\(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx\) [210]

Optimal result

Integrand size = 31, antiderivative size = 183 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) \text {arctanh}(\sin (e+f x))}{2 a f}-\frac {(3 c-4 d) d (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a f}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {d \left (4 \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right )+d \left (6 c^2-20 c d+9 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a f} \]

[Out]

1/2*d*(8*c^3-12*c^2*d+12*c*d^2-3*d^3)*arctanh(sin(f*x+e))/a/f-1/3*(3*c-4*d)*d*(c+d*sec(f*x+e))^2*tan(f*x+e)/a/
f+(c-d)*(c+d*sec(f*x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x+e))-1/6*d*(12*c^3-64*c^2*d+48*c*d^2-16*d^3+d*(6*c^2-20*c*
d+9*d^2)*sec(f*x+e))*tan(f*x+e)/a/f

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4072, 100, 158, 152, 65, 223, 209} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {d \tan (e+f x) \left (d \left (6 c^2-20 c d+9 d^2\right ) \sec (e+f x)+4 \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right )\right )}{6 a f}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{f (a \sec (e+f x)+a)}-\frac {d (3 c-4 d) \tan (e+f x) (c+d \sec (e+f x))^2}{3 a f} \]

[In]

Int[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x]),x]

[Out]

(d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f
*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((3*c - 4*d)*d*(c + d*Sec[e + f*x])^2*Tan[e + f*x
])/(3*a*f) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) - (d*(4*(3*c^3 - 16*c^2*d
+ 12*c*d^2 - 4*d^3) + d*(6*c^2 - 20*c*d + 9*d^2)*Sec[e + f*x])*Tan[e + f*x])/(6*a*f)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 158

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 4072

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[a^2*g*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]
])), Subst[Int[(g*x)^(p - 1)*(a + b*x)^(m - 1/2)*((c + d*x)^n/Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; Free
Q[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p,
1] || IntegerQ[m - 1/2])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^4}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x)^2 \left (-a^2 (4 c-3 d) d+a^2 (3 c-4 d) d x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {(3 c-4 d) d (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a f}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x) \left (a^4 d \left (12 c^2-15 c d+8 d^2\right )-a^4 d \left (6 c^2-20 c d+9 d^2\right ) x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{3 a^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {(3 c-4 d) d (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a f}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {d \left (4 \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right )+d \left (6 c^2-20 c d+9 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a f}-\frac {\left (a d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {(3 c-4 d) d (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a f}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {d \left (4 \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right )+d \left (6 c^2-20 c d+9 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a f}+\frac {\left (d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {(3 c-4 d) d (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a f}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {d \left (4 \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right )+d \left (6 c^2-20 c d+9 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a f}+\frac {\left (d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d \left (8 c^3-12 c^2 d+12 c d^2-3 d^3\right ) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-4 d) d (c+d \sec (e+f x))^2 \tan (e+f x)}{3 a f}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {d \left (4 \left (3 c^3-16 c^2 d+12 c d^2-4 d^3\right )+d \left (6 c^2-20 c d+9 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1243\) vs. \(2(183)=366\).

