3.196 \(\int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx\)
Optimal. Leaf size=412 \[ -\frac{(b c-a d) \left (a^2 \left (-28 c^2 d^3+34 c^4 d+9 d^5\right )-a b c \left (17 c^2 d^2+18 c^4-5 d^4\right )+b^2 c^2 d \left (13 c^2+2 d^2\right )\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)}-\frac{\left (-a^2 b \left (9 c^5 d^2+6 c^7\right )+a^3 \left (7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right )+3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{c^4 f \sqrt{c-d} \sqrt{c+d} \left (c^2-d^2\right )^3}+\frac{a^3 x}{c^4}+\frac{(b c-a d)^2 \left (-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^2}-\frac{d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^3} \]
[Out]
(a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d
- 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^4*Sqrt[c - d]*Sqrt[
c + d]*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e
+ f*x])^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(
d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*
(34*c^4*d - 28*c^2*d^3 + 9*d^5))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))
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Rubi [A] time = 1.0623, antiderivative size = 412, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used =
{3941, 2989, 3031, 3021, 2735, 2659, 208} \[ -\frac{(b c-a d) \left (a^2 \left (-28 c^2 d^3+34 c^4 d+9 d^5\right )-a b c \left (17 c^2 d^2+18 c^4-5 d^4\right )+b^2 c^2 d \left (13 c^2+2 d^2\right )\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)}-\frac{\left (-a^2 b \left (9 c^5 d^2+6 c^7\right )+a^3 \left (7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right )+3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{c^4 f \sqrt{c-d} \sqrt{c+d} \left (c^2-d^2\right )^3}+\frac{a^3 x}{c^4}+\frac{(b c-a d)^2 \left (-8 a c^2 d+3 a d^3+3 b c^3+2 b c d^2\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^2}-\frac{d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4,x]
[Out]
(a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d
- 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^4*Sqrt[c - d]*Sqrt[
c + d]*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e
+ f*x])^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(
d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*
(34*c^4*d - 28*c^2*d^3 + 9*d^5))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))
Rule 3941
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Int[
((b + a*Sin[e + f*x])^m*(d + c*Sin[e + f*x])^n)/Sin[e + f*x]^(m + n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
&& NeQ[b*c - a*d, 0] && IntegerQ[m] && IntegerQ[n] && LeQ[-2, m + n, 0]
Rule 2989
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((b*c - a*d)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)
*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[
e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (
A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)
*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2,
0] && GtQ[m, 1] && LtQ[n, -1]
Rule 3031
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]
Rule 3021
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Rule 2735
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]
Rule 2659
