3.341 \(\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx\)
Optimal. Leaf size=292 \[ -\frac{\left (-8 a^4 A b^3+7 a^2 A b^5+8 a^6 A b-3 a^5 b^2 B-2 a^7 B-2 A b^7\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left (-17 a^2 A b^3+26 a^4 A b-4 a^3 b^2 B-11 a^5 B+6 A b^5\right ) \tan (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{b \left (8 a^2 A b-5 a^3 B-3 A b^3\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{A x}{a^4} \]
[Out]
(A*x)/a^4 - ((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTanh[(Sqrt[a - b]*Ta
n[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b
^2)*d*(a + b*Sec[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b
*Sec[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*Tan[c + d*x])/(6*a^3*(a^
2 - b^2)^3*d*(a + b*Sec[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 1.06764, antiderivative size = 292, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used =
{3923, 4060, 3919, 3831, 2659, 208} \[ -\frac{\left (-8 a^4 A b^3+7 a^2 A b^5+8 a^6 A b-3 a^5 b^2 B-2 a^7 B-2 A b^7\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left (-17 a^2 A b^3+26 a^4 A b-4 a^3 b^2 B-11 a^5 B+6 A b^5\right ) \tan (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{b \left (8 a^2 A b-5 a^3 B-3 A b^3\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{A x}{a^4} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4,x]
[Out]
(A*x)/a^4 - ((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTanh[(Sqrt[a - b]*Ta
n[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b
^2)*d*(a + b*Sec[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b
*Sec[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*Tan[c + d*x])/(6*a^3*(a^
2 - b^2)^3*d*(a + b*Sec[c + d*x]))
Rule 3923
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(b*(
b*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 -
b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b
*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m,
-1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
Rule 4060
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(a*f*(m + 1
)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Rule 3919
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]
Rule 3831
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 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 (c+d x)}{(a+b \sec (c+d x))^4} \, dx &=\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\int \frac{-3 A \left (a^2-b^2\right )+3 a (A b-a B) \sec (c+d x)-2 b (A b-a B) \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\int \frac{6 A \left (a^2-b^2\right )^2-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \sec (c+d x)+b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\int \frac{-6 A \left (a^2-b^2\right )^3+3 a \left (6 a^4 A b-2 a^2 A b^3+A b^5-2 a^5 B-3 a^3 b^2 B\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac{A x}{a^4}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{A x}{a^4}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 a^4 b \left (a^2-b^2\right )^3}\\ &=\frac{A x}{a^4}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 b \left (a^2-b^2\right )^3 d}\\ &=\frac{A x}{a^4}-\frac{\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [B] time = 3.37973, size = 769, normalized size = 2.63 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left (\frac{36 a^7 A b^2 \sin (c+d x)+36 a^7 A b^2 \sin (3 (c+d x))+120 a^6 A b^3 \sin (2 (c+d x))+72 a^5 A b^4 \sin (c+d x)-32 a^5 A b^4 \sin (3 (c+d x))-90 a^4 A b^5 \sin (2 (c+d x))-57 a^3 A b^6 \sin (c+d x)+11 a^3 A b^6 \sin (3 (c+d x))+30 a^2 A b^7 \sin (2 (c+d x))-18 a^7 A b^2 c \cos (3 (c+d x))-18 a^7 A b^2 d x \cos (3 (c+d