3.194 \(\int \frac{(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^4} \, dx\)
Optimal. Leaf size=377 \[ -\frac{\left (-a^2 d^2 \left (-28 c^2 d^2+34 c^4+9 d^4\right )+2 a b c d \left (-5 c^2 d^2+18 c^4+2 d^4\right )+b^2 \left (-\left (10 c^4 d^2-c^2 d^4+6 c^6\right )\right )\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)}-\frac{\left (a^2 \left (7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right )-a b \left (6 c^5 d^2+4 c^7\right )+b^2 c^4 d \left (4 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{c^4 f (c-d)^{7/2} (c+d)^{7/2}}+\frac{a^2 x}{c^4}-\frac{d (b c-a d) \left (-8 a c^2 d+3 a d^3+6 b c^3-b c d^2\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^2}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^3} \]
[Out]
(a^2*x)/c^4 - ((b^2*c^4*d*(4*c^2 + d^2) - a*b*(4*c^7 + 6*c^5*d^2) + a^2*(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*(c - d)^(7/2)*(c + d)^(7/2)*f) + (d^2*(b + a*Co
s[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) - (d*(b*c - a*d)*(6*b*c^3 - 8*a*c^2*d -
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) - ((2*a*b*c*d*(18*c^4 - 5*c^2
*d^2 + 2*d^4) - a^2*d^2*(34*c^4 - 28*c^2*d^2 + 9*d^4) - b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*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 = 1.99896, antiderivative size = 377, 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, 3048, 3031, 3021, 2735, 2659, 208} \[ -\frac{\left (-a^2 d^2 \left (-28 c^2 d^2+34 c^4+9 d^4\right )+2 a b c d \left (-5 c^2 d^2+18 c^4+2 d^4\right )+b^2 \left (-\left (10 c^4 d^2-c^2 d^4+6 c^6\right )\right )\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^3 (c \cos (e+f x)+d)}-\frac{\left (a^2 \left (7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right )-a b \left (6 c^5 d^2+4 c^7\right )+b^2 c^4 d \left (4 c^2+d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{c^4 f (c-d)^{7/2} (c+d)^{7/2}}+\frac{a^2 x}{c^4}-\frac{d (b c-a d) \left (-8 a c^2 d+3 a d^3+6 b c^3-b c d^2\right ) \sin (e+f x)}{6 c^3 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)^2}+\frac{d^2 \sin (e+f x) (a \cos (e+f x)+b)^2}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4,x]
[Out]
(a^2*x)/c^4 - ((b^2*c^4*d*(4*c^2 + d^2) - a*b*(4*c^7 + 6*c^5*d^2) + a^2*(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*(c - d)^(7/2)*(c + d)^(7/2)*f) + (d^2*(b + a*Co
s[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) - (d*(b*c - a*d)*(6*b*c^3 - 8*a*c^2*d -
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) - ((2*a*b*c*d*(18*c^4 - 5*c^2
*d^2 + 2*d^4) - a^2*d^2*(34*c^4 - 28*c^2*d^2 + 9*d^4) - b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*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 3048
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(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
- 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(
m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && 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))^2}{(c+d \sec (e+f x))^4} \, dx &=\int \frac{\cos ^2(e+f x) (b+a \cos (e+f x))^2}{(d+c \cos (e+f x))^4} \, dx\\ &=\frac{d^2 (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 (-d (3 b c-2 a d)+\left (3 b c^2-3 a c d-b d^2\right ) \cos (e+f x)+3 a \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^2 (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{d (b c-a d) \left (6 b c^3-8 a c^2 d-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 \left (a^2 d^2 \left (8 c^2-3 d^2\right )-2 a b c d \left (6 c^2-d^2\right )+b^2 \left (3 c^4+2 c^2 d^2\right )\right )+\left (b^2 c^2 d \left (6 c^2-d^2\right )-2 a b c \left (6 c^4-3 c^2 d^2+2 d^4\right )+a^2 \left (12 c^4 d-10 c^2 d^3+3 d^5\right )\right ) \cos (e+f x)-6 a^2 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^2 (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{d (b c-a d) \left (6 b c^3-8 a c^2 d-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{\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 \left (b^2 c^2 d \left (4 c^2+d^2\right )-2 a b c^3 \left (2 c^2+3 d^2\right )+a^2 \left (6 c^4 d-2 c^2 d^3+d^5\right )\right )-6 a^2 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^2 x}{c^4}+\frac{d^2 (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{d (b c-a d) \left (6 b c^3-8 a c^2 d-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{\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 (b^2 c^4 d \left (4 c^2+d^2\right )-2 a b c^5 \left (2 c^2+3 d^2\right )+a^2 \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^2 x}{c^4}+\frac{d^2 (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{d (b c-a d) \left (6 b c^3-8 a c^2 d-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{\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 (b^2 c^4 d \left (4 c^2+d^2\right )-2 a b c^5 \left (2 c^2+3 d^2\right )+a^2 \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^2 x}{c^4}+\frac{\left (4 a b c^7-8 a^2 c^6 d-4 b^2 c^6 d+6 a b c^5 d^2+8 a^2 c^4 d^3-b^2 c^4 d^3-7 a^2 c^2 d^5+2 a^2 d^7\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{c^4 (c-d)^{7/2} (c+d)^{7/2} f}+\frac{d^2 (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{d (b c-a d) \left (6 b c^3-8 a c^2 d-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{\left (2 a b c d \left (18 c^4-5 c^2 d^2+2 d^4\right )-a^2 d^2 \left (34 c^4-28 c^2 d^2+9 d^4\right )-b^2 \left (6 c^6+10 c^4 d^2-c^2 d^4\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 = 3.32157, size = 438, normalized size = 1.16 \[ \frac{\sec ^2(e+f x) (a+b \sec (e+f x))^2 (c \cos (e+f x)+d) \left (\frac{c \left (a^2 d^2 \left (-32 c^2 d^2+36 c^4+11 d^4\right )-2 a b c d \left (-5 c^2 d^2+18 c^4+2 d^4\right )+b^2 \left (10 c^4 d^2-c^2 d^4+6 c^6\right )\right ) \sin (e+f x) (c \cos (e+f x)+d)^2}{\left (c^2-d^2\right )^3}-\frac{c d \left (a^2 d^2 \left (12 c^2-7 d^2\right )+a b \left (8 c d^3-18 c^3 d\right )+b^2 \left (6 c^4-c^2 d^2\right )\right ) \sin (e+f x) (c \cos (e+f x)+d)}{\left (c^2-d^2\right )^2}+\frac{6 \left (a^2 \left (7 c^2 d^5-8 c^4 d^3+8 c^6 d-2 d^7\right )-2 a b c^5 \left (2 c^2+3 d^2\right )+b^2 c^4 d \left (4 c^2+d^2\right )\right ) (c \cos (e+f x)+d)^3 \tanh ^{-1}\left (\frac{(d-c) \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+6 a^2 (e+f x) (c \cos (e+f x)+d)^3+\frac{2 c d^2 (b c-a d)^2 \sin (e+f x)}{c^2-d^2}\right )}{6 c^4 f (a \cos (e+f x)+b)^2 (c+d \sec (e+f x))^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4,x]
[Out]
((d + c*Cos[e + f*x])*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^2*(6*a^2*(e + f*x)*(d + c*Cos[e + f*x])^3 + (6*(b^2*
c^4*d*(4*c^2 + d^2) - 2*a*b*c^5*(2*c^2 + 3*d^2) + a^2*(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^2*(b*c - a*d)^2*Sin
[e + f*x])/(c^2 - d^2) - (c*d*(a^2*d^2*(12*c^2 - 7*d^2) + b^2*(6*c^4 - c^2*d^2) + a*b*(-18*c^3*d + 8*c*d^3))*(
d + c*Cos[e + f*x])*Sin[e + f*x])/(c^2 - d^2)^2 + (c*(-2*a*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) + a^2*d^2*(36*c^
4 - 32*c^2*d^2 + 11*d^4) + b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*(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])^2*(c + d*Sec[e + f*x])^4)
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Maple [B] time = 0.113, size = 3293, normalized size = 8.7 \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))^2/(c+d*sec(f*x+e))^4,x)
[Out]
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)
^5*a^2*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^2*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^2*d^6-2/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^2*d-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)^5*b^2*d^2+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^2*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^2*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^2*d^6+28/3/f*c/(ta
n(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^2*d^
