3.1.21 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx\) [21]

Optimal. Leaf size=227 \[ \frac {55 a^3 c^6 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac {25 a^3 c^6 \sec (e+f x) \tan (e+f x)}{128 f}-\frac {15 a^3 c^6 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac {5 a^3 c^6 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^6 \sec ^3(e+f x) \tan ^3(e+f x)}{16 f}-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac {a^3 c^6 \tan ^9(e+f x)}{9 f} \]

[Out]

55/128*a^3*c^6*arctanh(sin(f*x+e))/f-25/128*a^3*c^6*sec(f*x+e)*tan(f*x+e)/f-15/64*a^3*c^6*sec(f*x+e)^3*tan(f*x
+e)/f+5/24*a^3*c^6*sec(f*x+e)*tan(f*x+e)^3/f+5/16*a^3*c^6*sec(f*x+e)^3*tan(f*x+e)^3/f-1/6*a^3*c^6*sec(f*x+e)*t
an(f*x+e)^5/f-3/8*a^3*c^6*sec(f*x+e)^3*tan(f*x+e)^5/f+4/7*a^3*c^6*tan(f*x+e)^7/f+1/9*a^3*c^6*tan(f*x+e)^9/f

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Rubi [A]
time = 0.26, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {4043, 2691, 3855, 2687, 30, 3853, 14} \begin {gather*} \frac {a^3 c^6 \tan ^9(e+f x)}{9 f}+\frac {4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac {55 a^3 c^6 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac {3 a^3 c^6 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}+\frac {5 a^3 c^6 \tan ^3(e+f x) \sec ^3(e+f x)}{16 f}-\frac {15 a^3 c^6 \tan (e+f x) \sec ^3(e+f x)}{64 f}-\frac {a^3 c^6 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac {5 a^3 c^6 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac {25 a^3 c^6 \tan (e+f x) \sec (e+f x)}{128 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(55*a^3*c^6*ArcTanh[Sin[e + f*x]])/(128*f) - (25*a^3*c^6*Sec[e + f*x]*Tan[e + f*x])/(128*f) - (15*a^3*c^6*Sec[
e + f*x]^3*Tan[e + f*x])/(64*f) + (5*a^3*c^6*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) + (5*a^3*c^6*Sec[e + f*x]^3*T
an[e + f*x]^3)/(16*f) - (a^3*c^6*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) - (3*a^3*c^6*Sec[e + f*x]^3*Tan[e + f*x]^5
)/(8*f) + (4*a^3*c^6*Tan[e + f*x]^7)/(7*f) + (a^3*c^6*Tan[e + f*x]^9)/(9*f)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 3853

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 3855

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

Rule 4043

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_.), x_Symbol] :> Dist[((-a)*c)^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n
- m), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m,
n] && GeQ[n - m, 0] && GtQ[m*n, 0]

Rubi steps

\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx &=-\left (\left (a^3 c^3\right ) \int \left (c^3 \sec (e+f x) \tan ^6(e+f x)-3 c^3 \sec ^2(e+f x) \tan ^6(e+f x)+3 c^3 \sec ^3(e+f x) \tan ^6(e+f x)-c^3 \sec ^4(e+f x) \tan ^6(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c^6\right ) \int \sec (e+f x) \tan ^6(e+f x) \, dx\right )+\left (a^3 c^6\right ) \int \sec ^4(e+f x) \tan ^6(e+f x) \, dx+\left (3 a^3 c^6\right ) \int \sec ^2(e+f x) \tan ^6(e+f x) \, dx-\left (3 a^3 c^6\right ) \int \sec ^3(e+f x) \tan ^6(e+f x) \, dx\\ &=-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {1}{6} \left (5 a^3 c^6\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx+\frac {1}{8} \left (15 a^3 c^6\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx+\frac {\left (a^3 c^6\right ) \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}+\frac {\left (3 a^3 c^6\right ) \text {Subst}\left (\int x^6 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {5 a^3 c^6 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^6 \sec ^3(e+f x) \tan ^3(e+f x)}{16 f}-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {3 a^3 c^6 \tan ^7(e+f x)}{7 f}-\frac {1}{8} \left (5 a^3 c^6\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx-\frac {1}{16} \left (15 a^3 c^6\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx+\frac {\left (a^3 c^6\right ) \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {5 a^3 c^6 \sec (e+f x) \tan (e+f x)}{16 f}-\frac {15 a^3 c^6 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac {5 a^3 c^6 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^6 \sec ^3(e+f x) \tan ^3(e+f x)}{16 f}-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac {a^3 c^6 \tan ^9(e+f x)}{9 f}+\frac {1}{64} \left (15 a^3 c^6\right ) \int \sec ^3(e+f x) \, dx+\frac {1}{16} \left (5 a^3 c^6\right ) \int \sec (e+f x) \, dx\\ &=\frac {5 a^3 c^6 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {25 a^3 c^6 \sec (e+f x) \tan (e+f x)}{128 f}-\frac {15 a^3 c^6 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac {5 a^3 c^6 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^6 \sec ^3(e+f x) \tan ^3(e+f x)}{16 f}-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac {a^3 c^6 \tan ^9(e+f x)}{9 f}+\frac {1}{128} \left (15 a^3 c^6\right ) \int \sec (e+f x) \, dx\\ &=\frac {55 a^3 c^6 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac {25 a^3 c^6 \sec (e+f x) \tan (e+f x)}{128 f}-\frac {15 a^3 c^6 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac {5 a^3 c^6 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^6 \sec ^3(e+f x) \tan ^3(e+f x)}{16 f}-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac {a^3 c^6 \tan ^9(e+f x)}{9 f}\\ \end {align*}

