3.52 \(\int \frac {\sec (e+f x) (c-c \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx\)

Optimal. Leaf size=215 \[ \frac {77 c^6 \tan ^3(e+f x)}{5 a^3 f}+\frac {924 c^6 \tan (e+f x)}{5 a^3 f}-\frac {231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac {693 c^6 \tan (e+f x) \sec (e+f x)}{10 a^3 f}+\frac {66 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^3}{5 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {22 c^2 \tan (e+f x) (c-c \sec (e+f x))^4}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^5}{5 f (a \sec (e+f x)+a)^3} \]

[Out]

-231/2*c^6*arctanh(sin(f*x+e))/a^3/f+924/5*c^6*tan(f*x+e)/a^3/f-693/10*c^6*sec(f*x+e)*tan(f*x+e)/a^3/f-22/15*c
^2*(c-c*sec(f*x+e))^4*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^2+2/5*c*(c-c*sec(f*x+e))^5*tan(f*x+e)/f/(a+a*sec(f*x+e))
^3+66/5*(c^2-c^2*sec(f*x+e))^3*tan(f*x+e)/f/(a^3+a^3*sec(f*x+e))+77/5*c^6*tan(f*x+e)^3/a^3/f

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Rubi [A]  time = 0.34, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3957, 3791, 3770, 3767, 8, 3768} \[ \frac {77 c^6 \tan ^3(e+f x)}{5 a^3 f}+\frac {924 c^6 \tan (e+f x)}{5 a^3 f}-\frac {231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac {693 c^6 \tan (e+f x) \sec (e+f x)}{10 a^3 f}+\frac {66 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^3}{5 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {22 c^2 \tan (e+f x) (c-c \sec (e+f x))^4}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^5}{5 f (a \sec (e+f x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*(c - c*Sec[e + f*x])^6)/(a + a*Sec[e + f*x])^3,x]

[Out]

(-231*c^6*ArcTanh[Sin[e + f*x]])/(2*a^3*f) + (924*c^6*Tan[e + f*x])/(5*a^3*f) - (693*c^6*Sec[e + f*x]*Tan[e +
f*x])/(10*a^3*f) - (22*c^2*(c - c*Sec[e + f*x])^4*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + (2*c*(c - c*
Sec[e + f*x])^5*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + (66*(c^2 - c^2*Sec[e + f*x])^3*Tan[e + f*x])/(5*f
*(a^3 + a^3*Sec[e + f*x])) + (77*c^6*Tan[e + f*x]^3)/(5*a^3*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3791

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rule 3957

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

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx &=\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {(11 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx}{5 a}\\ &=-\frac {22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\left (33 c^2\right ) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx}{5 a^2}\\ &=-\frac {22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (231 c^3\right ) \int \sec (e+f x) (c-c \sec (e+f x))^3 \, dx}{5 a^3}\\ &=-\frac {22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (231 c^3\right ) \int \left (c^3 \sec (e+f x)-3 c^3 \sec ^2(e+f x)+3 c^3 \sec ^3(e+f x)-c^3 \sec ^4(e+f x)\right ) \, dx}{5 a^3}\\ &=-\frac {22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (231 c^6\right ) \int \sec (e+f x) \, dx}{5 a^3}+\frac {\left (231 c^6\right ) \int \sec ^4(e+f x) \, dx}{5 a^3}+\frac {\left (693 c^6\right ) \int \sec ^2(e+f x) \, dx}{5 a^3}-\frac {\left (693 c^6\right ) \int \sec ^3(e+f x) \, dx}{5 a^3}\\ &=-\frac {231 c^6 \tanh ^{-1}(\sin (e+f x))}{5 a^3 f}-\frac {693 c^6 \sec (e+f x) \tan (e+f x)}{10 a^3 f}-\frac {22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (693 c^6\right ) \int \sec (e+f x) \, dx}{10 a^3}-\frac {\left (231 c^6\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{5 a^3 f}-\frac {\left (693 c^6\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{5 a^3 f}\\ &=-\frac {231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}+\frac {924 c^6 \tan (e+f x)}{5 a^3 f}-\frac {693 c^6 \sec (e+f x) \tan (e+f x)}{10 a^3 f}-\frac {22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {77 c^6 \tan ^3(e+f x)}{5 a^3 f}\\ \end {align*}