2-12/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^2*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^2*d^4+6/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*b*d^2-1/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^2*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^2*d^6+2/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)*b^2*d-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)*b^2*d^2+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*b*d^3-8/3/f/(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^2-2*c*d+d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*a*b*d^3+2/f*a^2/c^4*arctan(ta
n(1/2*f*x+1/2*e))+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)*ta
n(1/2*f*x+1/2*e)*a*b*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^2*d^2-1/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))*b^2*d^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^2*d^3+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)^5*a*b*d^2-24/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*b*d+12/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*d-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*b*d^2-4/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))*b^2*d-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^2*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
^2*d^7-2/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*b^2+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*b^2-2/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)*b^2-7/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^2*d^5+4/f*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/((c+d)*(c-d))^(1/2)*arctanh(ta
n(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))*a*b-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^2*d^3-1/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^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)*a^2*d^3+1/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^2*d^3+12/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*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))^2/(c+d*sec(f*x+e))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.10881, size = 5010, normalized size = 13.29 \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))^2/(c+d*sec(f*x+e))^4,x, algorithm="fricas")
[Out]
[1/12*(12*(a^2*c^11 - 4*a^2*c^9*d^2 + 6*a^2*c^7*d^4 - 4*a^2*c^5*d^6 + a^2*c^3*d^8)*f*x*cos(f*x + e)^3 + 36*(a^
2*c^10*d - 4*a^2*c^8*d^3 + 6*a^2*c^6*d^5 - 4*a^2*c^4*d^7 + a^2*c^2*d^9)*f*x*cos(f*x + e)^2 + 36*(a^2*c^9*d^2 -
4*a^2*c^7*d^4 + 6*a^2*c^5*d^6 - 4*a^2*c^3*d^8 + a^2*c*d^10)*f*x*cos(f*x + e) + 12*(a^2*c^8*d^3 - 4*a^2*c^6*d^
5 + 6*a^2*c^4*d^7 - 4*a^2*c^2*d^9 + a^2*d^11)*f*x - 3*(4*a*b*c^7*d^3 + 6*a*b*c^5*d^5 - 7*a^2*c^2*d^8 + 2*a^2*d
^10 - 4*(2*a^2 + b^2)*c^6*d^4 + (8*a^2 - b^2)*c^4*d^6 + (4*a*b*c^10 + 6*a*b*c^8*d^2 - 7*a^2*c^5*d^5 + 2*a^2*c^
3*d^7 - 4*(2*a^2 + b^2)*c^9*d + (8*a^2 - b^2)*c^7*d^3)*cos(f*x + e)^3 + 3*(4*a*b*c^9*d + 6*a*b*c^7*d^3 - 7*a^2
*c^4*d^6 + 2*a^2*c^2*d^8 - 4*(2*a^2 + b^2)*c^8*d^2 + (8*a^2 - b^2)*c^6*d^4)*cos(f*x + e)^2 + 3*(4*a*b*c^8*d^2
+ 6*a*b*c^6*d^4 - 7*a^2*c^3*d^7 + 2*a^2*c*d^9 - 4*(2*a^2 + b^2)*c^7*d^3 + (8*a^2 - 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*(2*b^2*c^9*d^2 - 22*a*b*c^
8*d^3 + 14*a*b*c^6*d^5 + 8*a*b*c^4*d^7 + 23*a^2*c^3*d^8 - 6*a^2*c*d^10 + (26*a^2 + 11*b^2)*c^7*d^4 - (43*a^2 +
13*b^2)*c^5*d^6 + (6*b^2*c^11 - 36*a*b*c^10*d + 46*a*b*c^8*d^3 - 14*a*b*c^6*d^5 + 4*a*b*c^4*d^7 - 11*a^2*c^3*
d^8 + 4*(9*a^2 + b^2)*c^9*d^2 - (68*a^2 + 11*b^2)*c^7*d^4 + (43*a^2 + b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*(2*b^2*
c^10*d - 18*a*b*c^9*d^2 + 16*a*b*c^7*d^4 + 2*a*b*c^5*d^6 - 5*a^2*c^2*d^9 + (20*a^2 + 7*b^2)*c^8*d^3 - 5*(7*a^2
+ 2*b^2)*c^6*d^5 + (20*a^2 + 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^2*c^11 - 4*a^2*c^9*d^2 + 6*a^2*c^7*d^4 - 4*a^2*c^5*d^6 + a^2*c^3
*d^8)*f*x*cos(f*x + e)^3 + 18*(a^2*c^10*d - 4*a^2*c^8*d^3 + 6*a^2*c^6*d^5 - 4*a^2*c^4*d^7 + a^2*c^2*d^9)*f*x*c
os(f*x + e)^2 + 18*(a^2*c^9*d^2 - 4*a^2*c^7*d^4 + 6*a^2*c^5*d^6 - 4*a^2*c^3*d^8 + a^2*c*d^10)*f*x*cos(f*x + e)
+ 6*(a^2*c^8*d^3 - 4*a^2*c^6*d^5 + 6*a^2*c^4*d^7 - 4*a^2*c^2*d^9 + a^2*d^11)*f*x + 3*(4*a*b*c^7*d^3 + 6*a*b*c
