3.1.64 \(\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^5} \, dx\) [64]

Optimal. Leaf size=172 \[ \frac {2 a^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{c^5 f}+\frac {2 a^2 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{c^5 f}-\frac {2 a \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 c^5 f}+\frac {2 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{5 c^5 f}+\frac {8 \cot ^9(e+f x) (a+a \sec (e+f x))^{9/2}}{9 a^2 c^5 f} \]

[Out]

2*a^(5/2)*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/c^5/f-2/3*a*cot(f*x+e)^3*(a+a*sec(f*x+e))^(3/2)/c^
5/f+2/5*cot(f*x+e)^5*(a+a*sec(f*x+e))^(5/2)/c^5/f+8/9*cot(f*x+e)^9*(a+a*sec(f*x+e))^(9/2)/a^2/c^5/f+2*a^2*cot(
f*x+e)*(a+a*sec(f*x+e))^(1/2)/c^5/f

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Rubi [A]
time = 0.13, antiderivative size = 172, normalized size of antiderivative = 1.00, 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 = {3989, 3972, 472, 209} \begin {gather*} \frac {2 a^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{c^5 f}+\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 \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} \end {gather*}

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 472

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 3972

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 3989

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])

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 \text {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 \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 3.60, size = 205, normalized size = 1.19 \begin {gather*} \frac {a^2 \sqrt {a (1+\sec (e+f x))} \left ((109+108 \cos (e+f x)+63 \cos (2 (e+f x))) \, _2F_1\left (-\frac {9}{2},-\frac {5}{2};-\frac {3}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )-15 (2+\cos (e+f x)-2 \cos (2 (e+f x))-\cos (3 (e+f x))) \, _3F_2\left (-\frac {7}{2},-\frac {3}{2},2;-\frac {1}{2},1;2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )+240 (1+2 \cos (e+f x)) \, _2F_1\left (-\frac {7}{2},-\frac {3}{2};-\frac {1}{2};2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2(e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{315 c^5 f \cos ^{\frac {9}{2}}(e+f x) (-1+\sec (e+f x))^5} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(491\) vs. \(2(152)=304\).
time = 0.30, size = 492, normalized size = 2.86

method	result	size
default	\(-\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right ) \left (-45 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}+180 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}-270 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}+178 \left (\cos ^{5}\left (f x +e \right )\right )+180 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}-486 \left (\cos ^{4}\left (f x +e \right )\right )-45 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+648 \left (\cos ^{3}\left (f x +e \right )\right )-390 \left (\cos ^{2}\left (f x +e \right )\right )+90 \cos \left (f x +e \right )\right ) a^{2}}{45 c^{5} f \sin \left (f x +e \right )^{3} \left (-1+\cos \left (f x +e \right )\right )^{3}}\)	\(492\)

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,method=_RETURNVERBOSE)

[Out]

-1/45/c^5/f*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)*(cos(f*x+e)+1)*(-45*cos(f*x+e)^4*sin(f*x+e)*(-2*cos(f*x+e)/(co
s(f*x+e)+1))^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*2^(1/2)+180
*cos(f*x+e)^3*sin(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)
*sin(f*x+e)/cos(f*x+e)*2^(1/2))*2^(1/2)-270*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(c
os(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*cos(f*x+e)^2*sin(f*x+e)*2^(1/2)+178*cos(f*x+e)^5+180*sin(f*
x+e)*cos(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+
e)/cos(f*x+e)*2^(1/2))*2^(1/2)-486*cos(f*x+e)^4-45*2^(1/2)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*si
n(f*x+e)/cos(f*x+e)*2^(1/2))*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+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*a^2

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 4.35, size = 649, normalized size = 3.77 \begin {gather*} \left [\frac {45 \, {\left (a^{2} \cos \left (f x + e\right )^{4} - 4 \, a^{2} \cos \left (f x + e\right )^{3} + 6 \, a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} - 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 4 \, {\left (89 \, a^{2} \cos \left (f x + e\right )^{5} - 243 \, a^{2} \cos \left (f x + e\right )^{4} + 324 \, a^{2} \cos \left (f x + e\right )^{3} - 195 \, a^{2} \cos \left (f x + e\right )^{2} + 45 \, 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 )}}}{90 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )}, \frac {45 \, {\left (a^{2} \cos \left (f x + e\right )^{4} - 4 \, a^{2} \cos \left (f x + e\right )^{3} + 6 \, a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (89 \, a^{2} \cos \left (f x + e\right )^{5} - 243 \, a^{2} \cos \left (f x + e\right )^{4} + 324 \, a^{2} \cos \left (f x + e\right )^{3} - 195 \, a^{2} \cos \left (f x + e\right )^{2} + 45 \, 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 )}}}{45 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )}\right ] \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (152) = 304\).
time = 1.84, size = 672, normalized size = 3.91 \begin {gather*} -\frac {\frac {45 \, \sqrt {-a} a^{3} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{c^{5} {\left | a \right |}} + \frac {4 \, {\left (45 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{16} \sqrt {-a} a^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 270 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{14} \sqrt {-a} a^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 900 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{12} \sqrt {-a} a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 1575 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{10} \sqrt {-a} a^{6} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1953 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{8} \sqrt {-a} a^{7} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 1452 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{6} \sqrt {-a} a^{8} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 738 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{4} \sqrt {-a} a^{9} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 207 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} \sqrt {-a} a^{10} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 28 \, \sqrt {2} \sqrt {-a} a^{11} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - a\right )}^{9} c^{5}}}{45 \, f} \end {gather*}

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]

-1/45*(45*sqrt(-a)*a^3*log(abs(2*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 4*s
qrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 4*sqrt(2)
*abs(a) - 6*a))*sgn(cos(f*x + e))/(c^5*abs(a)) + 4*(45*sqrt(2)*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/
2*f*x + 1/2*e)^2 + a))^16*sqrt(-a)*a^3*sgn(cos(f*x + e)) - 270*sqrt(2)*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-
a*tan(1/2*f*x + 1/2*e)^2 + a))^14*sqrt(-a)*a^4*sgn(cos(f*x + e)) + 900*sqrt(2)*(sqrt(-a)*tan(1/2*f*x + 1/2*e)
- sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^12*sqrt(-a)*a^5*sgn(cos(f*x + e)) - 1575*sqrt(2)*(sqrt(-a)*tan(1/2*f*x
+ 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^10*sqrt(-a)*a^6*sgn(cos(f*x + e)) + 1953*sqrt(2)*(sqrt(-a)*tan
(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^8*sqrt(-a)*a^7*sgn(cos(f*x + e)) - 1452*sqrt(2)*(sqrt
(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^6*sqrt(-a)*a^8*sgn(cos(f*x + e)) + 738*sqrt(2
)*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(-a)*a^9*sgn(cos(f*x + e)) - 207
*sqrt(2)*(sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(-a)*a^10*sgn(cos(f*x + e
)) + 28*sqrt(2)*sqrt(-a)*a^11*sgn(cos(f*x + e)))/(((sqrt(-a)*tan(1/2*f*x + 1/2*e) - sqrt(-a*tan(1/2*f*x + 1/2*
e)^2 + a))^2 - a)^9*c^5))/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^(5/2)/(c - c/cos(e + f*x))^5,x)

[Out]

int((a + a/cos(e + f*x))^(5/2)/(c - c/cos(e + f*x))^5, x)

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