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{(a+b \sec (e+f x))^3}{(c+d \sec (e+f x))^4} \, dx &=\int \frac{\cos (e+f x) (b+a \cos (e+f x))^3}{(d+c \cos (e+f x))^4} \, dx\\ &=-\frac{d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac{\int \frac{(b+a \cos (e+f x)) \left ((3 b c-2 a d) (b c-a d)-\left (3 a^2 c d+2 b^2 c d-a b \left (6 c^2-d^2\right )\right ) \cos (e+f x)+3 a^2 \left (c^2-d^2\right ) \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^3} \, dx}{3 c \left (c^2-d^2\right )}\\ &=-\frac{d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac{(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac{\int \frac{-2 c (b c-a d) \left (9 a b c^3-8 a^2 c^2 d-5 b^2 c^2 d+a b c d^2+3 a^2 d^3\right )+\left (3 a b^2 c^2 d \left (6 c^2-d^2\right )-b^3 c^3 \left (3 c^2+2 d^2\right )-a^2 b c \left (18 c^4-7 c^2 d^2+4 d^4\right )+a^3 \left (12 c^4 d-10 c^2 d^3+3 d^5\right )\right ) \cos (e+f x)-6 a^3 c \left (c^2-d^2\right )^2 \cos ^2(e+f x)}{(d+c \cos (e+f x))^2} \, dx}{6 c^3 \left (c^2-d^2\right )^2}\\ &=-\frac{d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac{(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac{(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}-\frac{\int \frac{-3 c^2 (b c-a d) \left (6 a^2 c^4+b^2 c^4-11 a b c^3 d-2 a^2 c^2 d^2+4 b^2 c^2 d^2+a b c d^3+a^2 d^4\right )-6 a^3 c \left (c^2-d^2\right )^3 \cos (e+f x)}{d+c \cos (e+f x)} \, dx}{6 c^4 \left (c^2-d^2\right )^3}\\ &=\frac{a^3 x}{c^4}-\frac{d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac{(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac{(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}-\frac{\left (3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \int \frac{1}{d+c \cos (e+f x)} \, dx}{2 c^4 \left (c^2-d^2\right )^3}\\ &=\frac{a^3 x}{c^4}-\frac{d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac{(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac{(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}-\frac{\left (3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+d+(-c+d) x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{c^4 \left (c^2-d^2\right )^3 f}\\ &=\frac{a^3 x}{c^4}-\frac{\left (3 a b^2 c^4 d \left (4 c^2+d^2\right )-b^3 c^5 \left (c^2+4 d^2\right )-a^2 b \left (6 c^7+9 c^5 d^2\right )+a^3 \left (8 c^6 d-8 c^4 d^3+7 c^2 d^5-2 d^7\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{c^4 \sqrt{c-d} \sqrt{c+d} \left (c^2-d^2\right )^3 f}-\frac{d (b c-a d) (b+a \cos (e+f x))^2 \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3}+\frac{(b c-a d)^2 \left (3 b c^3-8 a c^2 d+2 b c d^2+3 a d^3\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2}-\frac{(b c-a d) \left (b^2 c^2 d \left (13 c^2+2 d^2\right )-a b c \left (18 c^4+17 c^2 d^2-5 d^4\right )+a^2 \left (34 c^4 d-28 c^2 d^3+9 d^5\right )\right ) \sin (e+f x)}{6 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x))}\\ \end{align*}
Mathematica [A] time = 3.80536, size = 459, normalized size = 1.11 \[ \frac{\sec (e+f x) (a+b \sec (e+f x))^3 (c \cos (e+f x)+d) \left (\frac{c \left (-3 a^2 b c d \left (-5 c^2 d^2+18 c^4+2 d^4\right )+a^3 \left (-32 c^2 d^4+36 c^4 d^2+11 d^6\right )+3 a b^2 c^2 \left (10 c^2 d^2+6 c^4-d^4\right )-b^3 c^3 d \left (13 c^2+2 d^2\right )\right ) \sin (e+f x) (c \cos (e+f x)+d)^2}{\left (c^2-d^2\right )^3}-\frac{6 \left (a^2 b \left (9 c^5 d^2+6 c^7\right )+a^3 \left (-7 c^2 d^5+8 c^4 d^3-8 c^6 d+2 d^7\right )-3 a b^2 c^4 d \left (4 c^2+d^2\right )+b^3 c^5 \left (c^2+4 d^2\right )\right ) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left (\frac{(d-c) \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+6 a^3 (e+f x) (c \cos (e+f x)+d)^3-\frac{2 c d (b c-a d)^3 \sin (e+f x)}{c^2-d^2}+\frac{c \left (-12 a c^2 d+7 a d^3+3 b c^3+2 b c d^2\right ) (b c-a d)^2 \sin (e+f x) (c \cos (e+f x)+d)}{\left (c^2-d^2\right )^2}\right )}{6 c^4 f (a \cos (e+f x)+b)^3 (c+d \sec (e+f x))^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4,x]
[Out]
((d + c*Cos[e + f*x])*Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(6*a^3*(e + f*x)*(d + c*Cos[e + f*x])^3 - (6*(-3*a*b
