3.86 \(\int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx\)

Optimal. Leaf size=185 \[ -\frac{a c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}} \]

[Out]

(a*c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c^3*Sqrt[c -
c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/
(2*f*Sqrt[a + a*Sec[e + f*x]]) - (a*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])

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Rubi [A]  time = 0.372323, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3906, 3905, 3475} \[ -\frac{a c^3 \tan (e+f x) \sqrt{c-c \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a}}-\frac{a c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt{a \sec (e+f x)+a}}+\frac{a c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2),x]

[Out]

(a*c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c^3*Sqrt[c -
c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/
(2*f*Sqrt[a + a*Sec[e + f*x]]) - (a*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])

Rule 3906

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp
[(2*a*c*Cot[e + f*x]*(c + d*Csc[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[c, Int[Sq
rt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d,
0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1/2]

Rule 3905

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(m_), x_Symbol] :> Dist
[((-(a*c))^(m + 1/2)*Cot[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Int[Cot[e + f*x]^(2*m)
, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m + 1/2]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx &=-\frac{a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+c \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx\\ &=-\frac{a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}-\frac{a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+c^2 \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac{a c^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}-\frac{a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+c^3 \int \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{a c^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}-\frac{a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}-\frac{\left (a c^4 \tan (e+f x)\right ) \int \tan (e+f x) \, dx}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{a c^4 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}-\frac{a c^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt{a+a \sec (e+f x)}}-\frac{a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 5.81818, size = 149, normalized size = 0.81 \[ \frac{c^3 \csc \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)} \left (-18 \cos (2 (e+f x))+3 i f x \cos (3 (e+f x))+9 \left (-\log \left (1+e^{2 i (e+f x)}\right )+i f x+2\right ) \cos (e+f x)-3 \log \left (1+e^{2 i (e+f x)}\right ) \cos (3 (e+f x))-22\right )}{24 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2),x]

[Out]

(c^3*Csc[(e + f*x)/2]*(-22 - 18*Cos[2*(e + f*x)] + (3*I)*f*x*Cos[3*(e + f*x)] + 9*Cos[e + f*x]*(2 + I*f*x - Lo
g[1 + E^((2*I)*(e + f*x))]) - 3*Cos[3*(e + f*x)]*Log[1 + E^((2*I)*(e + f*x))])*Sec[(e + f*x)/2]*Sec[e + f*x]^2
*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(24*f)

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Maple [A]  time = 0.356, size = 194, normalized size = 1.1 \begin{align*}{\frac{\cos \left ( fx+e \right ) }{6\,f\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3}} \left ( 6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) -6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -6\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -29\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-18\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+9\,\cos \left ( fx+e \right ) -2 \right ) \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{7}{2}}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sec(f*x+e))^(7/2)*(a+a*sec(f*x+e))^(1/2),x)

[Out]

1/6/f*(6*cos(f*x+e)^3*ln(2/(1+cos(f*x+e)))-6*cos(f*x+e)^3*ln((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-6*cos(f*x+e
)^3*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-29*cos(f*x+e)^3-18*cos(f*x+e)^2+9*cos(f*x+e)-2)*cos(f*x+e)*(c*(
-1+cos(f*x+e))/cos(f*x+e))^(7/2)*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)/sin(f*x+e)/(-1+cos(f*x+e))^3

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Maxima [B]  time = 2.14074, size = 1740, normalized size = 9.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^(7/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

