3.64 \(\int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^5} \, dx\)
Optimal. Leaf size=172 \[ \frac{8 \cot ^9(e+f x) (a \sec (e+f x)+a)^{9/2}}{9 a^2 c^5 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^5 f}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c^5 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^5 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^5 f} \]
[Out]
(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^5*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*S
ec[e + f*x]])/(c^5*f) - (2*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^5*f) + (2*Cot[e + f*x]^5*(a + a*S
ec[e + f*x])^(5/2))/(5*c^5*f) + (8*Cot[e + f*x]^9*(a + a*Sec[e + f*x])^(9/2))/(9*a^2*c^5*f)
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Rubi [A] time = 0.198272, antiderivative size = 172, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{3904, 3887, 461, 203} \[ \frac{8 \cot ^9(e+f x) (a \sec (e+f x)+a)^{9/2}}{9 a^2 c^5 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^5 f}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c^5 f}+\frac{2 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{5 c^5 f}-\frac{2 a \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 c^5 f} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^5,x]
[Out]
(2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^5*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*S
ec[e + f*x]])/(c^5*f) - (2*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^5*f) + (2*Cot[e + f*x]^5*(a + a*S
ec[e + f*x])^(5/2))/(5*c^5*f) + (8*Cot[e + f*x]^9*(a + a*Sec[e + f*x])^(9/2))/(9*a^2*c^5*f)
Rule 3904
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[(-(a*c))^m, Int[Cot[e + f*x]^(2*m)*(c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !(IntegerQ[n] && GtQ[m - n, 0])
Rule 3887
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[(-2*a^(m/2 +
n + 1/2))/d, Subst[Int[(x^m*(2 + a*x^2)^(m/2 + n - 1/2))/(1 + a*x^2), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]
Rule 461
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^5} \, dx &=-\frac{\int \cot ^{10}(e+f x) (a+a \sec (e+f x))^{15/2} \, dx}{a^5 c^5}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (2+a x^2\right )^2}{x^{10} \left (1+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^2 c^5 f}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{4}{x^{10}}+\frac{a^2}{x^6}-\frac{a^3}{x^4}+\frac{a^4}{x^2}-\frac{a^5}{1+a x^2}\right ) \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a^2 c^5 f}\\ &=\frac{2 a^2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c^5 f}-\frac{2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^5 f}+\frac{2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 c^5 f}+\frac{8 \cot ^9(e+f x) (a+a \sec (e+f x))^{9/2}}{9 a^2 c^5 f}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^5 f}\\ &=\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^5 f}+\frac{2 a^2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c^5 f}-\frac{2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^5 f}+\frac{2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 c^5 f}+\frac{8 \cot ^9(e+f x) (a+a \sec (e+f x))^{9/2}}{9 a^2 c^5 f}\\ \end{align*}
Mathematica [C] time = 3.47695, size = 205, normalized size = 1.19 \[ \frac{a^2 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \left (-15 (\cos (e+f x)-2 \cos (2 (e+f x))-\cos (3 (e+f x))+2) \text{HypergeometricPFQ}\left (\left \{-\frac{7}{2},-\frac{3}{2},2\right \},\left \{-\frac{1}{2},1\right \},2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )+240 \sin ^2(e+f x) (2 \cos (e+f x)+1) \text{Hypergeometric2F1}\left (-\frac{7}{2},-\frac{3}{2},-\frac{1}{2},2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )+(108 \cos (e+f x)+63 \cos (2 (e+f x))+109) \text{Hypergeometric2F1}\left (-\frac{9}{2},-\frac{5}{2},-\frac{3}{2},2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )}{315 c^5 f \cos ^{\frac{9}{2}}(e+f x) (\sec (e+f x)-1)^5} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^5,x]
[Out]
(a^2*Sqrt[a*(1 + Sec[e + f*x])]*((109 + 108*Cos[e + f*x] + 63*Cos[2*(e + f*x)])*Hypergeometric2F1[-9/2, -5/2,
-3/2, 2*Sin[(e + f*x)/2]^2] - 15*(2 + Cos[e + f*x] - 2*Cos[2*(e + f*x)] - Cos[3*(e + f*x)])*HypergeometricPFQ[
{-7/2, -3/2, 2}, {-1/2, 1}, 2*Sin[(e + f*x)/2]^2] + 240*(1 + 2*Cos[e + f*x])*Hypergeometric2F1[-7/2, -3/2, -1/
2, 2*Sin[(e + f*x)/2]^2]*Sin[e + f*x]^2)*Tan[(e + f*x)/2])/(315*c^5*f*Cos[e + f*x]^(9/2)*(-1 + Sec[e + f*x])^5
)
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Maple [B] time = 0.357, size = 492, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^5,x)
[Out]
1/45/c^5/f*a^2*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)*(1+cos(f*x+e))*(45*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*c
os(f*x+e)^4*sin(f*x+e)*2^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))
-180*2^(1/2)*cos(f*x+e)^3*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(
1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))+270*cos(f*x+e)^2*sin(f*x+e)*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))
^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))-180*2^(1/2)*cos(f*x+e)*
sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f
*x+e)/cos(f*x+e))-178*cos(f*x+e)^5+45*2^(1/2)*sin(f*x+e)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1
/2)*sin(f*x+e)/cos(f*x+e))*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)+486*cos(f*x+e)^4-648*cos(f*x+e)^3+390*cos(f*x+
e)^2-90*cos(f*x+e))/sin(f*x+e)^3/(-1+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((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^5,x, algorithm="maxima")
[Out]
Timed out
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Fricas [A] time = 1.6696, size = 1512, normalized size = 8.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^5,x, algorithm="fricas")
[Out]
[1/90*(45*(a^2*cos(f*x + e)^4 - 4*a^2*cos(f*x + e)^3 + 6*a^2*cos(f*x + e)^2 - 4*a^2*cos(f*x + e) + a^2)*sqrt(-
a)*log(-(8*a*cos(f*x + e)^3 - 4*(2*cos(f*x + e)^2 - cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x +
e))*sin(f*x + e) - 7*a*cos(f*x + e) + a)/(cos(f*x + e) + 1))*sin(f*x + e) + 4*(89*a^2*cos(f*x + e)^5 - 243*a^
2*cos(f*x + e)^4 + 324*a^2*cos(f*x + e)^3 - 195*a^2*cos(f*x + e)^2 + 45*a^2*cos(f*x + e))*sqrt((a*cos(f*x + e)
+ a)/cos(f*x + e)))/((c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 + 6*c^5*f*cos(f*x + e)^2 - 4*c^5*f*cos(f*
x + e) + c^5*f)*sin(f*x + e)), 1/45*(45*(a^2*cos(f*x + e)^4 - 4*a^2*cos(f*x + e)^3 + 6*a^2*cos(f*x + e)^2 - 4*
a^2*cos(f*x + e) + a^2)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x
+ e)/(2*a*cos(f*x + e)^2 + a*cos(f*x + e) - a))*sin(f*x + e) + 2*(89*a^2*cos(f*x + e)^5 - 243*a^2*cos(f*x + e)
^4 + 324*a^2*cos(f*x + e)^3 - 195*a^2*cos(f*x + e)^2 + 45*a^2*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x
+ e)))/((c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 + 6*c^5*f*cos(f*x + e)^2 - 4*c^5*f*cos(f*x + e) + c^5*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((a+a*sec(f*x+e))**(5/2)/(c-c*sec(f*x+e))**5,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((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^5,x, algorithm="giac")
[Out]
Timed out