3.80 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{99 f}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3}{693 f \sqrt{c-c \sec (e+f x)}}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f} \]

[Out]

(-64*c^3*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(693*f*Sqrt[c - c*Sec[e + f*x]]) - (16*c^2*(a + a*Sec[e + f*x])^
3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(99*f) - (2*c*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)*Tan[e
+ f*x])/(11*f)

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Rubi [A]  time = 0.323442, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3955, 3953} \[ -\frac{16 c^2 \tan (e+f x) (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}}{99 f}-\frac{64 c^3 \tan (e+f x) (a \sec (e+f x)+a)^3}{693 f \sqrt{c-c \sec (e+f x)}}-\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{3/2}}{11 f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2),x]

[Out]

(-64*c^3*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(693*f*Sqrt[c - c*Sec[e + f*x]]) - (16*c^2*(a + a*Sec[e + f*x])^
3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(99*f) - (2*c*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)*Tan[e
+ f*x])/(11*f)

Rule 3955

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

Rule 3953

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

Rubi steps

\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2} \, dx &=-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{11 f}+\frac{1}{11} (8 c) \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac{16 c^2 (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{99 f}-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{11 f}+\frac{1}{99} \left (32 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \, dx\\ &=-\frac{64 c^3 (a+a \sec (e+f x))^3 \tan (e+f x)}{693 f \sqrt{c-c \sec (e+f x)}}-\frac{16 c^2 (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \tan (e+f x)}{99 f}-\frac{2 c (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{11 f}\\ \end{align*}

Mathematica [A]  time = 1.54198, size = 78, normalized size = 0.61 \[ \frac{8 a^3 c^2 \cos ^6\left (\frac{1}{2} (e+f x)\right ) (-364 \cos (e+f x)+151 \cos (2 (e+f x))+277) \cot \left (\frac{1}{2} (e+f x)\right ) \sec ^5(e+f x) \sqrt{c-c \sec (e+f x)}}{693 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2),x]

[Out]

(8*a^3*c^2*Cos[(e + f*x)/2]^6*(277 - 364*Cos[e + f*x] + 151*Cos[2*(e + f*x)])*Cot[(e + f*x)/2]*Sec[e + f*x]^5*
Sqrt[c - c*Sec[e + f*x]])/(693*f)

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Maple [A]  time = 0.208, size = 75, normalized size = 0.6 \begin{align*}{\frac{2\,{a}^{3} \left ( 151\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-182\,\cos \left ( fx+e \right ) +63 \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{7}}{693\,f \left ( -1+\cos \left ( fx+e \right ) \right ) ^{6} \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x)

[Out]

2/693*a^3/f*(151*cos(f*x+e)^2-182*cos(f*x+e)+63)*sin(f*x+e)^7*(c*(-1+cos(f*x+e))/cos(f*x+e))^(5/2)/(-1+cos(f*x
+e))^6/cos(f*x+e)^3

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Maxima [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(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.488722, size = 355, normalized size = 2.77 \begin{align*} \frac{2 \,{\left (151 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} + 422 \, a^{3} c^{2} \cos \left (f x + e\right )^{5} + 241 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} - 236 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} - 199 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{3} c^{2} \cos \left (f x + e\right ) + 63 \, a^{3} c^{2}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{693 \, f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

2/693*(151*a^3*c^2*cos(f*x + e)^6 + 422*a^3*c^2*cos(f*x + e)^5 + 241*a^3*c^2*cos(f*x + e)^4 - 236*a^3*c^2*cos(
f*x + e)^3 - 199*a^3*c^2*cos(f*x + e)^2 + 70*a^3*c^2*cos(f*x + e) + 63*a^3*c^2)*sqrt((c*cos(f*x + e) - c)/cos(
f*x + e))/(f*cos(f*x + e)^5*sin(f*x + e))

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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(sec(f*x+e)*(a+a*sec(f*x+e))**3*(c-c*sec(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 3.54377, size = 116, normalized size = 0.91 \begin{align*} \frac{64 \, \sqrt{2}{\left (99 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{2} c^{5} + 154 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} c^{6} + 63 \, c^{7}\right )} a^{3} c}{693 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{11}{2}} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

64/693*sqrt(2)*(99*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^5 + 154*(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^6 + 63*c^7)*a^3
*c/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(11/2)*f)