3.113 \(\int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \, dx\)
Optimal. Leaf size=89 \[ \frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{7/2}}{10 f \sqrt {a \sec (e+f x)+a}}+\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}{5 f} \]
[Out]
1/10*a^2*(c-c*sec(f*x+e))^(7/2)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+1/5*a*(c-c*sec(f*x+e))^(7/2)*(a+a*sec(f*x+
e))^(1/2)*tan(f*x+e)/f
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Rubi [A] time = 0.28, antiderivative size = 89, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used =
{3955, 3953} \[ \frac {a^2 \tan (e+f x) (c-c \sec (e+f x))^{7/2}}{10 f \sqrt {a \sec (e+f x)+a}}+\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}{5 f} \]
Antiderivative was successfully verified.
[In]
Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(7/2),x]
[Out]
(a^2*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(10*f*Sqrt[a + a*Sec[e + f*x]]) + (a*Sqrt[a + a*Sec[e + f*x]]*(c
- c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(5*f)
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)]
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])
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{7/2} \, dx &=\frac {a \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{5 f}+\frac {1}{5} (2 a) \int \sec (e+f x) \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx\\ &=\frac {a^2 (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{10 f \sqrt {a+a \sec (e+f x)}}+\frac {a \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \tan (e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 108, normalized size = 1.21 \[ \frac {a c^3 (-10 \cos (e+f x)+20 \cos (2 (e+f x))-10 \cos (3 (e+f x))+5 \cos (4 (e+f x))+7) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^4(e+f x) \sqrt {a (\sec (e+f x)+1)} \sqrt {c-c \sec (e+f x)}}{80 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(7/2),x]
[Out]
(a*c^3*(7 - 10*Cos[e + f*x] + 20*Cos[2*(e + f*x)] - 10*Cos[3*(e + f*x)] + 5*Cos[4*(e + f*x)])*Csc[(e + f*x)/2]
*Sec[(e + f*x)/2]*Sec[e + f*x]^4*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])/(80*f)
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fricas [A] time = 0.47, size = 112, normalized size = 1.26 \[ \frac {{\left (10 \, a c^{3} \cos \left (f x + e\right )^{4} - 10 \, a c^{3} \cos \left (f x + e\right )^{3} + 5 \, a c^{3} \cos \left (f x + e\right ) - 2 \, a c^{3}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{10 \, f \cos \left (f x + e\right )^{4} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="fricas")
[Out]
1/10*(10*a*c^3*cos(f*x + e)^4 - 10*a*c^3*cos(f*x + e)^3 + 5*a*c^3*cos(f*x + e) - 2*a*c^3)*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)^4*sin(f*x + e))
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)8/5*a*c*sqrt(-a*c)*(-15*c^5*(c*tan(1/2*(f*x+exp(1)))^2-c)-4*c^6-10*c^3*(c*tan(1/2*(f*x+exp(1)))^2-c)^3-20*