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Mathematica [A]  time = 2.17, size = 406, normalized size = 1.89 \[ \frac {c^6 \cos \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) \left (\sec \left (\frac {e}{2}\right ) \sec (e) \left (-130340 \sin \left (e-\frac {f x}{2}\right )+75600 \sin \left (e+\frac {f x}{2}\right )-120176 \sin \left (2 e+\frac {f x}{2}\right )-34230 \sin \left (e+\frac {3 f x}{2}\right )+82278 \sin \left (2 e+\frac {3 f x}{2}\right )-79450 \sin \left (3 e+\frac {3 f x}{2}\right )+91670 \sin \left (e+\frac {5 f x}{2}\right )-14730 \sin \left (2 e+\frac {5 f x}{2}\right )+61920 \sin \left (3 e+\frac {5 f x}{2}\right )-44480 \sin \left (4 e+\frac {5 f x}{2}\right )+53593 \sin \left (2 e+\frac {7 f x}{2}\right )-1735 \sin \left (3 e+\frac {7 f x}{2}\right )+38123 \sin \left (4 e+\frac {7 f x}{2}\right )-17205 \sin \left (5 e+\frac {7 f x}{2}\right )+23735 \sin \left (3 e+\frac {9 f x}{2}\right )+2455 \sin \left (4 e+\frac {9 f x}{2}\right )+17785 \sin \left (5 e+\frac {9 f x}{2}\right )-3495 \sin \left (6 e+\frac {9 f x}{2}\right )+5446 \sin \left (4 e+\frac {11 f x}{2}\right )+1190 \sin \left (5 e+\frac {11 f x}{2}\right )+4256 \sin \left (6 e+\frac {11 f x}{2}\right )-65436 \sin \left (\frac {f x}{2}\right )+127498 \sin \left (\frac {3 f x}{2}\right )\right ) \sec ^3(e+f x)+887040 \cos ^5\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )}{960 a^3 f (\sec (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*(c - c*Sec[e + f*x])^6)/(a + a*Sec[e + f*x])^3,x]

[Out]

(c^6*Cos[(e + f*x)/2]*Sec[e + f*x]^3*(887040*Cos[(e + f*x)/2]^5*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Lo
g[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) + Sec[e/2]*Sec[e]*Sec[e + f*x]^3*(-65436*Sin[(f*x)/2] + 127498*Sin[(3*
f*x)/2] - 130340*Sin[e - (f*x)/2] + 75600*Sin[e + (f*x)/2] - 120176*Sin[2*e + (f*x)/2] - 34230*Sin[e + (3*f*x)
/2] + 82278*Sin[2*e + (3*f*x)/2] - 79450*Sin[3*e + (3*f*x)/2] + 91670*Sin[e + (5*f*x)/2] - 14730*Sin[2*e + (5*
f*x)/2] + 61920*Sin[3*e + (5*f*x)/2] - 44480*Sin[4*e + (5*f*x)/2] + 53593*Sin[2*e + (7*f*x)/2] - 1735*Sin[3*e
+ (7*f*x)/2] + 38123*Sin[4*e + (7*f*x)/2] - 17205*Sin[5*e + (7*f*x)/2] + 23735*Sin[3*e + (9*f*x)/2] + 2455*Sin
[4*e + (9*f*x)/2] + 17785*Sin[5*e + (9*f*x)/2] - 3495*Sin[6*e + (9*f*x)/2] + 5446*Sin[4*e + (11*f*x)/2] + 1190
*Sin[5*e + (11*f*x)/2] + 4256*Sin[6*e + (11*f*x)/2])))/(960*a^3*f*(1 + Sec[e + f*x])^3)