x))+18 a^5 A b^4 c \cos (3 (c+d x))+18 a^5 A b^4 d x \cos (3 (c+d x))-6 a^3 A b^6 c \cos (3 (c+d x))-6 a^3 A b^6 d x \cos (3 (c+d x))+36 a^2 A b \left (a^2-b^2\right )^3 (c+d x) \cos (2 (c+d x))+18 a A \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (c+d x) \cos (c+d x)-84 a^6 A b^3 c+36 a^4 A b^5 c+36 a^2 A b^7 c-84 a^6 A b^3 d x+36 a^4 A b^5 d x+36 a^2 A b^7 d x+36 a^8 A b c+36 a^8 A b d x+6 a^9 A c \cos (3 (c+d x))+6 a^9 A d x \cos (3 (c+d x))-54 a^7 b^2 B \sin (2 (c+d x))-39 a^6 b^3 B \sin (c+d x)+5 a^6 b^3 B \sin (3 (c+d x))-6 a^5 b^4 B \sin (2 (c+d x))-18 a^4 b^5 B \sin (c+d x)-2 a^4 b^5 B \sin (3 (c+d x))-18 a^8 b B \sin (c+d x)-18 a^8 b B \sin (3 (c+d x))+24 a A b^8 \sin (c+d x)-24 A b^9 c-24 A b^9 d x}{\left (a^2-b^2\right )^3}-\frac{24 \left (8 a^4 A b^3-7 a^2 A b^5-8 a^6 A b+3 a^5 b^2 B+2 a^7 B+2 A b^7\right ) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}\right )}{24 a^4 d (a+b \sec (c+d x))^4 (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4,x]
[Out]
((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((-24*(-8*a^6*A*b + 8*a^4*A*b^3 - 7*a^2*A*b^5 + 2*A*
b^7 + 2*a^7*B + 3*a^5*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2
- b^2)^(7/2) + (36*a^8*A*b*c - 84*a^6*A*b^3*c + 36*a^4*A*b^5*c + 36*a^2*A*b^7*c - 24*A*b^9*c + 36*a^8*A*b*d*x
- 84*a^6*A*b^3*d*x + 36*a^4*A*b^5*d*x + 36*a^2*A*b^7*d*x - 24*A*b^9*d*x + 18*a*A*(a^2 - b^2)^3*(a^2 + 4*b^2)*
(c + d*x)*Cos[c + d*x] + 36*a^2*A*b*(a^2 - b^2)^3*(c + d*x)*Cos[2*(c + d*x)] + 6*a^9*A*c*Cos[3*(c + d*x)] - 18
*a^7*A*b^2*c*Cos[3*(c + d*x)] + 18*a^5*A*b^4*c*Cos[3*(c + d*x)] - 6*a^3*A*b^6*c*Cos[3*(c + d*x)] + 6*a^9*A*d*x
*Cos[3*(c + d*x)] - 18*a^7*A*b^2*d*x*Cos[3*(c + d*x)] + 18*a^5*A*b^4*d*x*Cos[3*(c + d*x)] - 6*a^3*A*b^6*d*x*Co
s[3*(c + d*x)] + 36*a^7*A*b^2*Sin[c + d*x] + 72*a^5*A*b^4*Sin[c + d*x] - 57*a^3*A*b^6*Sin[c + d*x] + 24*a*A*b^
8*Sin[c + d*x] - 18*a^8*b*B*Sin[c + d*x] - 39*a^6*b^3*B*Sin[c + d*x] - 18*a^4*b^5*B*Sin[c + d*x] + 120*a^6*A*b
^3*Sin[2*(c + d*x)] - 90*a^4*A*b^5*Sin[2*(c + d*x)] + 30*a^2*A*b^7*Sin[2*(c + d*x)] - 54*a^7*b^2*B*Sin[2*(c +
d*x)] - 6*a^5*b^4*B*Sin[2*(c + d*x)] + 36*a^7*A*b^2*Sin[3*(c + d*x)] - 32*a^5*A*b^4*Sin[3*(c + d*x)] + 11*a^3*
A*b^6*Sin[3*(c + d*x)] - 18*a^8*b*B*Sin[3*(c + d*x)] + 5*a^6*b^3*B*Sin[3*(c + d*x)] - 2*a^4*b^5*B*Sin[3*(c + d
*x)])/(a^2 - b^2)^3))/(24*a^4*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^4)
________________________________________________________________________________________
Maple [B] time = 0.108, size = 2242, normalized size = 7.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x)
[Out]
3/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)
)*B*a-8/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(
1/2))*A*a^2-12/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*t
an(1/2*d*x+1/2*c)^3*B+6/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a+b)/(a^3-3*a^2*b+3*a*b
^2-b^3)*tan(1/2*d*x+1/2*c)*B+6/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^4/(a-b)/(a^3+3*a^2*
b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+3/d*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^2/(a-b)/(a
^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-12/d*b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*
a/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+6/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-
a-b)^3*a^2/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-12/d*b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b-a-b)^3*a/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+24/d*b^2/(tan(1/2*d*x+1/2*c)^2*a-tan
(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d/a^3/(tan(1/2*d*x+1/2*c
