3.1.11 \(\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx\) [11]

Optimal. Leaf size=188 \[ a^3 c^5 x-\frac {5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {a^3 c^5 \tan (e+f x)}{f}+\frac {5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f} \]

[Out]

a^3*c^5*x-5/8*a^3*c^5*arctanh(sin(f*x+e))/f-a^3*c^5*tan(f*x+e)/f+5/8*a^3*c^5*sec(f*x+e)*tan(f*x+e)/f+1/3*a^3*c
^5*tan(f*x+e)^3/f-5/12*a^3*c^5*sec(f*x+e)*tan(f*x+e)^3/f-1/5*a^3*c^5*tan(f*x+e)^5/f+1/3*a^3*c^5*sec(f*x+e)*tan
(f*x+e)^5/f-1/7*a^3*c^5*tan(f*x+e)^7/f

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Rubi [A]
time = 0.18, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3989, 3971, 3554, 8, 2691, 3855, 2687, 30} \begin {gather*} -\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^5 \tan (e+f x)}{f}-\frac {5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{3 f}-\frac {5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{12 f}+\frac {5 a^3 c^5 \tan (e+f x) \sec (e+f x)}{8 f}+a^3 c^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*c^5*x - (5*a^3*c^5*ArcTanh[Sin[e + f*x]])/(8*f) - (a^3*c^5*Tan[e + f*x])/f + (5*a^3*c^5*Sec[e + f*x]*Tan[e
+ f*x])/(8*f) + (a^3*c^5*Tan[e + f*x]^3)/(3*f) - (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(12*f) - (a^3*c^5*Ta
n[e + f*x]^5)/(5*f) + (a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^5)/(3*f) - (a^3*c^5*Tan[e + f*x]^7)/(7*f)

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3855

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

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

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 (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx &=-\left (\left (a^3 c^3\right ) \int (c-c \sec (e+f x))^2 \tan ^6(e+f x) \, dx\right )\\ &=-\left (\left (a^3 c^3\right ) \int \left (c^2 \tan ^6(e+f x)-2 c^2 \sec (e+f x) \tan ^6(e+f x)+c^2 \sec ^2(e+f x) \tan ^6(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c^5\right ) \int \tan ^6(e+f x) \, dx\right )-\left (a^3 c^5\right ) \int \sec ^2(e+f x) \tan ^6(e+f x) \, dx+\left (2 a^3 c^5\right ) \int \sec (e+f x) \tan ^6(e+f x) \, dx\\ &=-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}+\left (a^3 c^5\right ) \int \tan ^4(e+f x) \, dx-\frac {1}{3} \left (5 a^3 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx-\frac {\left (a^3 c^5\right ) \text {Subst}\left (\int x^6 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\left (a^3 c^5\right ) \int \tan ^2(e+f x) \, dx+\frac {1}{4} \left (5 a^3 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac {a^3 c^5 \tan (e+f x)}{f}+\frac {5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac {1}{8} \left (5 a^3 c^5\right ) \int \sec (e+f x) \, dx+\left (a^3 c^5\right ) \int 1 \, dx\\ &=a^3 c^5 x-\frac {5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {a^3 c^5 \tan (e+f x)}{f}+\frac {5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}\\ \end {align*}

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Mathematica [A]
time = 2.33, size = 189, normalized size = 1.01 \begin {gather*} \frac {a^3 c^5 \sec ^7(e+f x) \left (14700 (e+f x) \cos (e+f x)-16800 \tanh ^{-1}(\sin (e+f x)) \cos ^7(e+f x)+8820 e \cos (3 (e+f x))+8820 f x \cos (3 (e+f x))+2940 e \cos (5 (e+f x))+2940 f x \cos (5 (e+f x))+420 e \cos (7 (e+f x))+420 f x \cos (7 (e+f x))-4200 \sin (e+f x)+2975 \sin (2 (e+f x))-2184 \sin (3 (e+f x))+980 \sin (4 (e+f x))-2408 \sin (5 (e+f x))+1155 \sin (6 (e+f x))-584 \sin (7 (e+f x))\right )}{26880 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c^5*Sec[e + f*x]^7*(14700*(e + f*x)*Cos[e + f*x] - 16800*ArcTanh[Sin[e + f*x]]*Cos[e + f*x]^7 + 8820*e*Co
s[3*(e + f*x)] + 8820*f*x*Cos[3*(e + f*x)] + 2940*e*Cos[5*(e + f*x)] + 2940*f*x*Cos[5*(e + f*x)] + 420*e*Cos[7
*(e + f*x)] + 420*f*x*Cos[7*(e + f*x)] - 4200*Sin[e + f*x] + 2975*Sin[2*(e + f*x)] - 2184*Sin[3*(e + f*x)] + 9
80*Sin[4*(e + f*x)] - 2408*Sin[5*(e + f*x)] + 1155*Sin[6*(e + f*x)] - 584*Sin[7*(e + f*x)]))/(26880*f)