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fricas [A]  time = 0.48, size = 263, normalized size = 1.22 \[ -\frac {3465 \, {\left (c^{6} \cos \left (f x + e\right )^{6} + 3 \, c^{6} \cos \left (f x + e\right )^{5} + 3 \, c^{6} \cos \left (f x + e\right )^{4} + c^{6} \cos \left (f x + e\right )^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3465 \, {\left (c^{6} \cos \left (f x + e\right )^{6} + 3 \, c^{6} \cos \left (f x + e\right )^{5} + 3 \, c^{6} \cos \left (f x + e\right )^{4} + c^{6} \cos \left (f x + e\right )^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (5446 \, c^{6} \cos \left (f x + e\right )^{5} + 12843 \, c^{6} \cos \left (f x + e\right )^{4} + 8711 \, c^{6} \cos \left (f x + e\right )^{3} + 815 \, c^{6} \cos \left (f x + e\right )^{2} - 105 \, c^{6} \cos \left (f x + e\right ) + 10 \, c^{6}\right )} \sin \left (f x + e\right )}{60 \, {\left (a^{3} f \cos \left (f x + e\right )^{6} + 3 \, a^{3} f \cos \left (f x + e\right )^{5} + 3 \, a^{3} f \cos \left (f x + e\right )^{4} + a^{3} f \cos \left (f x + e\right )^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^6/(a+a*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/60*(3465*(c^6*cos(f*x + e)^6 + 3*c^6*cos(f*x + e)^5 + 3*c^6*cos(f*x + e)^4 + c^6*cos(f*x + e)^3)*log(sin(f*
x + e) + 1) - 3465*(c^6*cos(f*x + e)^6 + 3*c^6*cos(f*x + e)^5 + 3*c^6*cos(f*x + e)^4 + c^6*cos(f*x + e)^3)*log
(-sin(f*x + e) + 1) - 2*(5446*c^6*cos(f*x + e)^5 + 12843*c^6*cos(f*x + e)^4 + 8711*c^6*cos(f*x + e)^3 + 815*c^
6*cos(f*x + e)^2 - 105*c^6*cos(f*x + e) + 10*c^6)*sin(f*x + e))/(a^3*f*cos(f*x + e)^6 + 3*a^3*f*cos(f*x + e)^5
+ 3*a^3*f*cos(f*x + e)^4 + a^3*f*cos(f*x + e)^3)

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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)*(c-c*sec(f*x+e))^6/(a+a*sec(f*x+e))^3,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*((8/5*tan((f*x+exp(1))/2)^5*c^6*a^12+32/3*tan((f*x+exp(1))
/2)^3*c^6*a^12+80*tan((f*x+exp(1))/2)*c^6*a^12)/a^15+(-267*tan((f*x+exp(1))/2)^5*c^6+472*tan((f*x+exp(1))/2)^3
*c^6-213*tan((f*x+exp(1))/2)*c^6)*1/6/a^3/(tan((f*x+exp(1))/2)^2-1)^3+231*c^6*1/4/a^3*ln(abs(tan((f*x+exp(1))/
2)-1))-231*c^6*1/4/a^3*ln(abs(tan((f*x+exp(1))/2)+1)))

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maple [A]  time = 0.82, size = 256, normalized size = 1.19 \[ \frac {16 c^{6} \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5 f \,a^{3}}+\frac {64 c^{6} \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 f \,a^{3}}+\frac {160 c^{6} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{3}}-\frac {c^{6}}{3 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{3}}-\frac {5 c^{6}}{f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{2}}-\frac {89 c^{6}}{2 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {231 c^{6} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{2 f \,a^{3}}-\frac {c^{6}}{3 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{3}}+\frac {5 c^{6}}{f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{2}}-\frac {89 c^{6}}{2 f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {231 c^{6} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{2 f \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(c-c*sec(f*x+e))^6/(a+a*sec(f*x+e))^3,x)

[Out]