)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^6/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+1/d/a^2/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-2/d/a^
3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^6/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)
^5*A+6/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d
*x+1/2*c)*A-3/d*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^2/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*ta
n(1/2*d*x+1/2*c)*B-1/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^5/(a+b)/(a^3-3*a^2*b+3*a*b^
2-b^3)*tan(1/2*d*x+1/2*c)*A-4/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^3/(a-b)/(a^3+3*a^2*b+3
*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+2/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^3/(a-b)/(a^3+3*
a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-4/3/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^3/(a^2
-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+4/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b
^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+2/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b
)^3*b^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-7/d/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b
))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^5+2/d/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a
+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^7+4/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-t
an(1/2*d*x+1/2*c)^2*b-a-b)^3*b^6/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-44/3/d/a/(tan(1/2*d*x+
1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^4/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+2/d*A/a^4*
arctan(tan(1/2*d*x+1/2*c))+2/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2
*c)/((a+b)*(a-b))^(1/2))*B*a^3+8/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2
*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A
________________________________________________________________________________________
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(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.04381, size = 4111, normalized size = 14.08 \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(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="fricas")
[Out]
[1/12*(12*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8)*d*x*cos(d*x + c)^3 + 36*(A*a^10*b - 4
*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*d*x*cos(d*x + c)^2 + 36*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5
*b^6 - 4*A*a^3*b^8 + A*a*b^10)*d*x*cos(d*x + c) + 12*(A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*
b^11)*d*x - 3*(2*B*a^7*b^3 - 8*A*a^6*b^4 + 3*B*a^5*b^5 + 8*A*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10 + (2*B*a^10 - 8*
A*a^9*b + 3*B*a^8*b^2 + 8*A*a^7*b^3 - 7*A*a^5*b^5 + 2*A*a^3*b^7)*cos(d*x + c)^3 + 3*(2*B*a^9*b - 8*A*a^8*b^2 +
3*B*a^7*b^3 + 8*A*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c)^2 + 3*(2*B*a^8*b^2 - 8*A*a^7*b^3 + 3*B*a^
6*b^4 + 8*A*a^5*b^5 - 7*A*a^3*b^7 + 2*A*a*b^9)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 -
2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2
+ 2*a*b*cos(d*x + c) + b^2)) - 2*(11*B*a^8*b^3 - 26*A*a^7*b^4 - 7*B*a^6*b^5 + 43*A*a^5*b^6 - 4*B*a^4*b^7 - 23
*A*a^3*b^8 + 6*A*a*b^10 + (18*B*a^10*b - 36*A*a^9*b^2 - 23*B*a^8*b^3 + 68*A*a^7*b^4 + 7*B*a^6*b^5 - 43*A*a^5*b
^6 - 2*B*a^4*b^7 + 11*A*a^3*b^8)*cos(d*x + c)^2 + 3*(9*B*a^9*b^2 - 20*A*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 -
B*a^5*b^6 - 20*A*a^4*b^7 + 5*A*a^2*b^9)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*
b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2
