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

Optimal. Leaf size=43 \[ \frac{a \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}} \]

[Out]

(a*(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 = 0.132595, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {3953} \[ \frac{a \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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) \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx &=\frac{a (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}

Mathematica [B]  time = 0.463943, size = 87, normalized size = 2.02 \[ \frac{c^2 (-6 \cos (e+f x)+3 \cos (2 (e+f x))+5) \csc \left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}}{12 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(5 - 6*Cos[e + f*x] + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[a*(1 + Se
c[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(12*f)

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Maple [B]  time = 0.3, size = 82, normalized size = 1.9 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( 7\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4\,\cos \left ( fx+e \right ) +1 \right ) }{3\,f \left ( -1+\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}}}\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(sec(f*x+e)*(c-c*sec(f*x+e))^(5/2)*(a+a*sec(f*x+e))^(1/2),x)

[Out]

-1/3/f*sin(f*x+e)*(7*cos(f*x+e)^2-4*cos(f*x+e)+1)*(c*(-1+cos(f*x+e))/cos(f*x+e))^(5/2)*(1/cos(f*x+e)*a*(1+cos(
f*x+e)))^(1/2)/(-1+cos(f*x+e))^3

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Maxima [B]  time = 1.87654, size = 861, normalized size = 20.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(5/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

2/3*(30*c^2*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 9*c^2*cos(2*f*x + 2*e)*sin(f*x + e) - 3*c^2*sin(f*x + e) - (3*
c^2*sin(5*f*x + 5*e) - 6*c^2*sin(4*f*x + 4*e) + 10*c^2*sin(3*f*x + 3*e) - 6*c^2*sin(2*f*x + 2*e) + 3*c^2*sin(f
*x + e))*cos(6*f*x + 6*e) + 9*(c^2*sin(4*f*x + 4*e) + c^2*sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - 3*(10*c^2*sin(3
*f*x + 3*e) + 3*c^2*sin(f*x + e))*cos(4*f*x + 4*e) + (3*c^2*cos(5*f*x + 5*e) - 6*c^2*cos(4*f*x + 4*e) + 10*c^2
*cos(3*f*x + 3*e) - 6*c^2*cos(2*f*x + 2*e) + 3*c^2*cos(f*x + e))*sin(6*f*x + 6*e) - 3*(3*c^2*cos(4*f*x + 4*e)
+ 3*c^2*cos(2*f*x + 2*e) + c^2)*sin(5*f*x + 5*e) + 3*(10*c^2*cos(3*f*x + 3*e) + 3*c^2*cos(f*x + e) + 2*c^2)*si
n(4*f*x + 4*e) - 10*(3*c^2*cos(2*f*x + 2*e) + c^2)*sin(3*f*x + 3*e) + 3*(3*c^2*cos(f*x + e) + 2*c^2)*sin(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)*cos(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*cos(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 [B]  time = 0.473063, size = 225, normalized size = 5.23 \begin{align*} \frac{{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\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 )}}}{3 \, f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(5/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/3*(3*c^2*cos(f*x + e)^2 - 3*c^2*cos(f*x + e) + c^2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x
+ e) - c)/cos(f*x + e))/(f*cos(f*x + e)^2*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)*(c-c*sec(f*x+e))**(5/2)*(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(sec(f*x+e)*(c-c*sec(f*x+e))^(5/2)*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out