3.106 \(\int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=246 \[ -\frac {63 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{128 \sqrt {2} a^3 c^{5/2} f}-\frac {63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}}-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3 \sec (e+f x)+a^3\right ) (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}} \]

[Out]

-63/256*arctan(1/2*c^(1/2)*tan(f*x+e)*2^(1/2)/(c-c*sec(f*x+e))^(1/2))/a^3/c^(5/2)/f*2^(1/2)-21/32*tan(f*x+e)/a
^3/f/(c-c*sec(f*x+e))^(5/2)+1/5*tan(f*x+e)/f/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(5/2)+3/10*tan(f*x+e)/a/f/(a+
a*sec(f*x+e))^2/(c-c*sec(f*x+e))^(5/2)+21/20*tan(f*x+e)/f/(a^3+a^3*sec(f*x+e))/(c-c*sec(f*x+e))^(5/2)-63/128*t
an(f*x+e)/a^3/c/f/(c-c*sec(f*x+e))^(3/2)

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Rubi [A]  time = 0.53, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3960, 3796, 3795, 203} \[ -\frac {63 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{128 \sqrt {2} a^3 c^{5/2} f}-\frac {63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}}-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3 \sec (e+f x)+a^3\right ) (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)),x]

[Out]

(-63*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(128*Sqrt[2]*a^3*c^(5/2)*f) - (21*Tan[
e + f*x])/(32*a^3*f*(c - c*Sec[e + f*x])^(5/2)) + Tan[e + f*x]/(5*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]
)^(5/2)) + (3*Tan[e + f*x])/(10*a*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)) + (21*Tan[e + f*x])/(20
*f*(a^3 + a^3*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)) - (63*Tan[e + f*x])/(128*a^3*c*f*(c - c*Sec[e + f*x])^
(3/2))

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])

Rule 3795

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]

Rule 3796

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(b*Cot[e + f*x]*(a
+ b*Csc[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(m + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(
m + 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && IntegerQ[2*m]

Rule 3960

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] +
Dist[(m + n + 1)/(a*(2*m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n, x], x] /
; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ((ILtQ[m, 0] && ILtQ[n - 1/2, 0
]) || (ILtQ[m - 1/2, 0] && ILtQ[n - 1/2, 0] && LtQ[m, n]))

Rubi steps

\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {9 \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}} \, dx}{10 a}\\ &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \int \frac {\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2}} \, dx}{20 a^2}\\ &=\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3+a^3 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}+\frac {21 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{5/2}} \, dx}{8 a^3}\\ &=-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3+a^3 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}+\frac {63 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}} \, dx}{64 a^3 c}\\ &=-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3+a^3 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}}+\frac {63 \int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}} \, dx}{256 a^3 c^2}\\ &=-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3+a^3 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}}-\frac {63 \operatorname {Subst}\left (\int \frac {1}{2 c+x^2} \, dx,x,\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}\right )}{128 a^3 c^2 f}\\ &=-\frac {63 \tan ^{-1}\left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{128 \sqrt {2} a^3 c^{5/2} f}-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3+a^3 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 6.66, size = 468, normalized size = 1.90 \[ \frac {\sin ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^6(e+f x) \left (-\frac {257 \sin \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )}{10 f}+\frac {257 \cos \left (\frac {e}{2}\right ) \cos \left (\frac {f x}{2}\right )}{10 f}-\frac {2 \sec ^5\left (\frac {e}{2}+\frac {f x}{2}\right )}{5 f}+\frac {22 \sec ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{5 f}-\frac {124 \sec \left (\frac {e}{2}+\frac {f x}{2}\right )}{5 f}-\frac {\cot \left (\frac {e}{2}\right ) \csc ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f}+\frac {23 \cot \left (\frac {e}{2}\right ) \csc \left (\frac {e}{2}+\frac {f x}{2}\right )}{4 f}+\frac {\csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f}-\frac {23 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{4 f}\right )}{(a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}-\frac {63 e^{-\frac {1}{2} i (e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt {1+e^{2 i (e+f x)}} \sin ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) \cos ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^{\frac {11}{2}}(e+f x) \tanh ^{-1}\left (\frac {1+e^{i (e+f x)}}{\sqrt {2} \sqrt {1+e^{2 i (e+f x)}}}\right )}{4 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)),x]

