\(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx\) [226]

Optimal result

Integrand size = 31, antiderivative size = 287 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {d^3 \left (20 c^2-30 c d+13 d^2\right ) \text {arctanh}(\sin (e+f x))}{2 a^3 f}-\frac {2 d \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right ) \tan (e+f x)}{15 a^3 f}-\frac {d^2 \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x) \tan (e+f x)}{30 a^3 f}+\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3} \]

[Out]

1/2*d^3*(20*c^2-30*c*d+13*d^2)*arctanh(sin(f*x+e))/a^3/f-2/15*d*(2*c^4+15*c^3*d+72*c^2*d^2-180*c*d^3+76*d^4)*t
an(f*x+e)/a^3/f-1/30*d^2*(4*c^3+30*c^2*d+146*c*d^2-195*d^3)*sec(f*x+e)*tan(f*x+e)/a^3/f+1/15*(c-d)*(2*c^2+15*c
*d+76*d^2)*(c+d*sec(f*x+e))^2*tan(f*x+e)/f/(a^3+a^3*sec(f*x+e))+1/15*(c-d)*(2*c+11*d)*(c+d*sec(f*x+e))^3*tan(f
*x+e)/a/f/(a+a*sec(f*x+e))^2+1/5*(c-d)*(c+d*sec(f*x+e))^4*tan(f*x+e)/f/(a+a*sec(f*x+e))^3

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4072, 100, 155, 152, 65, 223, 209} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) \tan (e+f x) (c+d \sec (e+f x))^2}{15 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {d \tan (e+f x) \left (d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)+4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )\right )}{30 a^3 f}+\frac {d^3 \left (20 c^2-30 c d+13 d^2\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^4}{5 f (a \sec (e+f x)+a)^3}+\frac {(c-d) (2 c+11 d) \tan (e+f x) (c+d \sec (e+f x))^3}{15 a f (a \sec (e+f x)+a)^2} \]

[In]

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

[Out]

(d^3*(20*c^2 - 30*c*d + 13*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a^2
*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*(c + d*Sec[e + f*x]
)^2*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])) + ((c - d)*(2*c + 11*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])
/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3)
- (d*(4*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4) + d*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Sec[
e + f*x])*Tan[e + f*x])/(30*a^3*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 155

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

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)^5}{\sqrt {a-a x} (a+a x)^{7/2}} \, 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))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x)^3 \left (-a^2 (2 c-d) (c+4 d)+a^2 (2 c-7 d) d x\right )}{\sqrt {a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x)^2 \left (-a^4 \left (2 c^3+9 c^2 d+37 c d^2-33 d^3\right )+a^4 d \left (4 c^2+24 c d-43 d^2\right ) x\right )}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{15 a^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x) \left (-a^6 d^2 \left (2 c^2+165 c d-152 d^2\right )+a^6 d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{15 a^7 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}-\frac {\left (d^3 \left (20 c^2-30 c d+13 d^2\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 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}+\frac {\left (d^3 \left (20 c^2-30 c d+13 d^2\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 )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f}+\frac {\left (d^3 \left (20 c^2-30 c d+13 d^2\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 )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d^3 \left (20 c^2-30 c d+13 d^2\right ) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {(c-d) \left (2 c^2+15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{15 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac {(c-d) (2 c+11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {(c-d) (c+d \sec (e+f x))^4 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {d \left (4 \left (2 c^4+15 c^3 d+72 c^2 d^2-180 c d^3+76 d^4\right )+d \left (4 c^3+30 c^2 d+146 c d^2-195 d^3\right ) \sec (e+f x)\right ) \tan (e+f x)}{30 a^3 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.51 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.53 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {-480 d^3 \left (20 c^2-30 c d+13 d^2\right ) \cos ^6\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (29 c^5+105 c^4 d+340 c^3 d^2-1940 c^2 d^3+3420 c d^4-1354 d^5+3 \left (12 c^5+90 c^4 d+120 c^3 d^2-1020 c^2 d^3+1910 c d^4-777 d^5\right ) \cos (e+f x)+6 \left (6 c^5+20 c^4 d+60 c^3 d^2-360 c^2 d^3+630 c d^4-261 d^5\right ) \cos (2 (e+f x))+12 c^5 \cos (3 (e+f x))+90 c^4 d \cos (3 (e+f x))+120 c^3 d^2 \cos (3 (e+f x))-1020 c^2 d^3 \cos (3 (e+f x))+1710 c d^4 \cos (3 (e+f x))-717 d^5 \cos (3 (e+f x))+7 c^5 \cos (4 (e+f x))+15 c^4 d \cos (4 (e+f x))+20 c^3 d^2 \cos (4 (e+f x))-220 c^2 d^3 \cos (4 (e+f x))+360 c d^4 \cos (4 (e+f x))-152 d^5 \cos (4 (e+f x))\right ) \sec ^2(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{120 a^3 f (1+\cos (e+f x))^3} \]