Time = 7.85 (sec) , antiderivative size = 1243, normalized size of antiderivative = 6.79 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {\left (-8 c^3 d+12 c^2 d^2-12 c d^3+3 d^4\right ) \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \cos ^3(e+f x) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) (c+d \sec (e+f x))^4}{f (d+c \cos (e+f x))^4 (a+a \sec (e+f x))}+\frac {\left (8 c^3 d-12 c^2 d^2+12 c d^3-3 d^4\right ) \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \cos ^3(e+f x) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) (c+d \sec (e+f x))^4}{f (d+c \cos (e+f x))^4 (a+a \sec (e+f x))}+\frac {\cos \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec (e) (c+d \sec (e+f x))^4 \left (-18 c^4 \sin \left (\frac {f x}{2}\right )+72 c^3 d \sin \left (\frac {f x}{2}\right )-36 c^2 d^2 \sin \left (\frac {f x}{2}\right )+24 c d^3 \sin \left (\frac {f x}{2}\right )+6 d^4 \sin \left (\frac {f x}{2}\right )+18 c^4 \sin \left (\frac {3 f x}{2}\right )-72 c^3 d \sin \left (\frac {3 f x}{2}\right )+180 c^2 d^2 \sin \left (\frac {3 f x}{2}\right )-108 c d^3 \sin \left (\frac {3 f x}{2}\right )+39 d^4 \sin \left (\frac {3 f x}{2}\right )-72 c^2 d^2 \sin \left (e-\frac {f x}{2}\right )+48 c d^3 \sin \left (e-\frac {f x}{2}\right )-24 d^4 \sin \left (e-\frac {f x}{2}\right )-36 c^2 d^2 \sin \left (e+\frac {f x}{2}\right )+24 c d^3 \sin \left (e+\frac {f x}{2}\right )-6 d^4 \sin \left (e+\frac {f x}{2}\right )-18 c^4 \sin \left (2 e+\frac {f x}{2}\right )+72 c^3 d \sin \left (2 e+\frac {f x}{2}\right )-144 c^2 d^2 \sin \left (2 e+\frac {f x}{2}\right )+96 c d^3 \sin \left (2 e+\frac {f x}{2}\right )-24 d^4 \sin \left (2 e+\frac {f x}{2}\right )+72 c^2 d^2 \sin \left (e+\frac {3 f x}{2}\right )-36 c d^3 \sin \left (e+\frac {3 f x}{2}\right )+21 d^4 \sin \left (e+\frac {3 f x}{2}\right )+18 c^4 \sin \left (2 e+\frac {3 f x}{2}\right )-72 c^3 d \sin \left (2 e+\frac {3 f x}{2}\right )+72 c^2 d^2 \sin \left (2 e+\frac {3 f x}{2}\right )-36 c d^3 \sin \left (2 e+\frac {3 f x}{2}\right )+9 d^4 \sin \left (2 e+\frac {3 f x}{2}\right )-36 c^2 d^2 \sin \left (3 e+\frac {3 f x}{2}\right )+36 c d^3 \sin \left (3 e+\frac {3 f x}{2}\right )-9 d^4 \sin \left (3 e+\frac {3 f x}{2}\right )+36 c^2 d^2 \sin \left (e+\frac {5 f x}{2}\right )-12 c d^3 \sin \left (e+\frac {5 f x}{2}\right )+7 d^4 \sin \left (e+\frac {5 f x}{2}\right )-6 c^4 \sin \left (2 e+\frac {5 f x}{2}\right )+24 c^3 d \sin \left (2 e+\frac {5 f x}{2}\right )+12 c d^3 \sin \left (2 e+\frac {5 f x}{2}\right )+d^4 \sin \left (2 e+\frac {5 f x}{2}\right )+12 c d^3 \sin \left (3 e+\frac {5 f x}{2}\right )-3 d^4 \sin \left (3 e+\frac {5 f x}{2}\right )-6 c^4 \sin \left (4 e+\frac {5 f x}{2}\right )+24 c^3 d \sin \left (4 e+\frac {5 f x}{2}\right )-36 c^2 d^2 \sin \left (4 e+\frac {5 f x}{2}\right )+36 c d^3 \sin \left (4 e+\frac {5 f x}{2}\right )-9 d^4 \sin \left (4 e+\frac {5 f x}{2}\right )+6 c^4 \sin \left (2 e+\frac {7 f x}{2}\right )-24 c^3 d \sin \left (2 e+\frac {7 f x}{2}\right )+72 c^2 d^2 \sin \left (2 e+\frac {7 f x}{2}\right )-48 c d^3 \sin \left (2 e+\frac {7 f x}{2}\right )+16 d^4 \sin \left (2 e+\frac {7 f x}{2}\right )+36 c^2 d^2 \sin \left (3 e+\frac {7 f x}{2}\right )-24 c d^3 \sin \left (3 e+\frac {7 f x}{2}\right )+10 d^4 \sin \left (3 e+\frac {7 f x}{2}\right )+6 c^4 \sin \left (4 e+\frac {7 f x}{2}\right )-24 c^3 d \sin \left (4 e+\frac {7 f x}{2}\right )+36 c^2 d^2 \sin \left (4 e+\frac {7 f x}{2}\right )-24 c d^3 \sin \left (4 e+\frac {7 f x}{2}\right )+6 d^4 \sin \left (4 e+\frac {7 f x}{2}\right )\right )}{48 f (d+c \cos (e+f x))^4 (a+a \sec (e+f x))} \]

[In]

Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x]),x]

[Out]