^2*c^4*d*(4*c^2 + d^2) + b^3*c^5*(c^2 + 4*d^2) + a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(-8*c^6*d + 8*c^4*d^3 - 7*c^2
*d^5 + 2*d^7))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^3)/(c^2 - d^2)^(7/2)
- (2*c*d*(b*c - a*d)^3*Sin[e + f*x])/(c^2 - d^2) + (c*(b*c - a*d)^2*(3*b*c^3 - 12*a*c^2*d + 2*b*c*d^2 + 7*a*d^
3)*(d + c*Cos[e + f*x])*Sin[e + f*x])/(c^2 - d^2)^2 + (c*(-(b^3*c^3*d*(13*c^2 + 2*d^2)) + 3*a*b^2*c^2*(6*c^4 +
10*c^2*d^2 - d^4) - 3*a^2*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) + a^3*(36*c^4*d^2 - 32*c^2*d^4 + 11*d^6))*(d + c
*Cos[e + f*x])^2*Sin[e + f*x])/(c^2 - d^2)^3))/(6*c^4*f*(b + a*Cos[e + f*x])^3*(c + d*Sec[e + f*x])^4)
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Maple [B] time = 0.167, size = 4330, normalized size = 10.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x)
[Out]
12/f*c^3/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2
*e)^3*a*b^2-28/3/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*t
an(1/2*f*x+1/2*e)^3*b^3*d-12/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*
d^2-d^3)*tan(1/2*f*x+1/2*e)*a^3*d^2+6/f/c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c
^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a^3*d^4-1/f/c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+
d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a^3*d^5-2/f/c^3/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d
-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a^3*d^6-6/f*c^3/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x
+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a*b^2+6/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-ta
n(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*b^3*d-2/f*c/(tan(1/2*f*x+1/2*e)
^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*b^3*d^2+9/f*c/(c^6-3*c^4
*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a^2*b*d^2-12/f*c
^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a*b
^2*d+2/f*a^3/c^4*arctan(tan(1/2*f*x+1/2*e))-9/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/
(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a^2*b*d^2+6/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-
c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a*b^2*d-18/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1
/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a*b^2*d^2+1/f*c^3/(c^6-3*c^4*d^2+3*c^2*d^4
-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*b^3+8/f/(c^6-3*c^4*d^2+3*c^2*d
^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a^3*d^3-12/f*c/(tan(1/2*f*x+
1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a^3*d^2+6/f/c/(t
an(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a^3*d
^4+1/f/c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1
/2*e)^5*a^3*d^5-2/f/c^3/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*
tan(1/2*f*x+1/2*e)^5*a^3*d^6-6/f*c^3/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+
3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a*b^2+6/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c
^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*b^3*d+2/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3
/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*b^3*d^2+6/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2
*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a^2*b*d^3-3/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f
*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a*b^2*d^3-4/f/(tan(1/2*f*x+1/2*e)^2*
c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*a^2*b*d^3+6/f/(tan(1/2*f*
x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a^2*b*d^3+3/f/(t