[In]

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

[Out]

(-480*d^3*(20*c^2 - 30*c*d + 13*d^2)*Cos[(e + f*x)/2]^6*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2]]) + 2*Cos[(e + f*x)/2]*(29*c^5 + 105*c^4*d + 340*c^3*d^2 - 1940*c^2*d^3 + 3420*c
*d^4 - 1354*d^5 + 3*(12*c^5 + 90*c^4*d + 120*c^3*d^2 - 1020*c^2*d^3 + 1910*c*d^4 - 777*d^5)*Cos[e + f*x] + 6*(
6*c^5 + 20*c^4*d + 60*c^3*d^2 - 360*c^2*d^3 + 630*c*d^4 - 261*d^5)*Cos[2*(e + f*x)] + 12*c^5*Cos[3*(e + f*x)]
+ 90*c^4*d*Cos[3*(e + f*x)] + 120*c^3*d^2*Cos[3*(e + f*x)] - 1020*c^2*d^3*Cos[3*(e + f*x)] + 1710*c*d^4*Cos[3*
(e + f*x)] - 717*d^5*Cos[3*(e + f*x)] + 7*c^5*Cos[4*(e + f*x)] + 15*c^4*d*Cos[4*(e + f*x)] + 20*c^3*d^2*Cos[4*
(e + f*x)] - 220*c^2*d^3*Cos[4*(e + f*x)] + 360*c*d^4*Cos[4*(e + f*x)] - 152*d^5*Cos[4*(e + f*x)])*Sec[e + f*x
]^2*Sin[(e + f*x)/2])/(120*a^3*f*(1 + Cos[e + f*x])^3)

