3.117 \(\int \frac {(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=215 \[ \frac {c^4 \tan (e+f x) \sec (e+f x)}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {8 c^4 \tan (e+f x)}{a f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {4 c^4 \tan (e+f x) \log (\sec (e+f x)+1)}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {c^4 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]
[Out]
c^4*ln(cos(f*x+e))*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)-4*c^4*ln(1+sec(f*x+e))*tan(f*x
+e)/a/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)+c^4*sec(f*x+e)*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(1/2)/(c-
c*sec(f*x+e))^(1/2)-8*c^4*tan(f*x+e)/a/f/(1+sec(f*x+e))/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)
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Rubi [A] time = 0.14, antiderivative size = 215, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used =
{3912, 88} \[ \frac {c^4 \tan (e+f x) \sec (e+f x)}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {8 c^4 \tan (e+f x)}{a f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {4 c^4 \tan (e+f x) \log (\sec (e+f x)+1)}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {c^4 \tan (e+f x) \log (\cos (e+f x))}{a f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
Int[(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(3/2),x]
[Out]
(c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Log[1 +
Sec[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c^4*Sec[e + f*x]*Tan[e
+ f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (8*c^4*Tan[e + f*x])/(a*f*(1 + Sec[e + f*x])
*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])
Rule 88
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rule 3912
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[(a*c*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Subst[Int[((a + b*x)^(m - 1/2)*(c
+ d*x)^(n - 1/2))/x, x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && E
qQ[a^2 - b^2, 0]
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{3/2}} \, dx &=-\frac {(a c \tan (e+f x)) \operatorname {Subst}\left (\int \frac {(c-c x)^3}{x (a+a x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {(a c \tan (e+f x)) \operatorname {Subst}\left (\int \left (-\frac {c^3}{a^2}+\frac {c^3}{a^2 x}-\frac {8 c^3}{a^2 (1+x)^2}+\frac {4 c^3}{a^2 (1+x)}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=\frac {c^4 \log (\cos (e+f x)) \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {4 c^4 \log (1+\sec (e+f x)) \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {c^4 \sec (e+f x) \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {8 c^4 \tan (e+f x)}{a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 2.27, size = 204, normalized size = 0.95 \[ \frac {c^3 \cot \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {c-c \sec (e+f x)} \left (8 \log \left (1+e^{i (e+f x)}\right )-5 \log \left (1+e^{2 i (e+f x)}\right )+2 \left (8 \log \left (1+e^{i (e+f x)}\right )-5 \log \left (1+e^{2 i (e+f x)}\right )+i f x-9\right ) \cos (e+f x)+\left (8 \log \left (1+e^{i (e+f x)}\right )-5 \log \left (1+e^{2 i (e+f x)}\right )+i f x\right ) \cos (2 (e+f x))+i f x-2\right )}{2 a f (\cos (e+f x)+1) \sqrt {a (\sec (e+f x)+1)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(3/2),x]
[Out]
(c^3*Cot[(e + f*x)/2]*(-2 + I*f*x + 8*Log[1 + E^(I*(e + f*x))] + 2*Cos[e + f*x]*(-9 + I*f*x + 8*Log[1 + E^(I*(
e + f*x))] - 5*Log[1 + E^((2*I)*(e + f*x))]) + Cos[2*(e + f*x)]*(I*f*x + 8*Log[1 + E^(I*(e + f*x))] - 5*Log[1
+ E^((2*I)*(e + f*x))]) - 5*Log[1 + E^((2*I)*(e + f*x))])*Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(2*a*f*(1 + C
os[e + f*x])*Sqrt[a*(1 + Sec[e + f*x])])