((-8*c^3*d + 12*c^2*d^2 - 12*c*d^3 + 3*d^4)*Cos[e/2 + (f*x)/2]^2*Cos[e + f*x]^3*Log[Cos[e/2 + (f*x)/2] - Sin[e
/2 + (f*x)/2]]*(c + d*Sec[e + f*x])^4)/(f*(d + c*Cos[e + f*x])^4*(a + a*Sec[e + f*x])) + ((8*c^3*d - 12*c^2*d^
2 + 12*c*d^3 - 3*d^4)*Cos[e/2 + (f*x)/2]^2*Cos[e + f*x]^3*Log[Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]]*(c + d*
Sec[e + f*x])^4)/(f*(d + c*Cos[e + f*x])^4*(a + a*Sec[e + f*x])) + (Cos[e/2 + (f*x)/2]*Sec[e/2]*Sec[e]*(c + d*
Sec[e + f*x])^4*(-18*c^4*Sin[(f*x)/2] + 72*c^3*d*Sin[(f*x)/2] - 36*c^2*d^2*Sin[(f*x)/2] + 24*c*d^3*Sin[(f*x)/2
] + 6*d^4*Sin[(f*x)/2] + 18*c^4*Sin[(3*f*x)/2] - 72*c^3*d*Sin[(3*f*x)/2] + 180*c^2*d^2*Sin[(3*f*x)/2] - 108*c*
d^3*Sin[(3*f*x)/2] + 39*d^4*Sin[(3*f*x)/2] - 72*c^2*d^2*Sin[e - (f*x)/2] + 48*c*d^3*Sin[e - (f*x)/2] - 24*d^4*
Sin[e - (f*x)/2] - 36*c^2*d^2*Sin[e + (f*x)/2] + 24*c*d^3*Sin[e + (f*x)/2] - 6*d^4*Sin[e + (f*x)/2] - 18*c^4*S
in[2*e + (f*x)/2] + 72*c^3*d*Sin[2*e + (f*x)/2] - 144*c^2*d^2*Sin[2*e + (f*x)/2] + 96*c*d^3*Sin[2*e + (f*x)/2]
- 24*d^4*Sin[2*e + (f*x)/2] + 72*c^2*d^2*Sin[e + (3*f*x)/2] - 36*c*d^3*Sin[e + (3*f*x)/2] + 21*d^4*Sin[e + (3
*f*x)/2] + 18*c^4*Sin[2*e + (3*f*x)/2] - 72*c^3*d*Sin[2*e + (3*f*x)/2] + 72*c^2*d^2*Sin[2*e + (3*f*x)/2] - 36*
c*d^3*Sin[2*e + (3*f*x)/2] + 9*d^4*Sin[2*e + (3*f*x)/2] - 36*c^2*d^2*Sin[3*e + (3*f*x)/2] + 36*c*d^3*Sin[3*e +
(3*f*x)/2] - 9*d^4*Sin[3*e + (3*f*x)/2] + 36*c^2*d^2*Sin[e + (5*f*x)/2] - 12*c*d^3*Sin[e + (5*f*x)/2] + 7*d^4
*Sin[e + (5*f*x)/2] - 6*c^4*Sin[2*e + (5*f*x)/2] + 24*c^3*d*Sin[2*e + (5*f*x)/2] + 12*c*d^3*Sin[2*e + (5*f*x)/
2] + d^4*Sin[2*e + (5*f*x)/2] + 12*c*d^3*Sin[3*e + (5*f*x)/2] - 3*d^4*Sin[3*e + (5*f*x)/2] - 6*c^4*Sin[4*e + (
5*f*x)/2] + 24*c^3*d*Sin[4*e + (5*f*x)/2] - 36*c^2*d^2*Sin[4*e + (5*f*x)/2] + 36*c*d^3*Sin[4*e + (5*f*x)/2] -
9*d^4*Sin[4*e + (5*f*x)/2] + 6*c^4*Sin[2*e + (7*f*x)/2] - 24*c^3*d*Sin[2*e + (7*f*x)/2] + 72*c^2*d^2*Sin[2*e +
(7*f*x)/2] - 48*c*d^3*Sin[2*e + (7*f*x)/2] + 16*d^4*Sin[2*e + (7*f*x)/2] + 36*c^2*d^2*Sin[3*e + (7*f*x)/2] -
24*c*d^3*Sin[3*e + (7*f*x)/2] + 10*d^4*Sin[3*e + (7*f*x)/2] + 6*c^4*Sin[4*e + (7*f*x)/2] - 24*c^3*d*Sin[4*e +
(7*f*x)/2] + 36*c^2*d^2*Sin[4*e + (7*f*x)/2] - 24*c*d^3*Sin[4*e + (7*f*x)/2] + 6*d^4*Sin[4*e + (7*f*x)/2]))/(4
8*f*(d + c*Cos[e + f*x])^4*(a + a*Sec[e + f*x]))