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.25

method	result	size
parallelrisch	\(\frac {-2400 \left (c^{2}-\frac {3}{2} c d +\frac {13}{20} d^{2}\right ) \left (1+\cos \left (2 f x +2 e \right )\right ) d^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+2400 \left (c^{2}-\frac {3}{2} c d +\frac {13}{20} d^{2}\right ) \left (1+\cos \left (2 f x +2 e \right )\right ) d^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+7 \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (6 \left (\frac {20}{7} c^{4} d +\frac {6}{7} c^{5}-\frac {261}{7} d^{5}+\frac {60}{7} c^{3} d^{2}-\frac {360}{7} c^{2} d^{3}+90 c \,d^{4}\right ) \cos \left (2 f x +2 e \right )+\frac {3 \left (4 c^{5}+30 c^{4} d +40 c^{3} d^{2}-340 c^{2} d^{3}+570 c \,d^{4}-239 d^{5}\right ) \cos \left (3 f x +3 e \right )}{7}+\left (c^{5}-\frac {152}{7} d^{5}+\frac {15}{7} c^{4} d +\frac {20}{7} c^{3} d^{2}-\frac {220}{7} c^{2} d^{3}+\frac {360}{7} c \,d^{4}\right ) \cos \left (4 f x +4 e \right )+3 \left (-111 d^{5}+\frac {120}{7} c^{3} d^{2}+\frac {12}{7} c^{5}+\frac {90}{7} c^{4} d -\frac {1020}{7} c^{2} d^{3}+\frac {1910}{7} c \,d^{4}\right ) \cos \left (f x +e \right )-\frac {1354 d^{5}}{7}+\frac {29 c^{5}}{7}+15 c^{4} d +\frac {340 c^{3} d^{2}}{7}-\frac {1940 c^{2} d^{3}}{7}+\frac {3420 c \,d^{4}}{7}\right )}{240 f \,a^{3} \left (1+\cos \left (2 f x +2 e \right )\right )}\)	\(360\)
derivativedivides	\(\frac {\frac {c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-c^{4} d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+2 c^{3} d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-2 c^{2} d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+c \,d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-\frac {d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-\frac {2 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {20 c^{3} d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-\frac {40 c^{2} d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+10 c \,d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-\frac {8 d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 c^{4} d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+10 c^{3} d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-70 c^{2} d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+85 c \,d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-31 d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {2 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-2 d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-\frac {2 d^{4} \left (10 c -7 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+2 d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {2 d^{4} \left (10 c -7 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{4 f \,a^{3}}\)	\(441\)
default	\(\frac {\frac {c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-c^{4} d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+2 c^{3} d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-2 c^{2} d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+c \,d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-\frac {d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-\frac {2 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {20 c^{3} d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-\frac {40 c^{2} d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+10 c \,d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-\frac {8 d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 c^{4} d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+10 c^{3} d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-70 c^{2} d^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+85 c \,d^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-31 d^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {2 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-2 d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-\frac {2 d^{4} \left (10 c -7 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+2 d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {2 d^{4} \left (10 c -7 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{4 f \,a^{3}}\)	\(441\)
norman	\(\frac {\frac {\left (c^{5}-5 c^{4} d +10 c^{3} d^{2}-10 c^{2} d^{3}+5 c \,d^{4}-d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{20 a f}-\frac {5 \left (c^{5}-3 c^{4} d +2 c^{3} d^{2}+2 c^{2} d^{3}-3 c \,d^{4}+d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{12 a f}-\frac {\left (c^{5}+5 c^{4} d +10 c^{3} d^{2}-70 c^{2} d^{3}+125 c \,d^{4}-51 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 a f}+\frac {\left (17 c^{5}+75 c^{4} d +130 c^{3} d^{2}-1010 c^{2} d^{3}+1725 c \,d^{4}-721 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{12 a f}+\frac {\left (19 c^{5}-15 c^{4} d -10 c^{3} d^{2}-70 c^{2} d^{3}+135 c \,d^{4}-59 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{12 a f}-\frac {\left (41 c^{5}+45 c^{4} d +10 c^{3} d^{2}-710 c^{2} d^{3}+1125 c \,d^{4}-475 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{12 a f}+\frac {\left (53 c^{5}+135 c^{4} d +130 c^{3} d^{2}-1730 c^{2} d^{3}+2745 c \,d^{4}-1165 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{12 a f}-\frac {\left (203 c^{5}+735 c^{4} d +1030 c^{3} d^{2}-9530 c^{2} d^{3}+15615 c \,d^{4}-6613 d^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{60 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5} a^{2}}-\frac {d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{3} f}+\frac {d^{3} \left (20 c^{2}-30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{3} f}\)	\(552\)
risch	\(\text {Expression too large to display}\)	\(915\)

[In]

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

[Out]

1/240*(-2400*(c^2-3/2*c*d+13/20*d^2)*(1+cos(2*f*x+2*e))*d^3*ln(tan(1/2*f*x+1/2*e)-1)+2400*(c^2-3/2*c*d+13/20*d
^2)*(1+cos(2*f*x+2*e))*d^3*ln(tan(1/2*f*x+1/2*e)+1)+7*sec(1/2*f*x+1/2*e)^4*tan(1/2*f*x+1/2*e)*(6*(20/7*c^4*d+6
/7*c^5-261/7*d^5+60/7*c^3*d^2-360/7*c^2*d^3+90*c*d^4)*cos(2*f*x+2*e)+3/7*(4*c^5+30*c^4*d+40*c^3*d^2-340*c^2*d^
3+570*c*d^4-239*d^5)*cos(3*f*x+3*e)+(c^5-152/7*d^5+15/7*c^4*d+20/7*c^3*d^2-220/7*c^2*d^3+360/7*c*d^4)*cos(4*f*
x+4*e)+3*(-111*d^5+120/7*c^3*d^2+12/7*c^5+90/7*c^4*d-1020/7*c^2*d^3+1910/7*c*d^4)*cos(f*x+e)-1354/7*d^5+29/7*c
^5+15*c^4*d+340/7*c^3*d^2-1940/7*c^2*d^3+3420/7*c*d^4))/f/a^3/(1+cos(2*f*x+2*e))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.75 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {15 \, {\left ({\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{5} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left ({\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{5} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (15 \, d^{5} + 2 \, {\left (7 \, c^{5} + 15 \, c^{4} d + 20 \, c^{3} d^{2} - 220 \, c^{2} d^{3} + 360 \, c d^{4} - 152 \, d^{5}\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (4 \, c^{5} + 30 \, c^{4} d + 40 \, c^{3} d^{2} - 340 \, c^{2} d^{3} + 570 \, c d^{4} - 239 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (4 \, c^{5} + 30 \, c^{4} d + 140 \, c^{3} d^{2} - 640 \, c^{2} d^{3} + 1170 \, c d^{4} - 479 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + 15 \, {\left (10 \, c d^{4} - 3 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{60 \, {\left (a^{3} f \cos \left (f x + e\right )^{5} + 3 \, a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + a^{3} f \cos \left (f x + e\right )^{2}\right )}} \]