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Mathematica [A]
time = 3.26, size = 122, normalized size = 0.54 \begin {gather*} \frac {a^3 c^6 \left (443520 \tanh ^{-1}(\sin (e+f x))-\sec ^9(e+f x) (-88704 \sin (e+f x)+88074 \sin (2 (e+f x))+37632 \sin (3 (e+f x))-2142 \sin (4 (e+f x))+2304 \sin (5 (e+f x))+39858 \sin (6 (e+f x))-7488 \sin (7 (e+f x))+4599 \sin (8 (e+f x))+1856 \sin (9 (e+f x)))\right )}{1032192 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c^6*(443520*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^9*(-88704*Sin[e + f*x] + 88074*Sin[2*(e + f*x)] + 37632*
Sin[3*(e + f*x)] - 2142*Sin[4*(e + f*x)] + 2304*Sin[5*(e + f*x)] + 39858*Sin[6*(e + f*x)] - 7488*Sin[7*(e + f*
x)] + 4599*Sin[8*(e + f*x)] + 1856*Sin[9*(e + f*x)])))/(1032192*f)

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Maple [A]
time = 0.35, size = 345, normalized size = 1.52

method	result	size
risch	\(\frac {i a^{3} c^{6} \left (4599 \,{\mathrm e}^{17 i \left (f x +e \right )}-24192 \,{\mathrm e}^{16 i \left (f x +e \right )}+39858 \,{\mathrm e}^{15 i \left (f x +e \right )}-64512 \,{\mathrm e}^{14 i \left (f x +e \right )}-2142 \,{\mathrm e}^{13 i \left (f x +e \right )}-118272 \,{\mathrm e}^{12 i \left (f x +e \right )}+88074 \,{\mathrm e}^{11 i \left (f x +e \right )}-322560 \,{\mathrm e}^{10 i \left (f x +e \right )}-145152 \,{\mathrm e}^{8 i \left (f x +e \right )}-88074 \,{\mathrm e}^{7 i \left (f x +e \right )}-193536 \,{\mathrm e}^{6 i \left (f x +e \right )}+2142 \,{\mathrm e}^{5 i \left (f x +e \right )}-69120 \,{\mathrm e}^{4 i \left (f x +e \right )}-39858 \,{\mathrm e}^{3 i \left (f x +e \right )}-9216 \,{\mathrm e}^{2 i \left (f x +e \right )}-4599 \,{\mathrm e}^{i \left (f x +e \right )}-3712\right )}{4032 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{9}}-\frac {55 a^{3} c^{6} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{128 f}+\frac {55 a^{3} c^{6} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{128 f}\)	\(253\)
derivativedivides	\(\frac {-a^{3} c^{6} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (f x +e \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (f x +e \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (f x +e \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (f x +e \right )\right )}{315}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \tan \left (f x +e \right )+a^{3} c^{6} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+8 a^{3} c^{6} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+6 a^{3} c^{6} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-6 a^{3} c^{6} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-8 a^{3} c^{6} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \left (-\left (-\frac {\left (\sec ^{7}\left (f x +e \right )\right )}{8}-\frac {7 \left (\sec ^{5}\left (f x +e \right )\right )}{48}-\frac {35 \left (\sec ^{3}\left (f x +e \right )\right )}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\)	\(345\)
default	\(\frac {-a^{3} c^{6} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (f x +e \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (f x +e \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (f x +e \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (f x +e \right )\right )}{315}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \tan \left (f x +e \right )+a^{3} c^{6} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+8 a^{3} c^{6} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+6 a^{3} c^{6} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-6 a^{3} c^{6} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-8 a^{3} c^{6} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \left (-\left (-\frac {\left (\sec ^{7}\left (f x +e \right )\right )}{8}-\frac {7 \left (\sec ^{5}\left (f x +e \right )\right )}{48}-\frac {35 \left (\sec ^{3}\left (f x +e \right )\right )}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\)	\(345\)