c^4*(c*tan(1/2*(f*x+exp(1)))^2-c)^2)*abs(c)*sign(tan(1/2*(f*x+exp(1)))^3+tan(1/2*(f*x+exp(1))))/(c*tan(1/2*(f*
x+exp(1)))^2-c)^5/f
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maple [A] time = 1.96, size = 103, normalized size = 1.16 \[ \frac {\left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \left (13 \left (\cos ^{3}\left (f x +e \right )\right )-16 \left (\cos ^{2}\left (f x +e \right )\right )+9 \cos \left (f x +e \right )-2\right ) \left (\sin ^{3}\left (f x +e \right )\right ) a}{10 f \left (-1+\cos \left (f x +e \right )\right )^{5} \cos \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)*(c-c*sec(f*x+e))^(7/2),x)
[Out]
1/10/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(7/2)*(a*(1+cos(f*x+e))/cos(f*x+e))^(1/2)*(13*cos(f*x+e)^3-16*cos(f*x+e)
^2+9*cos(f*x+e)-2)*sin(f*x+e)^3/(-1+cos(f*x+e))^5/cos(f*x+e)*a
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maxima [B] time = 0.91, size = 1680, normalized size = 18.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="maxima")
[Out]
2/5*(100*a*c^3*cos(3*f*x + 3*e)*sin(2*f*x + 2*e) - 25*a*c^3*cos(2*f*x + 2*e)*sin(f*x + e) - 5*a*c^3*sin(f*x +
e) - (5*a*c^3*sin(9*f*x + 9*e) - 10*a*c^3*sin(8*f*x + 8*e) + 20*a*c^3*sin(7*f*x + 7*e) - 10*a*c^3*sin(6*f*x +
6*e) + 14*a*c^3*sin(5*f*x + 5*e) - 10*a*c^3*sin(4*f*x + 4*e) + 20*a*c^3*sin(3*f*x + 3*e) - 10*a*c^3*sin(2*f*x
+ 2*e) + 5*a*c^3*sin(f*x + e))*cos(10*f*x + 10*e) + 25*(a*c^3*sin(8*f*x + 8*e) + 2*a*c^3*sin(6*f*x + 6*e) + 2*
a*c^3*sin(4*f*x + 4*e) + a*c^3*sin(2*f*x + 2*e))*cos(9*f*x + 9*e) - 5*(20*a*c^3*sin(7*f*x + 7*e) + 10*a*c^3*si
n(6*f*x + 6*e) + 14*a*c^3*sin(5*f*x + 5*e) + 10*a*c^3*sin(4*f*x + 4*e) + 20*a*c^3*sin(3*f*x + 3*e) + 5*a*c^3*s
in(f*x + e))*cos(8*f*x + 8*e) + 100*(2*a*c^3*sin(6*f*x + 6*e) + 2*a*c^3*sin(4*f*x + 4*e) + a*c^3*sin(2*f*x + 2
*e))*cos(7*f*x + 7*e) - 10*(14*a*c^3*sin(5*f*x + 5*e) + 20*a*c^3*sin(3*f*x + 3*e) - 5*a*c^3*sin(2*f*x + 2*e) +
5*a*c^3*sin(f*x + e))*cos(6*f*x + 6*e) + 70*(2*a*c^3*sin(4*f*x + 4*e) + a*c^3*sin(2*f*x + 2*e))*cos(5*f*x + 5
*e) - 50*(4*a*c^3*sin(3*f*x + 3*e) - a*c^3*sin(2*f*x + 2*e) + a*c^3*sin(f*x + e))*cos(4*f*x + 4*e) + (5*a*c^3*
cos(9*f*x + 9*e) - 10*a*c^3*cos(8*f*x + 8*e) + 20*a*c^3*cos(7*f*x + 7*e) - 10*a*c^3*cos(6*f*x + 6*e) + 14*a*c^
3*cos(5*f*x + 5*e) - 10*a*c^3*cos(4*f*x + 4*e) + 20*a*c^3*cos(3*f*x + 3*e) - 10*a*c^3*cos(2*f*x + 2*e) + 5*a*c
^3*cos(f*x + e))*sin(10*f*x + 10*e) - 5*(5*a*c^3*cos(8*f*x + 8*e) + 10*a*c^3*cos(6*f*x + 6*e) + 10*a*c^3*cos(4
*f*x + 4*e) + 5*a*c^3*cos(2*f*x + 2*e) + a*c^3)*sin(9*f*x + 9*e) + 5*(20*a*c^3*cos(7*f*x + 7*e) + 10*a*c^3*cos
(6*f*x + 6*e) + 14*a*c^3*cos(5*f*x + 5*e) + 10*a*c^3*cos(4*f*x + 4*e) + 20*a*c^3*cos(3*f*x + 3*e) + 5*a*c^3*co
s(f*x + e) + 2*a*c^3)*sin(8*f*x + 8*e) - 20*(10*a*c^3*cos(6*f*x + 6*e) + 10*a*c^3*cos(4*f*x + 4*e) + 5*a*c^3*c
os(2*f*x + 2*e) + a*c^3)*sin(7*f*x + 7*e) + 10*(14*a*c^3*cos(5*f*x + 5*e) + 20*a*c^3*cos(3*f*x + 3*e) - 5*a*c^