[In]

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

[Out]

1/60*(15*((20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)^5 + 3*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)^4
+ 3*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)^3 + (20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)^2)*log(sin
(f*x + e) + 1) - 15*((20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)^5 + 3*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(
f*x + e)^4 + 3*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)^3 + (20*c^2*d^3 - 30*c*d^4 + 13*d^5)*cos(f*x + e)
^2)*log(-sin(f*x + e) + 1) + 2*(15*d^5 + 2*(7*c^5 + 15*c^4*d + 20*c^3*d^2 - 220*c^2*d^3 + 360*c*d^4 - 152*d^5)
*cos(f*x + e)^4 + 3*(4*c^5 + 30*c^4*d + 40*c^3*d^2 - 340*c^2*d^3 + 570*c*d^4 - 239*d^5)*cos(f*x + e)^3 + (4*c^
5 + 30*c^4*d + 140*c^3*d^2 - 640*c^2*d^3 + 1170*c*d^4 - 479*d^5)*cos(f*x + e)^2 + 15*(10*c*d^4 - 3*d^5)*cos(f*
x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^5 + 3*a^3*f*cos(f*x + e)^4 + 3*a^3*f*cos(f*x + e)^3 + a^3*f*cos(f*x
+ e)^2)

Sympy [F]

\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {\int \frac {c^{5} \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{5} \sec ^{6}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {5 c d^{4} \sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {10 c^{2} d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {10 c^{3} d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {5 c^{4} d \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]

[In]

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

[Out]

(Integral(c**5*sec(e + f*x)/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(d**5*sec
(e + f*x)**6/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(5*c*d**4*sec(e + f*x)**
5/(sec(e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(10*c**2*d**3*sec(e + f*x)**4/(sec(
e + f*x)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(10*c**3*d**2*sec(e + f*x)**3/(sec(e + f*x
)**3 + 3*sec(e + f*x)**2 + 3*sec(e + f*x) + 1), x) + Integral(5*c**4*d*sec(e + f*x)**2/(sec(e + f*x)**3 + 3*se
c(e + f*x)**2 + 3*sec(e + f*x) + 1), x))/a**3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (275) = 550\).

Time = 0.24 (sec) , antiderivative size = 689, normalized size of antiderivative = 2.40 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {d^{5} {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {7 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac {2 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {390 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {390 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} - 15 \, c d^{4} {\left (\frac {40 \, \sin \left (f x + e\right )}{{\left (a^{3} - \frac {a^{3} \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 {\frac {85 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + 10 \, c^{2} d^{3} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} - \frac {10 \, c^{3} d^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {c^{5} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {15 \, c^{4} d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]

[In]

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

[Out]

-1/60*(d^5*(60*(5*sin(f*x + e)/(cos(f*x + e) + 1) - 7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^3 - 2*a^3*sin(f*
x + e)^2/(cos(f*x + e) + 1)^2 + a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4) + (465*sin(f*x + e)/(cos(f*x + e) + 1
) + 40*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 390*log(sin(f*x + e)
/(cos(f*x + e) + 1) + 1)/a^3 + 390*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) - 15*c*d^4*(40*sin(f*x + e)/(
(a^3 - a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1)) + (85*sin(f*x + e)/(cos(f*x + e) + 1) + 10
*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 60*log(sin(f*x + e)/(cos(f*x
+ e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) + 10*c^2*d^3*((105*sin(f*x + e)/(cos(f*
x + e) + 1) + 20*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3 - 60*log(sin
(f*x + e)/(cos(f*x + e) + 1) + 1)/a^3 + 60*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/a^3) - 10*c^3*d^2*(15*sin(
f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/
a^3 - c^5*(15*sin(f*x + e)/(cos(f*x + e) + 1) - 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos
(f*x + e) + 1)^5)/a^3 - 15*c^4*d*(5*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/a^3
)/f