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Maple [A]
time = 0.14, size = 288, normalized size = 1.53

method	result	size
risch	\(a^{3} c^{5} x -\frac {i c^{5} a^{3} \left (1155 \,{\mathrm e}^{13 i \left (f x +e \right )}+1680 \,{\mathrm e}^{12 i \left (f x +e \right )}+980 \,{\mathrm e}^{11 i \left (f x +e \right )}+10080 \,{\mathrm e}^{10 i \left (f x +e \right )}+2975 \,{\mathrm e}^{9 i \left (f x +e \right )}+16240 \,{\mathrm e}^{8 i \left (f x +e \right )}+24640 \,{\mathrm e}^{6 i \left (f x +e \right )}-2975 \,{\mathrm e}^{5 i \left (f x +e \right )}+14448 \,{\mathrm e}^{4 i \left (f x +e \right )}-980 \,{\mathrm e}^{3 i \left (f x +e \right )}+6496 \,{\mathrm e}^{2 i \left (f x +e \right )}-1155 \,{\mathrm e}^{i \left (f x +e \right )}+1168\right )}{420 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}+\frac {5 c^{5} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{8 f}-\frac {5 c^{5} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{8 f}\)	\(217\)
derivativedivides	\(\frac {c^{5} a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )+2 c^{5} a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-2 c^{5} a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-6 c^{5} a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+6 c^{5} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{5} a^{3} \tan \left (f x +e \right )-2 c^{5} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{3} \left (f x +e \right )}{f}\)	\(288\)
default	\(\frac {c^{5} a^{3} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )+2 c^{5} a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-2 c^{5} a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-6 c^{5} a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+6 c^{5} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{5} a^{3} \tan \left (f x +e \right )-2 c^{5} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{3} \left (f x +e \right )}{f}\)	\(288\)
norman	\(\frac {a^{3} c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a^{3} c^{5} x +7 a^{3} c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-21 a^{3} c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+35 a^{3} c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-35 a^{3} c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+21 a^{3} c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-7 a^{3} c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {3 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {19 c^{5} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {1409 c^{5} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{60 f}-\frac {1768 c^{5} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 f}+\frac {1413 c^{5} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 f}-\frac {23 c^{5} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {13 c^{5} a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}+\frac {5 c^{5} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}-\frac {5 c^{5} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

1/f*(c^5*a^3*(-16/35-1/7*sec(f*x+e)^6-6/35*sec(f*x+e)^4-8/35*sec(f*x+e)^2)*tan(f*x+e)+2*c^5*a^3*(-(-1/6*sec(f*
x+e)^5-5/24*sec(f*x+e)^3-5/16*sec(f*x+e))*tan(f*x+e)+5/16*ln(sec(f*x+e)+tan(f*x+e)))-2*c^5*a^3*(-8/15-1/5*sec(
f*x+e)^4-4/15*sec(f*x+e)^2)*tan(f*x+e)-6*c^5*a^3*(-(-1/4*sec(f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e)+3/8*ln(sec(f*
x+e)+tan(f*x+e)))+6*c^5*a^3*(1/2*sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+tan(f*x+e)))-2*c^5*a^3*tan(f*x+e)-2*c
^5*a^3*ln(sec(f*x+e)+tan(f*x+e))+c^5*a^3*(f*x+e))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (185) = 370\).
time = 0.29, size = 385, normalized size = 2.05 \begin {gather*} -\frac {48 \, {\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 224 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 1680 \, {\left (f x + e\right )} a^{3} c^{5} + 35 \, a^{3} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 630 \, a^{3} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 2520 \, a^{3} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 3360 \, a^{3} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 3360 \, a^{3} c^{5} \tan \left (f x + e\right )}{1680 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(48*(5*tan(f*x + e)^7 + 21*tan(f*x + e)^5 + 35*tan(f*x + e)^3 + 35*tan(f*x + e))*a^3*c^5 - 224*(3*tan(
f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*c^5 - 1680*(f*x + e)*a^3*c^5 + 35*a^3*c^5*(2*(15*sin(f*x
+ e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e))/(sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15
*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) - 1)) - 630*a^3*c^5*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f
*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(sin(f*x + e) - 1)) + 2520*a^3*c^5*(2*sin(f
*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 3360*a^3*c^5*log(sec(f*x + e)
+ tan(f*x + e)) + 3360*a^3*c^5*tan(f*x + e))/f