+ 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b^3 - 4*a^10*b^5 + 6*a^8
*b^7 - 4*a^6*b^9 + a^4*b^11)*d), 1/6*(6*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8)*d*x*cos
(d*x + c)^3 + 18*(A*a^10*b - 4*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*d*x*cos(d*x + c)^2 + 18*(A*a
^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6 - 4*A*a^3*b^8 + A*a*b^10)*d*x*cos(d*x + c) + 6*(A*a^8*b^3 - 4*A*a^6*b^5 + 6
*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*d*x + 3*(2*B*a^7*b^3 - 8*A*a^6*b^4 + 3*B*a^5*b^5 + 8*A*a^4*b^6 - 7*A*a^2*b^
8 + 2*A*b^10 + (2*B*a^10 - 8*A*a^9*b + 3*B*a^8*b^2 + 8*A*a^7*b^3 - 7*A*a^5*b^5 + 2*A*a^3*b^7)*cos(d*x + c)^3 +
3*(2*B*a^9*b - 8*A*a^8*b^2 + 3*B*a^7*b^3 + 8*A*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c)^2 + 3*(2*B*a
^8*b^2 - 8*A*a^7*b^3 + 3*B*a^6*b^4 + 8*A*a^5*b^5 - 7*A*a^3*b^7 + 2*A*a*b^9)*cos(d*x + c))*sqrt(-a^2 + b^2)*arc
tan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (11*B*a^8*b^3 - 26*A*a^7*b^4 - 7*B*a^
6*b^5 + 43*A*a^5*b^6 - 4*B*a^4*b^7 - 23*A*a^3*b^8 + 6*A*a*b^10 + (18*B*a^10*b - 36*A*a^9*b^2 - 23*B*a^8*b^3 +
68*A*a^7*b^4 + 7*B*a^6*b^5 - 43*A*a^5*b^6 - 2*B*a^4*b^7 + 11*A*a^3*b^8)*cos(d*x + c)^2 + 3*(9*B*a^9*b^2 - 20*A
*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 - B*a^5*b^6 - 20*A*a^4*b^7 + 5*A*a^2*b^9)*cos(d*x + c))*sin(d*x + c))/((
a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 -
4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*
x + c) + (a^12*b^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d)]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B \sec{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))**4,x)
[Out]
Integral((A + B*sec(c + d*x))/(a + b*sec(c + d*x))**4, x)
________________________________________________________________________________________
Giac [B] time = 1.47087, size = 1099, normalized size = 3.76 \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(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="giac")
[Out]
1/3*(3*(2*B*a^7 - 8*A*a^6*b + 3*B*a^5*b^2 + 8*A*a^4*b^3 - 7*A*a^2*b^5 + 2*A*b^7)*(pi*floor(1/2*(d*x + c)/pi +
1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^10 - 3
*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(-a^2 + b^2)) + 3*(d*x + c)*A/a^4 + (18*B*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 3
6*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*B*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 60*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5
+ 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^
5 - 45*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^2*b^6*tan(1/2*d*x + 1/2*c
)^5 + 15*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 - 6*A*b^8*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^7*b*tan(1/2*d*x + 1/2*c)^3 +
72*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 32*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 116*A*a^4*b^4*tan(1/2*d*x + 1/2*c
)^3 + 4*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^3 + 56*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^8*tan(1/2*d*x + 1/2*c)
^3 + 18*B*a^7*b*tan(1/2*d*x + 1/2*c) - 36*A*a^6*b^2*tan(1/2*d*x + 1/2*c) + 27*B*a^6*b^2*tan(1/2*d*x + 1/2*c) -
60*A*a^5*b^3*tan(1/2*d*x + 1/2*c) + 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c) + 6*A*a^4*b^4*tan(1/2*d*x + 1/2*c) + 3*B
*a^4*b^4*tan(1/2*d*x + 1/2*c) + 45*A*a^3*b^5*tan(1/2*d*x + 1/2*c) + 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c) + 6*A*a^2
*b^6*tan(1/2*d*x + 1/2*c) - 15*A*a*b^7*tan(1/2*d*x + 1/2*c) - 6*A*b^8*tan(1/2*d*x + 1/2*c))/((a^9 - 3*a^7*b^2
+ 3*a^5*b^4 - a^3*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3))/d