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.76 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {\frac {30 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac {60 \, {\left (10 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 7 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, d^{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 )}^{2} a^{3}} + \frac {3 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a^{12} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30 \, a^{12} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 30 \, a^{12} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15 \, a^{12} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, a^{12} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, a^{12} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 200 \, a^{12} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 150 \, a^{12} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 40 \, a^{12} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 75 \, a^{12} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 150 \, a^{12} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1050 \, a^{12} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1275 \, a^{12} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 465 \, a^{12} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15}}}{60 \, f} \]

[In]

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

[Out]

1/60*(30*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/a^3 - 30*(20*c^2*d^3 - 30*c*d^4 +
13*d^5)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/a^3 - 60*(10*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 7*d^5*tan(1/2*f*x + 1/
2*e)^3 - 10*c*d^4*tan(1/2*f*x + 1/2*e) + 5*d^5*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*a^3) + (3
*a^12*c^5*tan(1/2*f*x + 1/2*e)^5 - 15*a^12*c^4*d*tan(1/2*f*x + 1/2*e)^5 + 30*a^12*c^3*d^2*tan(1/2*f*x + 1/2*e)
^5 - 30*a^12*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 + 15*a^12*c*d^4*tan(1/2*f*x + 1/2*e)^5 - 3*a^12*d^5*tan(1/2*f*x +
1/2*e)^5 - 10*a^12*c^5*tan(1/2*f*x + 1/2*e)^3 + 100*a^12*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 - 200*a^12*c^2*d^3*tan
(1/2*f*x + 1/2*e)^3 + 150*a^12*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 40*a^12*d^5*tan(1/2*f*x + 1/2*e)^3 + 15*a^12*c^5
*tan(1/2*f*x + 1/2*e) + 75*a^12*c^4*d*tan(1/2*f*x + 1/2*e) + 150*a^12*c^3*d^2*tan(1/2*f*x + 1/2*e) - 1050*a^12
*c^2*d^3*tan(1/2*f*x + 1/2*e) + 1275*a^12*c*d^4*tan(1/2*f*x + 1/2*e) - 465*a^12*d^5*tan(1/2*f*x + 1/2*e))/a^15
)/f

Mupad [B] (verification not implemented)

Time = 13.86 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.88 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,{\left (c-d\right )}^5}{2\,a^3}-\frac {15\,\left (c+d\right )\,{\left (c-d\right )}^4}{4\,a^3}+\frac {5\,{\left (c+d\right )}^2\,{\left (c-d\right )}^3}{2\,a^3}\right )}{f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (10\,c\,d^4-5\,d^5\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (10\,c\,d^4-7\,d^5\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {{\left (c-d\right )}^5}{4\,a^3}-\frac {5\,\left (c+d\right )\,{\left (c-d\right )}^4}{12\,a^3}\right )}{f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\left (c-d\right )}^5}{20\,a^3\,f}+\frac {d^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (20\,c^2-30\,c\,d+13\,d^2\right )}{a^3\,f} \]

[In]

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

[Out]

(tan(e/2 + (f*x)/2)*((3*(c - d)^5)/(2*a^3) - (15*(c + d)*(c - d)^4)/(4*a^3) + (5*(c + d)^2*(c - d)^3)/(2*a^3))
)/f + (tan(e/2 + (f*x)/2)*(10*c*d^4 - 5*d^5) - tan(e/2 + (f*x)/2)^3*(10*c*d^4 - 7*d^5))/(f*(a^3*tan(e/2 + (f*x
)/2)^4 - 2*a^3*tan(e/2 + (f*x)/2)^2 + a^3)) + (tan(e/2 + (f*x)/2)^3*((c - d)^5/(4*a^3) - (5*(c + d)*(c - d)^4)
/(12*a^3)))/f + (tan(e/2 + (f*x)/2)^5*(c - d)^5)/(20*a^3*f) + (d^3*atanh(tan(e/2 + (f*x)/2))*(20*c^2 - 30*c*d
+ 13*d^2))/(a^3*f)