[Out]

(-63*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*Sqrt[1 + E^((2*I)*(e + f*x))]*ArcTanh[(1 + E^(I*(e + f*x)
))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(e + f*x))])]*Cos[e/2 + (f*x)/2]^6*Sec[e + f*x]^(11/2)*Sin[e/2 + (f*x)/2]^5)/(4*
E^((I/2)*(e + f*x))*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)) + (Cos[e/2 + (f*x)/2]^6*Sec[e + f*x]^
6*((257*Cos[e/2]*Cos[(f*x)/2])/(10*f) + (23*Cot[e/2]*Csc[e/2 + (f*x)/2])/(4*f) - (Cot[e/2]*Csc[e/2 + (f*x)/2]^
3)/(2*f) - (124*Sec[e/2 + (f*x)/2])/(5*f) + (22*Sec[e/2 + (f*x)/2]^3)/(5*f) - (2*Sec[e/2 + (f*x)/2]^5)/(5*f) -
(23*Csc[e/2]*Csc[e/2 + (f*x)/2]^2*Sin[(f*x)/2])/(4*f) + (Csc[e/2]*Csc[e/2 + (f*x)/2]^4*Sin[(f*x)/2])/(2*f) -
(257*Sin[e/2]*Sin[(f*x)/2])/(10*f))*Sin[e/2 + (f*x)/2]^5)/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2))

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fricas [A]  time = 0.62, size = 461, normalized size = 1.87 \[ \left [-\frac {315 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 4 \, {\left (257 \, \cos \left (f x + e\right )^{5} - 354 \, \cos \left (f x + e\right )^{4} - 588 \, \cos \left (f x + e\right )^{3} + 210 \, \cos \left (f x + e\right )^{2} + 315 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{2560 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )}, \frac {315 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (257 \, \cos \left (f x + e\right )^{5} - 354 \, \cos \left (f x + e\right )^{4} - 588 \, \cos \left (f x + e\right )^{3} + 210 \, \cos \left (f x + e\right )^{2} + 315 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{1280 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

[-1/2560*(315*sqrt(2)*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-c)*log((2*sqrt(2)*(cos(f*x + e)^2 + cos(f*
x + e))*sqrt(-c)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)) + (3*c*cos(f*x + e) + c)*sin(f*x + e))/((cos(f*x + e)
- 1)*sin(f*x + e)))*sin(f*x + e) + 4*(257*cos(f*x + e)^5 - 354*cos(f*x + e)^4 - 588*cos(f*x + e)^3 + 210*cos(
f*x + e)^2 + 315*cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/((a^3*c^3*f*cos(f*x + e)^4 - 2*a^3*c^3
*f*cos(f*x + e)^2 + a^3*c^3*f)*sin(f*x + e)), 1/1280*(315*sqrt(2)*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt
(c)*arctan(sqrt(2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/(sqrt(c)*sin(f*x + e)))*sin(f*x + e) -
2*(257*cos(f*x + e)^5 - 354*cos(f*x + e)^4 - 588*cos(f*x + e)^3 + 210*cos(f*x + e)^2 + 315*cos(f*x + e))*sqrt
((c*cos(f*x + e) - c)/cos(f*x + e)))/((a^3*c^3*f*cos(f*x + e)^4 - 2*a^3*c^3*f*cos(f*x + e)^2 + a^3*c^3*f)*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/(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/32*(-63/8*sqrt(c)*atan(sqrt(c*tan((f*x+exp(1))/2)^2-c)/sqr
t(c))+1/8*(17*c*sqrt(c*tan((f*x+exp(1))/2)^2-c)*(c*tan((f*x+exp(1))/2)^2-c)+15*c^2*sqrt(c*tan((f*x+exp(1))/2)^
2-c))/(c*tan((f*x+exp(1))/2)^2)^2+(-c^9*sqrt(c*tan((f*x+exp(1))/2)^2-c)*(c*tan((f*x+exp(1))/2)^2-c)+1/5*c^8*sq
rt(c*tan((f*x+exp(1))/2)^2-c)*(c*tan((f*x+exp(1))/2)^2-c)^2+6*c^10*sqrt(c*tan((f*x+exp(1))/2)^2-c))/c^10)/sqrt
(2)/a^3/c^3/sign(tan((f*x+exp(1))/2))/sign(tan((f*x+exp(1))/2)^2-1)