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.50

method	result	size
parallelrisch	\(\frac {-12 \left (c^{3}-\frac {3}{2} c^{2} d +\frac {3}{2} c \,d^{2}-\frac {3}{8} d^{3}\right ) \left (\cos \left (f x +e \right )+\frac {\cos \left (3 f x +3 e \right )}{3}\right ) d \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+12 \left (c^{3}-\frac {3}{2} c^{2} d +\frac {3}{2} c \,d^{2}-\frac {3}{8} d^{3}\right ) \left (\cos \left (f x +e \right )+\frac {\cos \left (3 f x +3 e \right )}{3}\right ) d \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (\frac {1}{3} c^{4}-\frac {4}{3} c^{3} d +4 c^{2} d^{2}-\frac {8}{3} c \,d^{3}+\frac {8}{9} d^{4}\right ) \cos \left (3 f x +3 e \right )+\left (4 c^{2} d^{2}-\frac {4}{3} c \,d^{3}+\frac {7}{9} d^{4}\right ) \cos \left (2 f x +2 e \right )+\left (c^{4}-4 c^{3} d +12 c^{2} d^{2}-\frac {16}{3} c \,d^{3}+\frac {22}{9} d^{4}\right ) \cos \left (f x +e \right )+4 c^{2} d^{2}-\frac {4 c \,d^{3}}{3}+\frac {11 d^{4}}{9}\right )}{a f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\)	\(275\)
derivativedivides	\(\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{3} d +6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2} d^{2}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c \,d^{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4}-\frac {d^{4}}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {d \left (8 c^{3}-12 c^{2} d +12 c \,d^{2}-3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}-\frac {d^{2} \left (12 c^{2}-12 c d +5 d^{2}\right )}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {d^{3} \left (2 c -d \right )}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {d^{4}}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {d \left (8 c^{3}-12 c^{2} d +12 c \,d^{2}-3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}-\frac {d^{2} \left (12 c^{2}-12 c d +5 d^{2}\right )}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {d^{3} \left (2 c -d \right )}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{f a}\)	\(309\)
default	\(\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{3} d +6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2} d^{2}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c \,d^{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4}-\frac {d^{4}}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {d \left (8 c^{3}-12 c^{2} d +12 c \,d^{2}-3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}-\frac {d^{2} \left (12 c^{2}-12 c d +5 d^{2}\right )}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {d^{3} \left (2 c -d \right )}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {d^{4}}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {d \left (8 c^{3}-12 c^{2} d +12 c \,d^{2}-3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}-\frac {d^{2} \left (12 c^{2}-12 c d +5 d^{2}\right )}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {d^{3} \left (2 c -d \right )}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{f a}\)	\(309\)
norman	\(\frac {\frac {\left (c^{4}-4 c^{3} d +6 c^{2} d^{2}-4 c \,d^{3}+d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{a f}+\frac {\left (c^{4}-4 c^{3} d +18 c^{2} d^{2}-8 c \,d^{3}+4 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {\left (4 c^{4}-16 c^{3} d +36 c^{2} d^{2}-28 c \,d^{3}+9 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f}-\frac {\left (12 c^{4}-48 c^{3} d +180 c^{2} d^{2}-108 c \,d^{3}+37 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}+\frac {\left (18 c^{4}-72 c^{3} d +216 c^{2} d^{2}-156 c \,d^{3}+49 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}-\frac {d \left (8 c^{3}-12 c^{2} d +12 c \,d^{2}-3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a f}+\frac {d \left (8 c^{3}-12 c^{2} d +12 c \,d^{2}-3 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a f}\)	\(347\)
risch	\(\frac {i \left (36 c^{2} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}+144 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-108 c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-72 c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}-24 c^{3} d \,{\mathrm e}^{6 i \left (f x +e \right )}-36 c \,d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-96 c \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+36 c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}-12 c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}-36 c \,d^{3} {\mathrm e}^{6 i \left (f x +e \right )}-72 c^{3} d \,{\mathrm e}^{4 i \left (f x +e \right )}+36 c^{2} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+72 c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-48 c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+180 c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+16 d^{4}+6 c^{4}+9 d^{4} {\mathrm e}^{5 i \left (f x +e \right )}+7 d^{4} {\mathrm e}^{i \left (f x +e \right )}+39 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{6 i \left (f x +e \right )}+24 d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+24 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{6 i \left (f x +e \right )}+18 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+18 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}-24 c^{3} d +72 c^{2} d^{2}-48 c \,d^{3}\right )}{3 f a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{3}}-\frac {4 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{3}}{a f}+\frac {6 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2}}{a f}-\frac {6 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c}{a f}+\frac {3 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a f}+\frac {4 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{3}}{a f}-\frac {6 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2}}{a f}+\frac {6 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c}{a f}-\frac {3 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a f}\)	\(633\)