-1/3*(3*(f*x + e)*c^3*cos(6*f*x + 6*e)^2 + 27*(f*x + e)*c^3*cos(4*f*x + 4*e)^2 + 27*(f*x + e)*c^3*cos(2*f*x +
2*e)^2 + 3*(f*x + e)*c^3*sin(6*f*x + 6*e)^2 + 27*(f*x + e)*c^3*sin(4*f*x + 4*e)^2 + 27*(f*x + e)*c^3*sin(2*f*x
+ 2*e)^2 + 18*(f*x + e)*c^3*cos(2*f*x + 2*e) + 3*(f*x + e)*c^3 + 18*c^3*sin(2*f*x + 2*e) - 3*(c^3*cos(6*f*x +
6*e)^2 + 9*c^3*cos(4*f*x + 4*e)^2 + 9*c^3*cos(2*f*x + 2*e)^2 + c^3*sin(6*f*x + 6*e)^2 + 9*c^3*sin(4*f*x + 4*e
)^2 + 18*c^3*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*c^3*sin(2*f*x + 2*e)^2 + 6*c^3*cos(2*f*x + 2*e) + c^3 + 2*(
3*c^3*cos(4*f*x + 4*e) + 3*c^3*cos(2*f*x + 2*e) + c^3)*cos(6*f*x + 6*e) + 6*(3*c^3*cos(2*f*x + 2*e) + c^3)*cos
(4*f*x + 4*e) + 6*(c^3*sin(4*f*x + 4*e) + c^3*sin(2*f*x + 2*e))*sin(6*f*x + 6*e))*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e) + 1) + 6*(3*(f*x + e)*c^3*cos(4*f*x + 4*e) + 3*(f*x + e)*c^3*cos(2*f*x + 2*e) + (f*x + e)*c^3 +
3*c^3*sin(4*f*x + 4*e) + 3*c^3*sin(2*f*x + 2*e))*cos(6*f*x + 6*e) + 18*(3*(f*x + e)*c^3*cos(2*f*x + 2*e) + (f
*x + e)*c^3)*cos(4*f*x + 4*e) + 18*(c^3*sin(6*f*x + 6*e) + 3*c^3*sin(4*f*x + 4*e) + 3*c^3*sin(2*f*x + 2*e))*co
s(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 44*(c^3*sin(6*f*x + 6*e) + 3*c^3*sin(4*f*x + 4*e) + 3*c^3
*sin(2*f*x + 2*e))*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 18*(c^3*sin(6*f*x + 6*e) + 3*c^3*sin
(4*f*x + 4*e) + 3*c^3*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 18*((f*x + e)*c
^3*sin(4*f*x + 4*e) + (f*x + e)*c^3*sin(2*f*x + 2*e) - c^3*cos(4*f*x + 4*e) - c^3*cos(2*f*x + 2*e))*sin(6*f*x
+ 6*e) + 18*(3*(f*x + e)*c^3*sin(2*f*x + 2*e) + c^3)*sin(4*f*x + 4*e) - 18*(c^3*cos(6*f*x + 6*e) + 3*c^3*cos(4
*f*x + 4*e) + 3*c^3*cos(2*f*x + 2*e) + c^3)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 44*(c^3*cos
(6*f*x + 6*e) + 3*c^3*cos(4*f*x + 4*e) + 3*c^3*cos(2*f*x + 2*e) + c^3)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) - 18*(c^3*cos(6*f*x + 6*e) + 3*c^3*cos(4*f*x + 4*e) + 3*c^3*cos(2*f*x + 2*e) + c^3)*sin(1/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((2*(3*cos(4*f*x + 4*e) + 3*cos(2*f*x + 2*e) + 1)*co
s(6*f*x + 6*e) + cos(6*f*x + 6*e)^2 + 6*(3*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 9*cos(4*f*x + 4*e)^2 + 9*c
os(2*f*x + 2*e)^2 + 6*(sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + sin(6*f*x + 6*e)^2 + 9*sin(4*f*
x + 4*e)^2 + 18*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 9*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e) + 1)*f)

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Fricas [A]  time = 1.69636, size = 1135, normalized size = 6.14 \begin{align*} \left [-\frac{{\left (11 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + 2 \, c^{3}\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 3 \,{\left (c^{3} \cos \left (f x + e\right )^{3} + c^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt{-a c} \log \left (\frac{a c \cos \left (f x + e\right )^{4} -{\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt{-a c} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{6 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, -\frac{{\left (11 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + 2 \, c^{3}\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 6 \,{\left (c^{3} \cos \left (f x + e\right )^{3} + c^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{6 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sec(f*x+e))^(7/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[-1/6*((11*c^3*cos(f*x + e)^2 - 7*c^3*cos(f*x + e) + 2*c^3)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*co
s(f*x + e) - c)/cos(f*x + e))*sin(f*x + e) - 3*(c^3*cos(f*x + e)^3 + c^3*cos(f*x + e)^2)*sqrt(-a*c)*log(1/2*(a
*c*cos(f*x + e)^4 - (cos(f*x + e)^3 + cos(f*x + e))*sqrt(-a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c
*cos(f*x + e) - c)/cos(f*x + e))*sin(f*x + e) + a*c)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 + f*cos(f*x + e)^2), -
1/6*((11*c^3*cos(f*x + e)^2 - 7*c^3*cos(f*x + e) + 2*c^3)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(
f*x + e) - c)/cos(f*x + e))*sin(f*x + e) - 6*(c^3*cos(f*x + e)^3 + c^3*cos(f*x + e)^2)*sqrt(a*c)*arctan(sqrt(a
*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e)/
(a*c*cos(f*x + e)^2 + a*c)))/(f*cos(f*x + e)^3 + f*cos(f*x + e)^2)]

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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((c-c*sec(f*x+e))**(7/2)*(a+a*sec(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [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((c-c*sec(f*x+e))^(7/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out