3.130 \(\int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx\)
Optimal. Leaf size=42 \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{6 f (c-c \sec (e+f x))^{7/2}} \]
[Out]
-((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(6*f*(c - c*Sec[e + f*x])^(7/2))
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Rubi [A] time = 0.145583, antiderivative size = 42, 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 =
{3950} \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^{5/2}}{6 f (c-c \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(7/2),x]
[Out]
-((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(6*f*(c - c*Sec[e + f*x])^(7/2))
Rule 3950
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
^(n_.), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/(a*f*(2*m + 1)), x] /
; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[2*m
+ 1, 0]
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{7/2}} \, dx &=-\frac{(a+a \sec (e+f x))^{5/2} \tan (e+f x)}{6 f (c-c \sec (e+f x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.550775, size = 76, normalized size = 1.81 \[ \frac{a^2 (3 \cos (2 (e+f x))+5) \csc ^5\left (\frac{1}{2} (e+f x)\right ) \sec \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)}}{48 c^3 f \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(7/2),x]
[Out]
(a^2*(5 + 3*Cos[2*(e + f*x)])*Csc[(e + f*x)/2]^5*Sec[(e + f*x)/2]*Sqrt[a*(1 + Sec[e + f*x])])/(48*c^3*f*Sqrt[c
- c*Sec[e + f*x]])
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Maple [B] time = 0.248, size = 75, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}}{6\,f \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{7}{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))^(5/2)/(c-c*sec(f*x+e))^(7/2),x)
[Out]
-1/6/f*a^2*sin(f*x+e)^5*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)/(-1+cos(f*x+e))^2/cos(f*x+e)^3/(c*(-1+cos(f*x+e)
)/cos(f*x+e))^(7/2)
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Maxima [B] time = 1.92277, size = 2450, normalized size = 58.33 \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)*(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")
[Out]
2/3*(208*a^2*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) + 48*a^2*cos(f*x + e)*sin(2*f*x + 2*e) - 48*a^2*cos(2*f*x + 2*e
)*sin(f*x + e) - 3*a^2*sin(f*x + e) - (3*a^2*sin(7*f*x + 7*e) + 13*a^2*sin(5*f*x + 5*e) + 13*a^2*sin(3*f*x + 3
*e) + 3*a^2*sin(f*x + e))*cos(8*f*x + 8*e) + 6*(8*a^2*sin(6*f*x + 6*e) + 15*a^2*sin(4*f*x + 4*e) + 8*a^2*sin(2
*f*x + 2*e))*cos(7*f*x + 7*e) - 16*(13*a^2*sin(5*f*x + 5*e) + 13*a^2*sin(3*f*x + 3*e) + 3*a^2*sin(f*x + e))*co
s(6*f*x + 6*e) + 26*(15*a^2*sin(4*f*x + 4*e) + 8*a^2*sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - 30*(13*a^2*sin(3*f*x
+ 3*e) + 3*a^2*sin(f*x + e))*cos(4*f*x + 4*e) + (3*a^2*cos(7*f*x + 7*e) + 13*a^2*cos(5*f*x + 5*e) + 13*a^2*co
s(3*f*x + 3*e) + 3*a^2*cos(f*x + e))*sin(8*f*x + 8*e) - 3*(16*a^2*cos(6*f*x + 6*e) + 30*a^2*cos(4*f*x + 4*e) +
16*a^2*cos(2*f*x + 2*e) + a^2)*sin(7*f*x + 7*e) + 16*(13*a^2*cos(5*f*x + 5*e) + 13*a^2*cos(3*f*x + 3*e) + 3*a
^2*cos(f*x + e))*sin(6*f*x + 6*e) - 13*(30*a^2*cos(4*f*x + 4*e) + 16*a^2*cos(2*f*x + 2*e) + a^2)*sin(5*f*x + 5
*e) + 30*(13*a^2*cos(3*f*x + 3*e) + 3*a^2*cos(f*x + e))*sin(4*f*x + 4*e) - 13*(16*a^2*cos(2*f*x + 2*e) + a^2)*
