3.21 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx\)

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

[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)

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Rubi [A]  time = 0.335235, antiderivative size = 227, normalized size of antiderivative = 1., 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 = {3958, 2611, 3770, 2607, 30, 3768, 14} \[ \frac{a^3 c^6 \tan ^9(e+f x)}{9 f}+\frac{4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac{55 a^3 c^6 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac{3 a^3 c^6 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}+\frac{5 a^3 c^6 \tan ^3(e+f x) \sec ^3(e+f x)}{16 f}-\frac{15 a^3 c^6 \tan (e+f x) \sec ^3(e+f x)}{64 f}-\frac{a^3 c^6 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^6 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{25 a^3 c^6 \tan (e+f x) \sec (e+f x)}{128 f} \]

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 3958

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]

Rule 2611

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 3770

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

Rule 2607

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 30

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

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 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]]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 3.39876, size = 122, normalized size = 0.54 \[ \frac{a^3 c^6 \left (443520 \tanh ^{-1}(\sin (e+f x))-(-88704 \sin (e+f x)+88074 \sin (2 (e+f x))+37632 \sin (3 (e+f x))-2142 \sin (4 (e+f x))+2304 \sin (5 (e+f x))+39858 \sin (6 (e+f x))-7488 \sin (7 (e+f x))+4599 \sin (8 (e+f x))+1856 \sin (9 (e+f x))) \sec ^9(e+f x)\right )}{1032192 f} \]

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 = 0.143, size = 242, normalized size = 1.1 \begin{align*} -{\frac{29\,{a}^{3}{c}^{6}\tan \left ( fx+e \right ) }{63\,f}}+{\frac{{a}^{3}{c}^{6}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{8}}{9\,f}}+{\frac{8\,{a}^{3}{c}^{6}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{6}}{63\,f}}-{\frac{22\,{a}^{3}{c}^{6}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{21\,f}}+{\frac{80\,{a}^{3}{c}^{6}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{63\,f}}+{\frac{55\,{a}^{3}{c}^{6}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{128\,f}}+{\frac{43\,{a}^{3}{c}^{6}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{5}}{48\,f}}-{\frac{73\,{a}^{3}{c}^{6} \left ( \sec \left ( fx+e \right ) \right ) ^{3}\tan \left ( fx+e \right ) }{192\,f}}-{\frac{73\,{a}^{3}{c}^{6}\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{128\,f}}-{\frac{3\,{a}^{3}{c}^{6}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{7}}{8\,f}} \end{align*}

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)

[Out]

-29/63/f*a^3*c^6*tan(f*x+e)+1/9/f*a^3*c^6*tan(f*x+e)*sec(f*x+e)^8+8/63/f*a^3*c^6*tan(f*x+e)*sec(f*x+e)^6-22/21
/f*a^3*c^6*tan(f*x+e)*sec(f*x+e)^4+80/63/f*a^3*c^6*tan(f*x+e)*sec(f*x+e)^2+55/128/f*a^3*c^6*ln(sec(f*x+e)+tan(
f*x+e))+43/48/f*a^3*c^6*tan(f*x+e)*sec(f*x+e)^5-73/192*a^3*c^6*sec(f*x+e)^3*tan(f*x+e)/f-73/128*a^3*c^6*sec(f*
x+e)*tan(f*x+e)/f-3/8/f*a^3*c^6*tan(f*x+e)*sec(f*x+e)^7

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Maxima [B]  time = 1.00825, size = 598, normalized size = 2.63 \begin{align*} \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{align*}

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 = 0.532053, size = 544, normalized size = 2.4 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} a^{3} c^{6} \left (\int \sec{\left (e + f x \right )}\, dx + \int - 3 \sec ^{2}{\left (e + f x \right )}\, dx + \int 8 \sec ^{4}{\left (e + f x \right )}\, dx + \int - 6 \sec ^{5}{\left (e + f x \right )}\, dx + \int - 6 \sec ^{6}{\left (e + f x \right )}\, dx + \int 8 \sec ^{7}{\left (e + f x \right )}\, dx + \int - 3 \sec ^{9}{\left (e + f x \right )}\, dx + \int \sec ^{10}{\left (e + f x \right )}\, dx\right ) \end{align*}

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 = 1.39888, size = 333, normalized size = 1.47 \begin{align*} \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{align*}

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