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maple [B]  time = 2.20, size = 631, normalized size = 2.57 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right )^{3} \left (45 \left (\cos ^{2}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {11}{2}}+20 \cos \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {11}{2}}+35 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {9}{2}} \left (\cos ^{2}\left (f x +e \right )\right )-25 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {11}{2}}-70 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {9}{2}} \cos \left (f x +e \right )-45 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \left (\cos ^{2}\left (f x +e \right )\right )+35 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {9}{2}}+90 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \cos \left (f x +e \right )+63 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\cos ^{2}\left (f x +e \right )\right )-45 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {7}{2}}-126 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \cos \left (f x +e \right )-105 \left (\cos ^{2}\left (f x +e \right )\right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+63 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}}+210 \cos \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}+315 \left (\cos ^{2}\left (f x +e \right )\right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )+315 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}-105 \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}}-630 \cos \left (f x +e \right ) \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )-630 \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+315 \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}}\right )+315 \sqrt {-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\right )}{160 a^{3} f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{5} \left (-\frac {2 \cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(5/2),x)

[Out]

-1/160/a^3/f*(-1+cos(f*x+e))^3*(45*cos(f*x+e)^2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(11/2)+20*cos(f*x+e)*(-2*cos(f*
x+e)/(1+cos(f*x+e)))^(11/2)+35*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(9/2)*cos(f*x+e)^2-25*(-2*cos(f*x+e)/(1+cos(f*x+
e)))^(11/2)-70*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(9/2)*cos(f*x+e)-45*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(7/2)*cos(f*x
+e)^2+35*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(9/2)+90*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(7/2)*cos(f*x+e)+63*(-2*cos(f*
x+e)/(1+cos(f*x+e)))^(5/2)*cos(f*x+e)^2-45*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(7/2)-126*(-2*cos(f*x+e)/(1+cos(f*x+
e)))^(5/2)*cos(f*x+e)-105*cos(f*x+e)^2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)+63*(-2*cos(f*x+e)/(1+cos(f*x+e)))^
(5/2)+210*cos(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)+315*cos(f*x+e)^2*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+
e)))^(1/2))+315*cos(f*x+e)^2*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)-105*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)-630
*cos(f*x+e)*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))-630*cos(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)
+315*arctan(1/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))+315*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2))/(c*(-1+cos(f*x+e
))/cos(f*x+e))^(5/2)/sin(f*x+e)^5/(-2*cos(f*x+e)/(1+cos(f*x+e)))^(5/2)

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maxima [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/(c-c*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(e + f*x)*(a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^(5/2)),x)

[Out]

int(1/(cos(e + f*x)*(a + a/cos(e + f*x))^3*(c - c/cos(e + f*x))^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{5}{\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{4}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(a+a*sec(f*x+e))**3/(c-c*sec(f*x+e))**(5/2),x)

[Out]

Integral(sec(e + f*x)/(c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**5 + c**2*sqrt(-c*sec(e + f*x) + c)*sec(e +
f*x)**4 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**2
+ c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**2*sqrt(-c*sec(e + f*x) + c)), x)/a**3

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