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Fricas [A]
time = 2.93, size = 208, normalized size = 1.11 \begin {gather*} \frac {1680 \, a^{3} c^{5} f x \cos \left (f x + e\right )^{7} - 525 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} \log \left (\sin \left (f x + e\right ) + 1\right ) + 525 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (1168 \, a^{3} c^{5} \cos \left (f x + e\right )^{6} - 1155 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 256 \, a^{3} c^{5} \cos \left (f x + e\right )^{4} + 910 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 192 \, a^{3} c^{5} \cos \left (f x + e\right )^{2} - 280 \, a^{3} c^{5} \cos \left (f x + e\right ) + 120 \, a^{3} c^{5}\right )} \sin \left (f x + e\right )}{1680 \, f \cos \left (f x + e\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(1680*a^3*c^5*f*x*cos(f*x + e)^7 - 525*a^3*c^5*cos(f*x + e)^7*log(sin(f*x + e) + 1) + 525*a^3*c^5*cos(f
*x + e)^7*log(-sin(f*x + e) + 1) - 2*(1168*a^3*c^5*cos(f*x + e)^6 - 1155*a^3*c^5*cos(f*x + e)^5 - 256*a^3*c^5*
cos(f*x + e)^4 + 910*a^3*c^5*cos(f*x + e)^3 - 192*a^3*c^5*cos(f*x + e)^2 - 280*a^3*c^5*cos(f*x + e) + 120*a^3*
c^5)*sin(f*x + e))/(f*cos(f*x + e)^7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - a^{3} c^{5} \left (\int \left (-1\right )\, dx + \int 2 \sec {\left (e + f x \right )}\, dx + \int 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 6 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 6 \sec ^{5}{\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{7}{\left (e + f x \right )}\right )\, dx + \int \sec ^{8}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**3*c**5*(Integral(-1, x) + Integral(2*sec(e + f*x), x) + Integral(2*sec(e + f*x)**2, x) + Integral(-6*sec(e
+ f*x)**3, x) + Integral(6*sec(e + f*x)**5, x) + Integral(-2*sec(e + f*x)**6, x) + Integral(-2*sec(e + f*x)**
7, x) + Integral(sec(e + f*x)**8, x))

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Giac [A]
time = 0.58, size = 210, normalized size = 1.12 \begin {gather*} \frac {840 \, {\left (f x + e\right )} a^{3} c^{5} - 525 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 525 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (1365 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 9660 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 29673 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 21216 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 9863 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2660 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 315 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{7}}}{840 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(840*(f*x + e)*a^3*c^5 - 525*a^3*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1)) + 525*a^3*c^5*log(abs(tan(1/2*f*
x + 1/2*e) - 1)) + 2*(1365*a^3*c^5*tan(1/2*f*x + 1/2*e)^13 - 9660*a^3*c^5*tan(1/2*f*x + 1/2*e)^11 + 29673*a^3*
c^5*tan(1/2*f*x + 1/2*e)^9 - 21216*a^3*c^5*tan(1/2*f*x + 1/2*e)^7 + 9863*a^3*c^5*tan(1/2*f*x + 1/2*e)^5 - 2660
*a^3*c^5*tan(1/2*f*x + 1/2*e)^3 + 315*a^3*c^5*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^7)/f

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Mupad [B]
time = 2.62, size = 259, normalized size = 1.38 \begin {gather*} \frac {\frac {13\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{4}-23\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+\frac {1413\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{20}-\frac {1768\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{35}+\frac {1409\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{60}-\frac {19\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+\frac {3\,a^3\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+a^3\,c^5\,x-\frac {5\,a^3\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{4\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1409*a^3*c^5*tan(e/2 + (f*x)/2)^5)/60 - (19*a^3*c^5*tan(e/2 + (f*x)/2)^3)/3 - (1768*a^3*c^5*tan(e/2 + (f*x)/
2)^7)/35 + (1413*a^3*c^5*tan(e/2 + (f*x)/2)^9)/20 - 23*a^3*c^5*tan(e/2 + (f*x)/2)^11 + (13*a^3*c^5*tan(e/2 + (
f*x)/2)^13)/4 + (3*a^3*c^5*tan(e/2 + (f*x)/2))/4)/(f*(7*tan(e/2 + (f*x)/2)^2 - 21*tan(e/2 + (f*x)/2)^4 + 35*ta
n(e/2 + (f*x)/2)^6 - 35*tan(e/2 + (f*x)/2)^8 + 21*tan(e/2 + (f*x)/2)^10 - 7*tan(e/2 + (f*x)/2)^12 + tan(e/2 +
(f*x)/2)^14 - 1)) + a^3*c^5*x - (5*a^3*c^5*atanh(tan(e/2 + (f*x)/2)))/(4*f)

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