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))^6,x,method=_RETURNVERBOSE)

[Out]

1/f*(-a^3*c^6*(-128/315-1/9*sec(f*x+e)^8-8/63*sec(f*x+e)^6-16/105*sec(f*x+e)^4-64/315*sec(f*x+e)^2)*tan(f*x+e)
-3*a^3*c^6*tan(f*x+e)+a^3*c^6*ln(sec(f*x+e)+tan(f*x+e))+8*a^3*c^6*(-(-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)))+6*a^3*c^6*(-8/15-1/5*sec(f*x+e)^4-4/15*sec(f*x+e)^2)*ta
n(f*x+e)-6*a^3*c^6*(-(-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)))-8*a^3*c^6*(-
2/3-1/3*sec(f*x+e)^2)*tan(f*x+e)-3*a^3*c^6*(-(-1/8*sec(f*x+e)^7-7/48*sec(f*x+e)^5-35/192*sec(f*x+e)^3-35/128*s
ec(f*x+e))*tan(f*x+e)+35/128*ln(sec(f*x+e)+tan(f*x+e))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (224) = 448\).
time = 0.28, size = 480, normalized size = 2.11 \begin {gather*} \frac {256 \, {\left (35 \, \tan \left (f x + e\right )^{9} + 180 \, \tan \left (f x + e\right )^{7} + 378 \, \tan \left (f x + e\right )^{5} + 420 \, \tan \left (f x + e\right )^{3} + 315 \, \tan \left (f x + e\right )\right )} a^{3} c^{6} - 32256 \, {\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^{6} + 215040 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{6} + 315 \, a^{3} c^{6} {\left (\frac {2 \, {\left (105 \, \sin \left (f x + e\right )^{7} - 385 \, \sin \left (f x + e\right )^{5} + 511 \, \sin \left (f x + e\right )^{3} - 279 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{8} - 4 \, \sin \left (f x + e\right )^{6} + 6 \, \sin \left (f x + e\right )^{4} - 4 \, \sin \left (f x + e\right )^{2} + 1} - 105 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 105 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 6720 \, a^{3} c^{6} {\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 )} + 30240 \, a^{3} c^{6} {\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 )} + 80640 \, a^{3} c^{6} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 241920 \, a^{3} c^{6} \tan \left (f x + e\right )}{80640 \, f} \end {gather*}

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))^6,x, algorithm="maxima")

[Out]

1/80640*(256*(35*tan(f*x + e)^9 + 180*tan(f*x + e)^7 + 378*tan(f*x + e)^5 + 420*tan(f*x + e)^3 + 315*tan(f*x +
e))*a^3*c^6 - 32256*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*c^6 + 215040*(tan(f*x + e)^3
+ 3*tan(f*x + e))*a^3*c^6 + 315*a^3*c^6*(2*(105*sin(f*x + e)^7 - 385*sin(f*x + e)^5 + 511*sin(f*x + e)^3 - 27
9*sin(f*x + e))/(sin(f*x + e)^8 - 4*sin(f*x + e)^6 + 6*sin(f*x + e)^4 - 4*sin(f*x + e)^2 + 1) - 105*log(sin(f*
x + e) + 1) + 105*log(sin(f*x + e) - 1)) - 6720*a^3*c^6*(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)) + 30240*a^3*c^6*(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)) + 80640*a^3*c^6*log(sec(f*x + e) + tan(f*x + e)) - 241920*a^3
*c^6*tan(f*x + e))/f

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Fricas [A]
time = 2.48, size = 223, normalized size = 0.98 \begin {gather*} \frac {3465 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3465 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (3712 \, a^{3} c^{6} \cos \left (f x + e\right )^{8} + 4599 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 10240 \, a^{3} c^{6} \cos \left (f x + e\right )^{6} + 3066 \, a^{3} c^{6} \cos \left (f x + e\right )^{5} + 8448 \, a^{3} c^{6} \cos \left (f x + e\right )^{4} - 7224 \, a^{3} c^{6} \cos \left (f x + e\right )^{3} - 1024 \, a^{3} c^{6} \cos \left (f x + e\right )^{2} + 3024 \, a^{3} c^{6} \cos \left (f x + e\right ) - 896 \, a^{3} c^{6}\right )} \sin \left (f x + e\right )}{16128 \, f \cos \left (f x + e\right )^{9}} \end {gather*}

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))^6,x, algorithm="fricas")

[Out]