16/5/f*c^6/a^3*tan(1/2*e+1/2*f*x)^5+64/3/f*c^6/a^3*tan(1/2*e+1/2*f*x)^3+160/f*c^6/a^3*tan(1/2*e+1/2*f*x)-1/3/f
*c^6/a^3/(tan(1/2*e+1/2*f*x)-1)^3-5/f*c^6/a^3/(tan(1/2*e+1/2*f*x)-1)^2-89/2/f*c^6/a^3/(tan(1/2*e+1/2*f*x)-1)+2
31/2/f*c^6/a^3*ln(tan(1/2*e+1/2*f*x)-1)-1/3/f*c^6/a^3/(tan(1/2*e+1/2*f*x)+1)^3+5/f*c^6/a^3/(tan(1/2*e+1/2*f*x)
+1)^2-89/2/f*c^6/a^3/(tan(1/2*e+1/2*f*x)+1)-231/2/f*c^6/a^3*ln(tan(1/2*e+1/2*f*x)+1)

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maxima [B]  time = 0.38, size = 935, normalized size = 4.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))^6/(a+a*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

1/60*(c^6*(20*(33*sin(f*x + e)/(cos(f*x + e) + 1) - 76*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 51*sin(f*x + e)^5
/(cos(f*x + e) + 1)^5)/(a^3 - 3*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*a^3*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 - a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6) + (735*sin(f*x + e)/(cos(f*x + e) + 1) + 50*sin(f*x + e)^3/(c
os(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 690*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)
/a^3 + 690*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) + 6*c^6*(60*(5*sin(f*x + e)/(cos(f*x + e) + 1) - 7*si
n(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^3 - 2*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^3*sin(f*x + e)^4/(cos(
f*x + e) + 1)^4) + (465*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x +
e)^5/(cos(f*x + e) + 1)^5)/a^3 - 390*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 390*log(sin(f*x + e)/(cos
(f*x + e) + 1) - 1)/a^3) + 45*c^6*(40*sin(f*x + e)/((a^3 - a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x +
e) + 1)) + (85*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f*x + e)^5/(cos
(f*x + e) + 1)^5)/a^3 - 60*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) +
1) - 1)/a^3) + 20*c^6*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 20*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f
*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 60*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(c
os(f*x + e) + 1) - 1)/a^3) + 15*c^6*(15*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 + c^6*(15*sin(f*x + e)/(cos(f*x + e) + 1) - 10*sin(f*x + e)^3/
(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 18*c^6*(5*sin(f*x + e)/(cos(f*x + e) + 1)
- sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3)/f

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mupad [B]  time = 1.65, size = 193, normalized size = 0.90 \[ \frac {160\,c^6\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}-\frac {89\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {472\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+71\,c^6\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^3\right )}+\frac {64\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^3\,f}+\frac {16\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}-\frac {231\,c^6\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - c/cos(e + f*x))^6/(cos(e + f*x)*(a + a/cos(e + f*x))^3),x)

[Out]

(160*c^6*tan(e/2 + (f*x)/2))/(a^3*f) - (89*c^6*tan(e/2 + (f*x)/2)^5 - (472*c^6*tan(e/2 + (f*x)/2)^3)/3 + 71*c^
6*tan(e/2 + (f*x)/2))/(f*(3*a^3*tan(e/2 + (f*x)/2)^2 - 3*a^3*tan(e/2 + (f*x)/2)^4 + a^3*tan(e/2 + (f*x)/2)^6 -
a^3)) + (64*c^6*tan(e/2 + (f*x)/2)^3)/(3*a^3*f) + (16*c^6*tan(e/2 + (f*x)/2)^5)/(5*a^3*f) - (231*c^6*atanh(ta
n(e/2 + (f*x)/2)))/(a^3*f)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{6} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {6 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {15 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {20 \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {15 \sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {6 \sec ^{6}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{7}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c-c*sec(f*x+e))**6/(a+a*sec(f*x+e))**3,x)

[Out]

c**6*(Integral(sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-6*sec(e
+ f*x)**2/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(15*sec(e + f*x)**3/(sec(e
+ f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(-20*sec(e + f*x)**4/(sec(e + f*x)**3 + 3*s
ec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(15*sec(e + f*x)**5/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 +
3*sec(e + f*x) + 1), x) + Integral(-6*sec(e + f*x)**6/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) +
1), x) + Integral(sec(e + f*x)**7/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x))/a**3

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