[In]

int(sec(f*x+e)*(c+d*sec(f*x+e))^4/(a+a*sec(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

3*(-4*(c^3-3/2*c^2*d+3/2*c*d^2-3/8*d^3)*(cos(f*x+e)+1/3*cos(3*f*x+3*e))*d*ln(tan(1/2*f*x+1/2*e)-1)+4*(c^3-3/2*
c^2*d+3/2*c*d^2-3/8*d^3)*(cos(f*x+e)+1/3*cos(3*f*x+3*e))*d*ln(tan(1/2*f*x+1/2*e)+1)+tan(1/2*f*x+1/2*e)*((1/3*c
^4-4/3*c^3*d+4*c^2*d^2-8/3*c*d^3+8/9*d^4)*cos(3*f*x+3*e)+(4*c^2*d^2-4/3*c*d^3+7/9*d^4)*cos(2*f*x+2*e)+(c^4-4*c
^3*d+12*c^2*d^2-16/3*c*d^3+22/9*d^4)*cos(f*x+e)+4*c^2*d^2-4/3*c*d^3+11/9*d^4))/a/f/(cos(3*f*x+3*e)+3*cos(f*x+e
))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.62 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {3 \, {\left ({\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + {\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left ({\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + {\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, d^{4} + 2 \, {\left (3 \, c^{4} - 12 \, c^{3} d + 36 \, c^{2} d^{2} - 24 \, c d^{3} + 8 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (36 \, c^{2} d^{2} - 12 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (12 \, c d^{3} - d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a f \cos \left (f x + e\right )^{4} + a f \cos \left (f x + e\right )^{3}\right )}} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^4/(a+a*sec(f*x+e)),x, algorithm="fricas")

[Out]

1/12*(3*((8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*cos(f*x + e)^4 + (8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*
cos(f*x + e)^3)*log(sin(f*x + e) + 1) - 3*((8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*cos(f*x + e)^4 + (8*c^3*d
- 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*cos(f*x + e)^3)*log(-sin(f*x + e) + 1) + 2*(2*d^4 + 2*(3*c^4 - 12*c^3*d + 36
*c^2*d^2 - 24*c*d^3 + 8*d^4)*cos(f*x + e)^3 + (36*c^2*d^2 - 12*c*d^3 + 7*d^4)*cos(f*x + e)^2 + (12*c*d^3 - d^4
)*cos(f*x + e))*sin(f*x + e))/(a*f*cos(f*x + e)^4 + a*f*cos(f*x + e)^3)

Sympy [F]

\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {\int \frac {c^{4} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{4} \sec ^{5}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {4 c d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {6 c^{2} d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {4 c^{3} d \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))**4/(a+a*sec(f*x+e)),x)

[Out]

(Integral(c**4*sec(e + f*x)/(sec(e + f*x) + 1), x) + Integral(d**4*sec(e + f*x)**5/(sec(e + f*x) + 1), x) + In
tegral(4*c*d**3*sec(e + f*x)**4/(sec(e + f*x) + 1), x) + Integral(6*c**2*d**2*sec(e + f*x)**3/(sec(e + f*x) +
1), x) + Integral(4*c**3*d*sec(e + f*x)**2/(sec(e + f*x) + 1), x))/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (174) = 348\).