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{3} \sec \left (f x + e\right )^{3} - 3 \, c^{3} \sec \left (f x + e\right )^{2} + 3 \, c^{3} \sec \left (f x + e\right ) - c^{3}\right )} \sqrt {a \sec \left (f x + e\right ) + a} \sqrt {-c \sec \left (f x + e\right ) + c}}{a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}, x\right ) \]
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))^(3/2),x, algorithm="fricas")
[Out]
integral(-(c^3*sec(f*x + e)^3 - 3*c^3*sec(f*x + e)^2 + 3*c^3*sec(f*x + e) - c^3)*sqrt(a*sec(f*x + e) + a)*sqrt
(-c*sec(f*x + e) + c)/(a^2*sec(f*x + e)^2 + 2*a^2*sec(f*x + e) + a^2), x)
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giac [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((c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(3/2),x, algorithm="giac")
[Out]
Timed out
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maple [A] time = 2.27, size = 278, normalized size = 1.29 \[ \frac {\left (5 \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+5 \ln \left (-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right )-\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+5 \cos \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+5 \cos \left (f x +e \right ) \ln \left (-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-3 \left (\cos ^{2}\left (f x +e \right )\right )-\cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+6 \cos \left (f x +e \right )+1\right ) \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}}{f \sin \left (f x +e \right )^{3} \left (-1+\cos \left (f x +e \right )\right )^{2} a^{2}} \]
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))^(3/2),x)
[Out]
1/f*(5*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))*cos(f*x+e)^2+5*ln(-(-sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))*co
s(f*x+e)^2-ln(2/(1+cos(f*x+e)))*cos(f*x+e)^2+5*cos(f*x+e)*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+5*cos(f*x
+e)*ln(-(-sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))-3*cos(f*x+e)^2-cos(f*x+e)*ln(2/(1+cos(f*x+e)))+6*cos(f*x+e)+1)*
(c*(-1+cos(f*x+e))/cos(f*x+e))^(7/2)*cos(f*x+e)^3*(a*(1+cos(f*x+e))/cos(f*x+e))^(1/2)/sin(f*x+e)^3/(-1+cos(f*x
+e))^2/a^2
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maxima [B] time = 1.48, size = 2393, normalized size = 11.13 \[ \text {result too large to display} \]
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))^(3/2),x, algorithm="maxima")
[Out]
-((f*x + e)*c^3*cos(4*f*x + 4*e)^2 + 4*(f*x + e)*c^3*cos(2*f*x + 2*e)^2 + (f*x + e)*c^3*sin(4*f*x + 4*e)^2 + 4
*(f*x + e)*c^3*sin(2*f*x + 2*e)^2 + 4*(f*x + e)*c^3*cos(2*f*x + 2*e) + (f*x + e)*c^3 - 4*c^3*sin(2*f*x + 2*e)
+ 4*((f*x + e)*c^3 - 5*c^3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1))*cos(3/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e)))^2 + 4*((f*x + e)*c^3 - 5*c^3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1))*cos(1/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*((f*x + e)*c^3 - 5*c^3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e
) + 1))*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*((f*x + e)*c^3 - 5*c^3*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e) + 1))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 5*(c^3*cos(4*f*x + 4*e)^
2 + 4*c^3*cos(2*f*x + 2*e)^2 + c^3*sin(4*f*x + 4*e)^2 + 4*c^3*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*c^3*sin(2*
f*x + 2*e)^2 + 4*c^3*cos(2*f*x + 2*e) + c^3 + 2*(2*c^3*cos(2*f*x + 2*e) + c^3)*cos(4*f*x + 4*e))*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e) + 1) + 8*(c^3*cos(4*f*x + 4*e)^2 + 4*c^3*cos(2*f*x + 2*e)^2 + 4*c^3*cos(3/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*c^3*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + c^
3*sin(4*f*x + 4*e)^2 + 4*c^3*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*c^3*sin(2*f*x + 2*e)^2 + 4*c^3*sin(3/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*c^3*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*
