) - (2*b)/(f*Sqrt[b*Sec[e + f*x]])

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

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.68 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {(-8939 \cos (e+f x)+887 \cos (3 (e+f x))-155 \cos (5 (e+f x))+15 \cos (7 (e+f x))) \sqrt {b \sec (e+f x)}}{6240 f} \]

[In]

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

[Out]

((-8939*Cos[e + f*x] + 887*Cos[3*(e + f*x)] - 155*Cos[5*(e + f*x)] + 15*Cos[7*(e + f*x)])*Sqrt[b*Sec[e + f*x]]
)/(6240*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(444\) vs. \(2(71)=142\).

Time = 1.01 (sec) , antiderivative size = 445, normalized size of antiderivative = 5.24

method	result	size
default	\(\frac {\left (60 \left (\cos ^{7}\left (f x +e \right )\right )-260 \left (\cos ^{5}\left (f x +e \right )\right )+468 \left (\cos ^{3}\left (f x +e \right )\right )-195 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )+195 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-195 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+195 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-780 \cos \left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{390 f}\)	\(445\)

[In]

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

[Out]

1/390/f*(60*cos(f*x+e)^7-260*cos(f*x+e)^5+468*cos(f*x+e)^3-195*ln((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)*cos(f*x+e)+195*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)*cos(f*x+e)-195*ln((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)+195*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)-780*cos(f*x+e))*(b*sec(f*x+e)
)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.66 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, \cos \left (f x + e\right )^{7} - 65 \, \cos \left (f x + e\right )^{5} + 117 \, \cos \left (f x + e\right )^{3} - 195 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{195 \, f} \]

[In]

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

[Out]

2/195*(15*cos(f*x + e)^7 - 65*cos(f*x + e)^5 + 117*cos(f*x + e)^3 - 195*cos(f*x + e))*sqrt(b/cos(f*x + e))/f

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.74 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, b^{6} - \frac {65 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac {117 \, b^{6}}{\cos \left (f x + e\right )^{4}} - \frac {195 \, b^{6}}{\cos \left (f x + e\right )^{6}}\right )} b}{195 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {13}{2}}} \]

[In]

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

[Out]

2/195*(15*b^6 - 65*b^6/cos(f*x + e)^2 + 117*b^6/cos(f*x + e)^4 - 195*b^6/cos(f*x + e)^6)*b/(f*(b/cos(f*x + e))
^(13/2))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{6} - 65 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{4} + 117 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{2} - 195 \, \sqrt {b \cos \left (f x + e\right )} b^{6}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{195 \, b^{6} f} \]

[In]

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

[Out]

2/195*(15*sqrt(b*cos(f*x + e))*b^6*cos(f*x + e)^6 - 65*sqrt(b*cos(f*x + e))*b^6*cos(f*x + e)^4 + 117*sqrt(b*co
s(f*x + e))*b^6*cos(f*x + e)^2 - 195*sqrt(b*cos(f*x + e))*b^6)*sgn(cos(f*x + e))/(b^6*f)

Mupad [F(-1)]

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

[In]

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

[Out]

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