sin(3*f*x + 3*e))*sqrt(a)*sqrt(c)/((c^4*cos(8*f*x + 8*e)^2 + 36*c^4*cos(7*f*x + 7*e)^2 + 256*c^4*cos(6*f*x + 6
*e)^2 + 676*c^4*cos(5*f*x + 5*e)^2 + 900*c^4*cos(4*f*x + 4*e)^2 + 676*c^4*cos(3*f*x + 3*e)^2 + 256*c^4*cos(2*f
*x + 2*e)^2 + 36*c^4*cos(f*x + e)^2 + c^4*sin(8*f*x + 8*e)^2 + 36*c^4*sin(7*f*x + 7*e)^2 + 256*c^4*sin(6*f*x +
6*e)^2 + 676*c^4*sin(5*f*x + 5*e)^2 + 900*c^4*sin(4*f*x + 4*e)^2 + 676*c^4*sin(3*f*x + 3*e)^2 + 256*c^4*sin(2
*f*x + 2*e)^2 - 192*c^4*sin(2*f*x + 2*e)*sin(f*x + e) + 36*c^4*sin(f*x + e)^2 - 12*c^4*cos(f*x + e) + c^4 - 2*
(6*c^4*cos(7*f*x + 7*e) - 16*c^4*cos(6*f*x + 6*e) + 26*c^4*cos(5*f*x + 5*e) - 30*c^4*cos(4*f*x + 4*e) + 26*c^4
*cos(3*f*x + 3*e) - 16*c^4*cos(2*f*x + 2*e) + 6*c^4*cos(f*x + e) - c^4)*cos(8*f*x + 8*e) - 12*(16*c^4*cos(6*f*
x + 6*e) - 26*c^4*cos(5*f*x + 5*e) + 30*c^4*cos(4*f*x + 4*e) - 26*c^4*cos(3*f*x + 3*e) + 16*c^4*cos(2*f*x + 2*
e) - 6*c^4*cos(f*x + e) + c^4)*cos(7*f*x + 7*e) - 32*(26*c^4*cos(5*f*x + 5*e) - 30*c^4*cos(4*f*x + 4*e) + 26*c
^4*cos(3*f*x + 3*e) - 16*c^4*cos(2*f*x + 2*e) + 6*c^4*cos(f*x + e) - c^4)*cos(6*f*x + 6*e) - 52*(30*c^4*cos(4*
f*x + 4*e) - 26*c^4*cos(3*f*x + 3*e) + 16*c^4*cos(2*f*x + 2*e) - 6*c^4*cos(f*x + e) + c^4)*cos(5*f*x + 5*e) -
60*(26*c^4*cos(3*f*x + 3*e) - 16*c^4*cos(2*f*x + 2*e) + 6*c^4*cos(f*x + e) - c^4)*cos(4*f*x + 4*e) - 52*(16*c^
4*cos(2*f*x + 2*e) - 6*c^4*cos(f*x + e) + c^4)*cos(3*f*x + 3*e) - 32*(6*c^4*cos(f*x + e) - c^4)*cos(2*f*x + 2*
e) - 4*(3*c^4*sin(7*f*x + 7*e) - 8*c^4*sin(6*f*x + 6*e) + 13*c^4*sin(5*f*x + 5*e) - 15*c^4*sin(4*f*x + 4*e) +
13*c^4*sin(3*f*x + 3*e) - 8*c^4*sin(2*f*x + 2*e) + 3*c^4*sin(f*x + e))*sin(8*f*x + 8*e) - 24*(8*c^4*sin(6*f*x
+ 6*e) - 13*c^4*sin(5*f*x + 5*e) + 15*c^4*sin(4*f*x + 4*e) - 13*c^4*sin(3*f*x + 3*e) + 8*c^4*sin(2*f*x + 2*e)
- 3*c^4*sin(f*x + e))*sin(7*f*x + 7*e) - 64*(13*c^4*sin(5*f*x + 5*e) - 15*c^4*sin(4*f*x + 4*e) + 13*c^4*sin(3*
f*x + 3*e) - 8*c^4*sin(2*f*x + 2*e) + 3*c^4*sin(f*x + e))*sin(6*f*x + 6*e) - 104*(15*c^4*sin(4*f*x + 4*e) - 13
*c^4*sin(3*f*x + 3*e) + 8*c^4*sin(2*f*x + 2*e) - 3*c^4*sin(f*x + e))*sin(5*f*x + 5*e) - 120*(13*c^4*sin(3*f*x
+ 3*e) - 8*c^4*sin(2*f*x + 2*e) + 3*c^4*sin(f*x + e))*sin(4*f*x + 4*e) - 104*(8*c^4*sin(2*f*x + 2*e) - 3*c^4*s
in(f*x + e))*sin(3*f*x + 3*e))*f)
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Fricas [B] time = 0.481269, size = 298, normalized size = 7.1 \begin{align*} \frac{{\left (3 \, a^{2} \cos \left (f x + e\right )^{3} + a^{2} \cos \left (f x + e\right )\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 \,{\left (c^{4} f \cos \left (f x + e\right )^{3} - 3 \, c^{4} f \cos \left (f x + e\right )^{2} + 3 \, c^{4} f \cos \left (f x + e\right ) - c^{4} f\right )} \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))^(5/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")
[Out]
1/3*(3*a^2*cos(f*x + e)^3 + a^2*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) - c
)/cos(f*x + e))/((c^4*f*cos(f*x + e)^3 - 3*c^4*f*cos(f*x + e)^2 + 3*c^4*f*cos(f*x + e) - c^4*f)*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))**(5/2)/(c-c*sec(f*x+e))**(7/2),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)*(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")
[Out]
Timed out