c^3*cos(2*f*x + 2*e) + c^3 + 2*(2*c^3*cos(2*f*x + 2*e) + c^3)*cos(4*f*x + 4*e) + 4*(c^3*cos(4*f*x + 4*e) + 2*c
^3*cos(2*f*x + 2*e) + 2*c^3*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^3)*cos(3/2*arctan2(sin(2*
f*x + 2*e), cos(2*f*x + 2*e))) + 4*(c^3*cos(4*f*x + 4*e) + 2*c^3*cos(2*f*x + 2*e) + c^3)*cos(1/2*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) + 4*(c^3*sin(4*f*x + 4*e) + 2*c^3*sin(2*f*x + 2*e) + 2*c^3*sin(1/2*arctan2(sin
(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*(c^3*sin(4*f*x + 4
*e) + 2*c^3*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*arctan2(sin(1/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))), cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 2*(2*(f*x + e)
*c^3*cos(2*f*x + 2*e) + (f*x + e)*c^3 - 2*c^3*sin(2*f*x + 2*e))*cos(4*f*x + 4*e) + 2*(2*(f*x + e)*c^3*cos(4*f*
x + 4*e) + 4*(f*x + e)*c^3*cos(2*f*x + 2*e) + 2*(f*x + e)*c^3 + 9*c^3*sin(4*f*x + 4*e) + 14*c^3*sin(2*f*x + 2*
e) - 10*(c^3*cos(4*f*x + 4*e) + 2*c^3*cos(2*f*x + 2*e) + c^3)*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)
+ 4*((f*x + e)*c^3 - 5*c^3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1))*cos(1/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))))*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*(f*x + e)*c^3*cos(4*f*x + 4*e)
+ 4*(f*x + e)*c^3*cos(2*f*x + 2*e) + 2*(f*x + e)*c^3 + 9*c^3*sin(4*f*x + 4*e) + 14*c^3*sin(2*f*x + 2*e) - 10*
(c^3*cos(4*f*x + 4*e) + 2*c^3*cos(2*f*x + 2*e) + c^3)*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1))*cos(1/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*((f*x + e)*c^3*sin(2*f*x + 2*e) + c^3*cos(2*f*x + 2*e))*sin(
4*f*x + 4*e) + 2*(2*(f*x + e)*c^3*sin(4*f*x + 4*e) + 4*(f*x + e)*c^3*sin(2*f*x + 2*e) - 9*c^3*cos(4*f*x + 4*e)
- 14*c^3*cos(2*f*x + 2*e) - 9*c^3 - 10*(c^3*sin(4*f*x + 4*e) + 2*c^3*sin(2*f*x + 2*e))*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e) + 1) + 4*((f*x + e)*c^3 - 5*c^3*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1))*sin(1/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*(2*(f*x
+ e)*c^3*sin(4*f*x + 4*e) + 4*(f*x + e)*c^3*sin(2*f*x + 2*e) - 9*c^3*cos(4*f*x + 4*e) - 14*c^3*cos(2*f*x + 2*
e) - 9*c^3 - 10*(c^3*sin(4*f*x + 4*e) + 2*c^3*sin(2*f*x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1
))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(c)/((a*cos(4*f*x + 4*e)^2 + 4*a*cos(2*f*x + 2*e)
^2 + 4*a*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*
f*x + 2*e)))^2 + a*sin(4*f*x + 4*e)^2 + 4*a*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*a*sin(2*f*x + 2*e)^2 + 4*a*s
in(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 4*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)
))^2 + 2*(2*a*cos(2*f*x + 2*e) + a)*cos(4*f*x + 4*e) + 4*a*cos(2*f*x + 2*e) + 4*(a*cos(4*f*x + 4*e) + 2*a*cos(
2*f*x + 2*e) + 2*a*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a)*cos(3/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) + 4*(a*cos(4*f*x + 4*e) + 2*a*cos(2*f*x + 2*e) + a)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) + 4*(a*sin(4*f*x + 4*e) + 2*a*sin(2*f*x + 2*e) + 2*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))))*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 4*(a*sin(4*f*x + 4*e) + 2*a*sin(2*f*x + 2*e
))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a)*sqrt(a)*f)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c - c/cos(e + f*x))^(7/2)/(a + a/cos(e + f*x))^(3/2),x)
[Out]
int((c - c/cos(e + f*x))^(7/2)/(a + a/cos(e + f*x))^(3/2), x)
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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((c-c*sec(f*x+e))**(7/2)/(a+a*sec(f*x+e))**(3/2),x)
[Out]
Timed out
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