an(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a*b^2*d
^3+24/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/
2*e)^3*a^3*d^2-44/3/f/c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*
tan(1/2*f*x+1/2*e)^3*a^3*d^4+4/f/c^3/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^
2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*a^3*d^6+2/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3
-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*b^3*d^3+4/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+
d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*a^3*d^3-8/f*c^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1
/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a^3*d+2/f/c^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(
c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a^3*d^7+4/f*c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/
((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*b^3*d^2+1/f*c^3/(tan(1/2*f*x+1/2*e)^
2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*b^3-1/f*c^3/(tan(1/2*f*
x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*tan(1/2*f*x+1/2*e)*b^3-7/f/c^2/(c^6
-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a^3*d^5+6/
f*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*
a^2*b-3/f/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/
2))*a*b^2*d^3-4/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/
2*f*x+1/2*e)^5*a^3*d^3+2/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^
3)*tan(1/2*f*x+1/2*e)^5*b^3*d^3-4/f/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2
+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*b^3*d^3-6/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(
c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a*b^2*d+9/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d
)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a^2*b*d^2+18/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*
x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a^2*b*d-18/f*c/(tan(1/2*f*x+1/2*e)^2*
c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c-d)/(c^3+3*c^2*d+3*c*d^2+d^3)*tan(1/2*f*x+1/2*e)^5*a*b^2*d^2-36/f*c^2/(tan(1
/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*a^2*b*d+2
8/f*c/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)
^3*a*b^2*d^2+18/f*c^2/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^3/(c+d)/(c^3-3*c^2*d+3*c*d^2-d^3)*ta
n(1/2*f*x+1/2*e)*a^2*b*d
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.27304, size = 5782, normalized size = 14.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x, algorithm="fricas")
[Out]
[1/12*(12*(a^3*c^11 - 4*a^3*c^9*d^2 + 6*a^3*c^7*d^4 - 4*a^3*c^5*d^6 + a^3*c^3*d^8)*f*x*cos(f*x + e)^3 + 36*(a^
3*c^10*d - 4*a^3*c^8*d^3 + 6*a^3*c^6*d^5 - 4*a^3*c^4*d^7 + a^3*c^2*d^9)*f*x*cos(f*x + e)^2 + 36*(a^3*c^9*d^2 -
4*a^3*c^7*d^4 + 6*a^3*c^5*d^6 - 4*a^3*c^3*d^8 + a^3*c*d^10)*f*x*cos(f*x + e) + 12*(a^3*c^8*d^3 - 4*a^3*c^6*d^
5 + 6*a^3*c^4*d^7 - 4*a^3*c^2*d^9 + a^3*d^11)*f*x + 3*(7*a^3*c^2*d^8 - 2*a^3*d^10 - (6*a^2*b + b^3)*c^7*d^3 +
4*(2*a^3 + 3*a*b^2)*c^6*d^4 - (9*a^2*b + 4*b^3)*c^5*d^5 - (8*a^3 - 3*a*b^2)*c^4*d^6 + (7*a^3*c^5*d^5 - 2*a^3*c
^3*d^7 - (6*a^2*b + b^3)*c^10 + 4*(2*a^3 + 3*a*b^2)*c^9*d - (9*a^2*b + 4*b^3)*c^8*d^2 - (8*a^3 - 3*a*b^2)*c^7*
d^3)*cos(f*x + e)^3 + 3*(7*a^3*c^4*d^6 - 2*a^3*c^2*d^8 - (6*a^2*b + b^3)*c^9*d + 4*(2*a^3 + 3*a*b^2)*c^8*d^2 -
(9*a^2*b + 4*b^3)*c^7*d^3 - (8*a^3 - 3*a*b^2)*c^6*d^4)*cos(f*x + e)^2 + 3*(7*a^3*c^3*d^7 - 2*a^3*c*d^9 - (6*a
^2*b + b^3)*c^8*d^2 + 4*(2*a^3 + 3*a*b^2)*c^7*d^3 - (9*a^2*b + 4*b^3)*c^6*d^4 - (8*a^3 - 3*a*b^2)*c^5*d^5)*cos