Time = 0.22 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.26 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {d^{4} {\left (\frac {2 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {16 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a - \frac {3 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {a \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 12 \, c d^{3} {\left (\frac {2 \, {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 36 \, c^{2} d^{2} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (f x + e\right )}{{\left (a - \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 24 \, c^{3} d {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + \frac {6 \, c^{4} \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{6 \, f} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^4/(a+a*sec(f*x+e)),x, algorithm="maxima")

[Out]

1/6*(d^4*(2*(9*sin(f*x + e)/(cos(f*x + e) + 1) - 16*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^5/(c
os(f*x + e) + 1)^5)/(a - 3*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - a
*sin(f*x + e)^6/(cos(f*x + e) + 1)^6) - 9*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a + 9*log(sin(f*x + e)/(cos
(f*x + e) + 1) - 1)/a + 6*sin(f*x + e)/(a*(cos(f*x + e) + 1))) - 12*c*d^3*(2*(sin(f*x + e)/(cos(f*x + e) + 1)
- 3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a - 2*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x + e)^4/(cos(
f*x + e) + 1)^4) - 3*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a + 3*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a
+ 2*sin(f*x + e)/(a*(cos(f*x + e) + 1))) - 36*c^2*d^2*(log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a - log(sin(f
*x + e)/(cos(f*x + e) + 1) - 1)/a - 2*sin(f*x + e)/((a - a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e)
+ 1)) - sin(f*x + e)/(a*(cos(f*x + e) + 1))) + 24*c^3*d*(log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a - log(sin(
f*x + e)/(cos(f*x + e) + 1) - 1)/a - sin(f*x + e)/(a*(cos(f*x + e) + 1))) + 6*c^4*sin(f*x + e)/(a*(cos(f*x + e
) + 1)))/f

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.88 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {\frac {3 \, {\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a} - \frac {3 \, {\left (8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a} + \frac {6 \, {\left (c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a} - \frac {2 \, {\left (36 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 36 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 72 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 48 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 16 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a}}{6 \, f} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^4/(a+a*sec(f*x+e)),x, algorithm="giac")

[Out]

1/6*(3*(8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a - 3*(8*c^3*d - 12*c^2*d^
2 + 12*c*d^3 - 3*d^4)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a + 6*(c^4*tan(1/2*f*x + 1/2*e) - 4*c^3*d*tan(1/2*f*x
+ 1/2*e) + 6*c^2*d^2*tan(1/2*f*x + 1/2*e) - 4*c*d^3*tan(1/2*f*x + 1/2*e) + d^4*tan(1/2*f*x + 1/2*e))/a - 2*(3
6*c^2*d^2*tan(1/2*f*x + 1/2*e)^5 - 36*c*d^3*tan(1/2*f*x + 1/2*e)^5 + 15*d^4*tan(1/2*f*x + 1/2*e)^5 - 72*c^2*d^
2*tan(1/2*f*x + 1/2*e)^3 + 48*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 16*d^4*tan(1/2*f*x + 1/2*e)^3 + 36*c^2*d^2*tan(1/
2*f*x + 1/2*e) - 12*c*d^3*tan(1/2*f*x + 1/2*e) + 9*d^4*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^3*a
))/f

Mupad [B] (verification not implemented)

Time = 14.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.15 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx=\frac {\left (12\,c^2\,d^2-12\,c\,d^3+5\,d^4\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-24\,c^2\,d^2+16\,c\,d^3-\frac {16\,d^4}{3}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (12\,c^2\,d^2-4\,c\,d^3+3\,d^4\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (-a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-3\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (c-d\right )}^4}{a\,f}+\frac {d\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (8\,c^3-12\,c^2\,d+12\,c\,d^2-3\,d^3\right )}{a\,f} \]

[In]

int((c + d/cos(e + f*x))^4/(cos(e + f*x)*(a + a/cos(e + f*x))),x)

[Out]

(tan(e/2 + (f*x)/2)*(3*d^4 - 4*c*d^3 + 12*c^2*d^2) + tan(e/2 + (f*x)/2)^5*(5*d^4 - 12*c*d^3 + 12*c^2*d^2) - ta
n(e/2 + (f*x)/2)^3*((16*d^4)/3 - 16*c*d^3 + 24*c^2*d^2))/(f*(a - 3*a*tan(e/2 + (f*x)/2)^2 + 3*a*tan(e/2 + (f*x
)/2)^4 - a*tan(e/2 + (f*x)/2)^6)) + (tan(e/2 + (f*x)/2)*(c - d)^4)/(a*f) + (d*atanh(tan(e/2 + (f*x)/2))*(12*c*
d^2 - 12*c^2*d + 8*c^3 - 3*d^3))/(a*f)