^5*d^5 - 7*a^2*c^2*d^8 + 2*a^2*d^10 - 4*(2*a^2 + b^2)*c^6*d^4 + (8*a^2 - b^2)*c^4*d^6 + (4*a*b*c^10 + 6*a*b*c^
8*d^2 - 7*a^2*c^5*d^5 + 2*a^2*c^3*d^7 - 4*(2*a^2 + b^2)*c^9*d + (8*a^2 - b^2)*c^7*d^3)*cos(f*x + e)^3 + 3*(4*a
*b*c^9*d + 6*a*b*c^7*d^3 - 7*a^2*c^4*d^6 + 2*a^2*c^2*d^8 - 4*(2*a^2 + b^2)*c^8*d^2 + (8*a^2 - b^2)*c^6*d^4)*co
s(f*x + e)^2 + 3*(4*a*b*c^8*d^2 + 6*a*b*c^6*d^4 - 7*a^2*c^3*d^7 + 2*a^2*c*d^9 - 4*(2*a^2 + b^2)*c^7*d^3 + (8*a
^2 - b^2)*c^5*d^5)*cos(f*x + e))*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*s
in(f*x + e))) + (2*b^2*c^9*d^2 - 22*a*b*c^8*d^3 + 14*a*b*c^6*d^5 + 8*a*b*c^4*d^7 + 23*a^2*c^3*d^8 - 6*a^2*c*d^
10 + (26*a^2 + 11*b^2)*c^7*d^4 - (43*a^2 + 13*b^2)*c^5*d^6 + (6*b^2*c^11 - 36*a*b*c^10*d + 46*a*b*c^8*d^3 - 14
*a*b*c^6*d^5 + 4*a*b*c^4*d^7 - 11*a^2*c^3*d^8 + 4*(9*a^2 + b^2)*c^9*d^2 - (68*a^2 + 11*b^2)*c^7*d^4 + (43*a^2
+ b^2)*c^5*d^6)*cos(f*x + e)^2 + 3*(2*b^2*c^10*d - 18*a*b*c^9*d^2 + 16*a*b*c^7*d^4 + 2*a*b*c^5*d^6 - 5*a^2*c^2
*d^9 + (20*a^2 + 7*b^2)*c^8*d^3 - 5*(7*a^2 + 2*b^2)*c^6*d^5 + (20*a^2 + 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)]
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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 )^{2}}{\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))**2/(c+d*sec(f*x+e))**4,x)
[Out]
Integral((a + b*sec(e + f*x))**2/(c + d*sec(e + f*x))**4, x)
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Giac [B] time = 1.63226, size = 1692, normalized size = 4.49 \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))^2/(c+d*sec(f*x+e))^4,x, algorithm="giac")
[Out]
1/3*(3*(4*a*b*c^7 - 8*a^2*c^6*d - 4*b^2*c^6*d + 6*a*b*c^5*d^2 + 8*a^2*c^4*d^3 - b^2*c^4*d^3 - 7*a^2*c^2*d^5 +
2*a^2*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)*sqrt(-c^2 + d^2)) + 3*(f*x + e)*a^2/c^
4 - (6*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 36*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 - 6*b^2*c^7*d*tan(1/2*f*x + 1/2*e)
^5 + 36*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 54*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 12*b^2*c^6*d^2*tan(1/2*f*
x + 1/2*e)^5 - 60*a^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 27*b^2*c^5*d^3*
tan(1/2*f*x + 1/2*e)^5 - 6*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 6*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 12*b^2*
c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 45*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^5
+ 3*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^5 - 6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 15*a^2*c*d^7*tan(1/2*f*x + 1/
2*e)^5 + 6*a^2*d^8*tan(1/2*f*x + 1/2*e)^5 - 12*b^2*c^8*tan(1/2*f*x + 1/2*e)^3 + 72*a*b*c^7*d*tan(1/2*f*x + 1/2
*e)^3 - 72*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 16*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 64*a*b*c^5*d^3*tan(1/2
*f*x + 1/2*e)^3 + 116*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 28*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 8*a*b*c^3*d
^5*tan(1/2*f*x + 1/2*e)^3 - 56*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 12*a^2*d^8*tan(1/2*f*x + 1/2*e)^3 + 6*b^2*
c^8*tan(1/2*f*x + 1/2*e) - 36*a*b*c^7*d*tan(1/2*f*x + 1/2*e) + 6*b^2*c^7*d*tan(1/2*f*x + 1/2*e) + 36*a^2*c^6*d
^2*tan(1/2*f*x + 1/2*e) - 54*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e) + 12*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e) + 60*a^2*c
^5*d^3*tan(1/2*f*x + 1/2*e) - 12*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e) + 27*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e) - 6*a^
2*c^4*d^4*tan(1/2*f*x + 1/2*e) - 6*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e) + 12*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) - 45
*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e) - 12*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e) - 3*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e) -
6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e) + 15*a^2*c*d^7*tan(1/2*f*x + 1/2*e) + 6*a^2*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