(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f
*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) + 2*(b^3*c^10*d + 6*
a*b^2*c^9*d^2 + 23*a^3*c^3*d^8 - 6*a^3*c*d^10 - 11*(3*a^2*b + b^3)*c^8*d^3 + (26*a^3 + 33*a*b^2)*c^7*d^4 + (21
*a^2*b + 4*b^3)*c^6*d^5 - (43*a^3 + 39*a*b^2)*c^5*d^6 + 6*(2*a^2*b + b^3)*c^4*d^7 + (18*a*b^2*c^11 + 6*a^2*b*c
^4*d^7 - 11*a^3*c^3*d^8 - (54*a^2*b + 13*b^3)*c^10*d + 12*(3*a^3 + a*b^2)*c^9*d^2 + (69*a^2*b + 11*b^3)*c^8*d^
3 - (68*a^3 + 33*a*b^2)*c^7*d^4 - (21*a^2*b - 2*b^3)*c^6*d^5 + (43*a^3 + 3*a*b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*
(b^3*c^11 + 6*a*b^2*c^10*d - 5*a^3*c^2*d^9 - (27*a^2*b + 10*b^3)*c^9*d^2 + (20*a^3 + 21*a*b^2)*c^8*d^3 + (24*a
^2*b + 7*b^3)*c^7*d^4 - 5*(7*a^3 + 6*a*b^2)*c^6*d^5 + (3*a^2*b + 2*b^3)*c^5*d^6 + (20*a^3 + 3*a*b^2)*c^4*d^7)*
cos(f*x + e))*sin(f*x + e))/((c^15 - 4*c^13*d^2 + 6*c^11*d^4 - 4*c^9*d^6 + c^7*d^8)*f*cos(f*x + e)^3 + 3*(c^14
*d - 4*c^12*d^3 + 6*c^10*d^5 - 4*c^8*d^7 + c^6*d^9)*f*cos(f*x + e)^2 + 3*(c^13*d^2 - 4*c^11*d^4 + 6*c^9*d^6 -
4*c^7*d^8 + c^5*d^10)*f*cos(f*x + e) + (c^12*d^3 - 4*c^10*d^5 + 6*c^8*d^7 - 4*c^6*d^9 + c^4*d^11)*f), 1/6*(6*(
a^3*c^11 - 4*a^3*c^9*d^2 + 6*a^3*c^7*d^4 - 4*a^3*c^5*d^6 + a^3*c^3*d^8)*f*x*cos(f*x + e)^3 + 18*(a^3*c^10*d -
4*a^3*c^8*d^3 + 6*a^3*c^6*d^5 - 4*a^3*c^4*d^7 + a^3*c^2*d^9)*f*x*cos(f*x + e)^2 + 18*(a^3*c^9*d^2 - 4*a^3*c^7*
d^4 + 6*a^3*c^5*d^6 - 4*a^3*c^3*d^8 + a^3*c*d^10)*f*x*cos(f*x + e) + 6*(a^3*c^8*d^3 - 4*a^3*c^6*d^5 + 6*a^3*c^
4*d^7 - 4*a^3*c^2*d^9 + a^3*d^11)*f*x - 3*(7*a^3*c^2*d^8 - 2*a^3*d^10 - (6*a^2*b + b^3)*c^7*d^3 + 4*(2*a^3 + 3
*a*b^2)*c^6*d^4 - (9*a^2*b + 4*b^3)*c^5*d^5 - (8*a^3 - 3*a*b^2)*c^4*d^6 + (7*a^3*c^5*d^5 - 2*a^3*c^3*d^7 - (6*
a^2*b + b^3)*c^10 + 4*(2*a^3 + 3*a*b^2)*c^9*d - (9*a^2*b + 4*b^3)*c^8*d^2 - (8*a^3 - 3*a*b^2)*c^7*d^3)*cos(f*x
+ e)^3 + 3*(7*a^3*c^4*d^6 - 2*a^3*c^2*d^8 - (6*a^2*b + b^3)*c^9*d + 4*(2*a^3 + 3*a*b^2)*c^8*d^2 - (9*a^2*b +
4*b^3)*c^7*d^3 - (8*a^3 - 3*a*b^2)*c^6*d^4)*cos(f*x + e)^2 + 3*(7*a^3*c^3*d^7 - 2*a^3*c*d^9 - (6*a^2*b + b^3)*
c^8*d^2 + 4*(2*a^3 + 3*a*b^2)*c^7*d^3 - (9*a^2*b + 4*b^3)*c^6*d^4 - (8*a^3 - 3*a*b^2)*c^5*d^5)*cos(f*x + e))*s
qrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + (b^3*c^10*d + 6*a*
b^2*c^9*d^2 + 23*a^3*c^3*d^8 - 6*a^3*c*d^10 - 11*(3*a^2*b + b^3)*c^8*d^3 + (26*a^3 + 33*a*b^2)*c^7*d^4 + (21*a
^2*b + 4*b^3)*c^6*d^5 - (43*a^3 + 39*a*b^2)*c^5*d^6 + 6*(2*a^2*b + b^3)*c^4*d^7 + (18*a*b^2*c^11 + 6*a^2*b*c^4
*d^7 - 11*a^3*c^3*d^8 - (54*a^2*b + 13*b^3)*c^10*d + 12*(3*a^3 + a*b^2)*c^9*d^2 + (69*a^2*b + 11*b^3)*c^8*d^3
- (68*a^3 + 33*a*b^2)*c^7*d^4 - (21*a^2*b - 2*b^3)*c^6*d^5 + (43*a^3 + 3*a*b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*(b
^3*c^11 + 6*a*b^2*c^10*d - 5*a^3*c^2*d^9 - (27*a^2*b + 10*b^3)*c^9*d^2 + (20*a^3 + 21*a*b^2)*c^8*d^3 + (24*a^2
*b + 7*b^3)*c^7*d^4 - 5*(7*a^3 + 6*a*b^2)*c^6*d^5 + (3*a^2*b + 2*b^3)*c^5*d^6 + (20*a^3 + 3*a*b^2)*c^4*d^7)*co
s(f*x + e))*sin(f*x + e))/((c^15 - 4*c^13*d^2 + 6*c^11*d^4 - 4*c^9*d^6 + c^7*d^8)*f*cos(f*x + e)^3 + 3*(c^14*d
- 4*c^12*d^3 + 6*c^10*d^5 - 4*c^8*d^7 + c^6*d^9)*f*cos(f*x + e)^2 + 3*(c^13*d^2 - 4*c^11*d^4 + 6*c^9*d^6 - 4*
c^7*d^8 + c^5*d^10)*f*cos(f*x + e) + (c^12*d^3 - 4*c^10*d^5 + 6*c^8*d^7 - 4*c^6*d^9 + c^4*d^11)*f)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sec{\left (e + f x \right )}\right )^{3}}{\left (c + d \sec{\left (e + f x \right )}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))**3/(c+d*sec(f*x+e))**4,x)
[Out]
Integral((a + b*sec(e + f*x))**3/(c + d*sec(e + f*x))**4, x)
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Giac [B] time = 1.7899, size = 2213, normalized size = 5.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sec(f*x+e))^3/(c+d*sec(f*x+e))^4,x, algorithm="giac")
