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

Optimal. Leaf size=85 \[ \frac {2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac {2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac {6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]

[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]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 = {2702, 276} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[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*} \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx &=\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]
time = 0.24, size = 58, normalized size = 0.68 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(516\) vs. \(2(71)=142\).
time = 2.74, size = 517, normalized size = 6.08

method	result	size
default	\(-\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (-60 \left (\cos ^{7}\left (f x +e \right )\right )+260 \left (\cos ^{5}\left (f x +e \right )\right )+195 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \ln \left (-\frac {2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1}{\sin \left (f x +e \right )^{2}}\right ) \cos \left (f x +e \right )-195 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \ln \left (-\frac {2 \left (2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1\right )}{\sin \left (f x +e \right )^{2}}\right ) \cos \left (f x +e \right )-468 \left (\cos ^{3}\left (f x +e \right )\right )+195 \ln \left (-\frac {2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1}{\sin \left (f x +e \right )^{2}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-195 \ln \left (-\frac {2 \left (2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1\right )}{\sin \left (f x +e \right )^{2}}\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 ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{390 f \sin \left (f x +e \right )^{4}}\)	\(517\)

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,method=_RETURNVERBOSE)

[Out]

-1/390/f*(-1+cos(f*x+e))^2*(-60*cos(f*x+e)^7+260*cos(f*x+e)^5+195*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*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)*cos(f*x+e)-195*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*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)*cos(f
*x+e)-468*cos(f*x+e)^3+195*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)^(1/2)-195*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)^(1/2)+780*cos(f*x+e))*(cos(f*x+e)+1)^2*(b/cos(f*x+e))^(1/2)/s
in(f*x+e)^4

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 67, normalized size = 0.79 \begin {gather*} \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}}} \end {gather*}

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]

2/195*(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))

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 61, normalized size = 0.72 \begin {gather*} \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} \end {gather*}

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]

2/195*(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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 5006 deep

________________________________________________________________________________________

Giac [A]
time = 2.66, size = 108, normalized size = 1.27 \begin {gather*} \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} \end {gather*}

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]

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]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\sin \left (e+f\,x\right )}^7\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________