1/16128*(3465*a^3*c^6*cos(f*x + e)^9*log(sin(f*x + e) + 1) - 3465*a^3*c^6*cos(f*x + e)^9*log(-sin(f*x + e) + 1
) - 2*(3712*a^3*c^6*cos(f*x + e)^8 + 4599*a^3*c^6*cos(f*x + e)^7 - 10240*a^3*c^6*cos(f*x + e)^6 + 3066*a^3*c^6
*cos(f*x + e)^5 + 8448*a^3*c^6*cos(f*x + e)^4 - 7224*a^3*c^6*cos(f*x + e)^3 - 1024*a^3*c^6*cos(f*x + e)^2 + 30
24*a^3*c^6*cos(f*x + e) - 896*a^3*c^6)*sin(f*x + e))/(f*cos(f*x + e)^9)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} c^{6} \left (\int \sec {\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int 8 \sec ^{4}{\left (e + f x \right )}\, dx + \int \left (- 6 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- 6 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int 8 \sec ^{7}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{9}{\left (e + f x \right )}\right )\, dx + \int \sec ^{10}{\left (e + f x \right )}\, dx\right ) \end {gather*}

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))**6,x)

[Out]

a**3*c**6*(Integral(sec(e + f*x), x) + Integral(-3*sec(e + f*x)**2, x) + Integral(8*sec(e + f*x)**4, x) + Inte
gral(-6*sec(e + f*x)**5, x) + Integral(-6*sec(e + f*x)**6, x) + Integral(8*sec(e + f*x)**7, x) + Integral(-3*s
ec(e + f*x)**9, x) + Integral(sec(e + f*x)**10, x))

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Giac [A]
time = 0.63, size = 235, normalized size = 1.04 \begin {gather*} \frac {3465 \, a^{3} c^{6} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3465 \, a^{3} c^{6} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3465 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{17} - 30030 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{15} + 115038 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 334602 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 360448 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 255222 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 115038 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30030 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3465 \, a^{3} c^{6} \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 )}^{9}}}{8064 \, f} \end {gather*}

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))^6,x, algorithm="giac")

[Out]

1/8064*(3465*a^3*c^6*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 3465*a^3*c^6*log(abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*
(3465*a^3*c^6*tan(1/2*f*x + 1/2*e)^17 - 30030*a^3*c^6*tan(1/2*f*x + 1/2*e)^15 + 115038*a^3*c^6*tan(1/2*f*x + 1
/2*e)^13 + 334602*a^3*c^6*tan(1/2*f*x + 1/2*e)^11 - 360448*a^3*c^6*tan(1/2*f*x + 1/2*e)^9 + 255222*a^3*c^6*tan
(1/2*f*x + 1/2*e)^7 - 115038*a^3*c^6*tan(1/2*f*x + 1/2*e)^5 + 30030*a^3*c^6*tan(1/2*f*x + 1/2*e)^3 - 3465*a^3*
c^6*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^9)/f

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Mupad [B]
time = 5.58, size = 316, normalized size = 1.39 \begin {gather*} \frac {55\,a^3\,c^6\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{64\,f}-\frac {\frac {55\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{17}}{64}-\frac {715\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}}{96}+\frac {913\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{32}+\frac {18589\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{224}-\frac {5632\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{63}+\frac {14179\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{224}-\frac {913\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{32}+\frac {715\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{96}-\frac {55\,a^3\,c^6\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{64}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{18}-9\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-84\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-36\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \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))^6)/cos(e + f*x),x)

[Out]

(55*a^3*c^6*atanh(tan(e/2 + (f*x)/2)))/(64*f) - ((715*a^3*c^6*tan(e/2 + (f*x)/2)^3)/96 - (913*a^3*c^6*tan(e/2
+ (f*x)/2)^5)/32 + (14179*a^3*c^6*tan(e/2 + (f*x)/2)^7)/224 - (5632*a^3*c^6*tan(e/2 + (f*x)/2)^9)/63 + (18589*
a^3*c^6*tan(e/2 + (f*x)/2)^11)/224 + (913*a^3*c^6*tan(e/2 + (f*x)/2)^13)/32 - (715*a^3*c^6*tan(e/2 + (f*x)/2)^
15)/96 + (55*a^3*c^6*tan(e/2 + (f*x)/2)^17)/64 - (55*a^3*c^6*tan(e/2 + (f*x)/2))/64)/(f*(9*tan(e/2 + (f*x)/2)^
2 - 36*tan(e/2 + (f*x)/2)^4 + 84*tan(e/2 + (f*x)/2)^6 - 126*tan(e/2 + (f*x)/2)^8 + 126*tan(e/2 + (f*x)/2)^10 -
84*tan(e/2 + (f*x)/2)^12 + 36*tan(e/2 + (f*x)/2)^14 - 9*tan(e/2 + (f*x)/2)^16 + tan(e/2 + (f*x)/2)^18 - 1))

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