[Out]
1/3*(3*(6*a^2*b*c^7 + b^3*c^7 - 8*a^3*c^6*d - 12*a*b^2*c^6*d + 9*a^2*b*c^5*d^2 + 4*b^3*c^5*d^2 + 8*a^3*c^4*d^3
- 3*a*b^2*c^4*d^3 - 7*a^3*c^2*d^5 + 2*a^3*d^7)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c
*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((c^10 - 3*c^8*d^2 + 3*c^6*d^4 - c^4*d^6)*s
qrt(-c^2 + d^2)) + 3*(f*x + e)*a^3/c^4 - (18*a*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 3*b^3*c^8*tan(1/2*f*x + 1/2*e)
^5 - 54*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 18*a*b^2*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 12*b^3*c^7*d*tan(1/2*f*x
+ 1/2*e)^5 + 36*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 81*a^2*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 36*a*b^2*c^6*d^
2*tan(1/2*f*x + 1/2*e)^5 + 27*b^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 - 60*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*
a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 81*a*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 12*b^3*c^5*d^3*tan(1/2*f*x +
1/2*e)^5 - 6*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 9*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 36*a*b^2*c^4*d^4*ta
n(1/2*f*x + 1/2*e)^5 + 6*b^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 45*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*b
*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 + 9*a*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 6*b^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^
5 - 6*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 15*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^5 + 6*a^3*d^8*tan(1/2*f*x + 1/2*e
)^5 - 36*a*b^2*c^8*tan(1/2*f*x + 1/2*e)^3 + 108*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 28*b^3*c^7*d*tan(1/2*f*x
+ 1/2*e)^3 - 72*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 48*a*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 96*a^2*b*c^5*d^
3*tan(1/2*f*x + 1/2*e)^3 - 16*b^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 + 116*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 84
*a*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 12*a^2*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 12*b^3*c^3*d^5*tan(1/2*f*x +
1/2*e)^3 - 56*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 12*a^3*d^8*tan(1/2*f*x + 1/2*e)^3 + 18*a*b^2*c^8*tan(1/2*f
*x + 1/2*e) + 3*b^3*c^8*tan(1/2*f*x + 1/2*e) - 54*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e) + 18*a*b^2*c^7*d*tan(1/2*f*
x + 1/2*e) - 12*b^3*c^7*d*tan(1/2*f*x + 1/2*e) + 36*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e) - 81*a^2*b*c^6*d^2*tan(1/
2*f*x + 1/2*e) + 36*a*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e) - 27*b^3*c^6*d^2*tan(1/2*f*x + 1/2*e) + 60*a^3*c^5*d^3*
tan(1/2*f*x + 1/2*e) - 18*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e) + 81*a*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e) - 12*b^3*
c^5*d^3*tan(1/2*f*x + 1/2*e) - 6*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e) - 9*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e) + 36*
a*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) - 6*b^3*c^4*d^4*tan(1/2*f*x + 1/2*e) - 45*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)
- 18*a^2*b*c^3*d^5*tan(1/2*f*x + 1/2*e) - 9*a*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e) - 6*b^3*c^3*d^5*tan(1/2*f*x + 1
/2*e) - 6*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e) + 15*a^3*c*d^7*tan(1/2*f*x + 1/2*e) + 6*a^3*d^8*tan(1/2*f*x + 1/2*e
))/((c^9 - 3*c^7*d^2 + 3*c^5*d^4 - c^3*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)^3))/
f