3*cos(2*f*x + 2*e) + 5*a*c^3*cos(f*x + e) + a*c^3)*sin(6*f*x + 6*e) - 14*(10*a*c^3*cos(4*f*x + 4*e) + 5*a*c^3*
cos(2*f*x + 2*e) + a*c^3)*sin(5*f*x + 5*e) + 10*(20*a*c^3*cos(3*f*x + 3*e) - 5*a*c^3*cos(2*f*x + 2*e) + 5*a*c^
3*cos(f*x + e) + a*c^3)*sin(4*f*x + 4*e) - 20*(5*a*c^3*cos(2*f*x + 2*e) + a*c^3)*sin(3*f*x + 3*e) + 5*(5*a*c^3
*cos(f*x + e) + 2*a*c^3)*sin(2*f*x + 2*e))*sqrt(a)*sqrt(c)/((2*(5*cos(8*f*x + 8*e) + 10*cos(6*f*x + 6*e) + 10*
cos(4*f*x + 4*e) + 5*cos(2*f*x + 2*e) + 1)*cos(10*f*x + 10*e) + cos(10*f*x + 10*e)^2 + 10*(10*cos(6*f*x + 6*e)
+ 10*cos(4*f*x + 4*e) + 5*cos(2*f*x + 2*e) + 1)*cos(8*f*x + 8*e) + 25*cos(8*f*x + 8*e)^2 + 20*(10*cos(4*f*x +
4*e) + 5*cos(2*f*x + 2*e) + 1)*cos(6*f*x + 6*e) + 100*cos(6*f*x + 6*e)^2 + 20*(5*cos(2*f*x + 2*e) + 1)*cos(4*
f*x + 4*e) + 100*cos(4*f*x + 4*e)^2 + 25*cos(2*f*x + 2*e)^2 + 10*(sin(8*f*x + 8*e) + 2*sin(6*f*x + 6*e) + 2*si
n(4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(10*f*x + 10*e) + sin(10*f*x + 10*e)^2 + 50*(2*sin(6*f*x + 6*e) + 2*sin(
4*f*x + 4*e) + sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 25*sin(8*f*x + 8*e)^2 + 100*(2*sin(4*f*x + 4*e) + sin(2*f*
x + 2*e))*sin(6*f*x + 6*e) + 100*sin(6*f*x + 6*e)^2 + 100*sin(4*f*x + 4*e)^2 + 100*sin(4*f*x + 4*e)*sin(2*f*x
+ 2*e) + 25*sin(2*f*x + 2*e)^2 + 10*cos(2*f*x + 2*e) + 1)*f)
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mupad [B] time = 6.15, size = 294, normalized size = 3.30 \[ \frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (\frac {a\,c^3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,28{}\mathrm {i}}{5\,f}-\frac {a\,c^3\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,8{}\mathrm {i}}{f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,16{}\mathrm {i}}{f}-\frac {a\,c^3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,8{}\mathrm {i}}{f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{f}\right )}{{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,4{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,6{}\mathrm {i}+{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,2{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + a/cos(e + f*x))^(3/2)*(c - c/cos(e + f*x))^(7/2))/cos(e + f*x),x)
[Out]
((c - c/cos(e + f*x))^(1/2)*((a*c^3*exp(e*5i + f*x*5i)*(a + a/cos(e + f*x))^(1/2)*28i)/(5*f) - (a*c^3*cos(e +
f*x)*exp(e*5i + f*x*5i)*(a + a/cos(e + f*x))^(1/2)*8i)/f + (a*c^3*exp(e*5i + f*x*5i)*cos(2*e + 2*f*x)*(a + a/c
os(e + f*x))^(1/2)*16i)/f - (a*c^3*exp(e*5i + f*x*5i)*cos(3*e + 3*f*x)*(a + a/cos(e + f*x))^(1/2)*8i)/f + (a*c
^3*exp(e*5i + f*x*5i)*cos(4*e + 4*f*x)*(a + a/cos(e + f*x))^(1/2)*4i)/f))/(exp(e*5i + f*x*5i)*sin(e + f*x)*4i
+ exp(e*5i + f*x*5i)*sin(3*e + 3*f*x)*6i + exp(e*5i + f*x*5i)*sin(5*e + 5*f*x)*2i)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(f*x+e)*(a+a*sec(f*x+e))**(3/2)*(c-c*sec(f*x+e))**(7/2),x)
[Out]
Timed out
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