\(\int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\) [222]

Optimal result

Integrand size = 33, antiderivative size = 261 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {7 (17 A-7 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(33 A-13 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {(33 A-13 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

1/6*(33*A-13*B)*sin(d*x+c)/a^3/d/sec(d*x+c)^(1/2)-1/5*(A-B)*sin(d*x+c)/d/(a+a*sec(d*x+c))^3/sec(d*x+c)^(1/2)-1
/3*(2*A-B)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2/sec(d*x+c)^(1/2)-7/30*(17*A-7*B)*sin(d*x+c)/d/(a^3+a^3*sec(d*x+c)
)/sec(d*x+c)^(1/2)-7/10*(17*A-7*B)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c
),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^3/d+1/6*(33*A-13*B)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/
2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^3/d

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4105, 3872, 3854, 3856, 2720, 2719} \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {(33 A-13 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}+\frac {(33 A-13 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}-\frac {7 (17 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3} \]

[In]

Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]

[Out]

(-7*(17*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B)*
Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((33*A - 13*B)*Sin[c + d*x])/(6*a
^3*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - ((2*A - B)
*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - (7*(17*A - 7*B)*Sin[c + d*x])/(30*d*Sqrt[Se
c[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4105

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(
2*m + 1))), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*
(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[
A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {\int \frac {\frac {1}{2} a (13 A-3 B)-\frac {7}{2} a (A-B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {\int \frac {\frac {3}{2} a^2 (23 A-8 B)-\frac {25}{2} a^2 (2 A-B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4} \\ & = -\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \frac {\frac {15}{4} a^3 (33 A-13 B)-\frac {21}{4} a^3 (17 A-7 B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^6} \\ & = -\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(33 A-13 B) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3}-\frac {(7 (17 A-7 B)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = \frac {(33 A-13 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(33 A-13 B) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}-\frac {\left (7 (17 A-7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = -\frac {7 (17 A-7 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(33 A-13 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left ((33 A-13 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3} \\ & = -\frac {7 (17 A-7 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(33 A-13 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {(33 A-13 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac {(2 A-B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac {7 (17 A-7 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 7.46 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.44 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {e^{-i d x} \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) (A+B \sec (c+d x)) (\cos (d x)+i \sin (d x)) \left (160 (33 A-13 B) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+7 i (17 A-7 B) e^{-\frac {3}{2} i (c+d x)} \left (1+e^{i (c+d x)}\right )^5 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+2 \cos (c+d x) \left (-210 i (17 A-7 B) \cos \left (\frac {1}{2} (c+d x)\right )-105 i (17 A-7 B) \cos \left (\frac {3}{2} (c+d x)\right )-357 i A \cos \left (\frac {5}{2} (c+d x)\right )+147 i B \cos \left (\frac {5}{2} (c+d x)\right )+352 A \sin \left (\frac {1}{2} (c+d x)\right )-142 B \sin \left (\frac {1}{2} (c+d x)\right )+545 A \sin \left (\frac {3}{2} (c+d x)\right )-205 B \sin \left (\frac {3}{2} (c+d x)\right )+227 A \sin \left (\frac {5}{2} (c+d x)\right )-87 B \sin \left (\frac {5}{2} (c+d x)\right )+10 A \sin \left (\frac {7}{2} (c+d x)\right )\right )\right )}{120 a^3 d (B+A \cos (c+d x)) (1+\sec (c+d x))^3} \]

[In]

Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]

[Out]

(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*(Cos[d*x] + I*Sin[d*x])*(160*(33*A - 13*B)*Cos[(c +
d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((7*I)*(17*A - 7*B)*(1 + E^(I*(c + d*x)))^5*Sqrt[1 +
E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(((3*I)/2)*(c + d*x)) + 2*Cos[c
+ d*x]*((-210*I)*(17*A - 7*B)*Cos[(c + d*x)/2] - (105*I)*(17*A - 7*B)*Cos[(3*(c + d*x))/2] - (357*I)*A*Cos[(5
*(c + d*x))/2] + (147*I)*B*Cos[(5*(c + d*x))/2] + 352*A*Sin[(c + d*x)/2] - 142*B*Sin[(c + d*x)/2] + 545*A*Sin[
(3*(c + d*x))/2] - 205*B*Sin[(3*(c + d*x))/2] + 227*A*Sin[(5*(c + d*x))/2] - 87*B*Sin[(5*(c + d*x))/2] + 10*A*
Sin[(7*(c + d*x))/2])))/(120*a^3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^3)

Maple [A] (verified)

Time = 11.12 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (160 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+468 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+330 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+714 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-348 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-130 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-294 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1058 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+578 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+474 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-264 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-47 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+37 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 A -3 B \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(465\)

[In]

int((A+B*sec(d*x+c))/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/60/a^3*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(160*A*cos(1/2*d*x+1/2*c)^10+468*A*cos(1/2*d
*x+1/2*c)^8+330*A*cos(1/2*d*x+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Elliptic
F(cos(1/2*d*x+1/2*c),2^(1/2))+714*A*cos(1/2*d*x+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2
+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-348*B*cos(1/2*d*x+1/2*c)^8-130*B*cos(1/2*d*x+1/2*c)^5*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-294*B*cos(1/2*d*x
+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))
-1058*A*cos(1/2*d*x+1/2*c)^6+578*B*cos(1/2*d*x+1/2*c)^6+474*A*cos(1/2*d*x+1/2*c)^4-264*B*cos(1/2*d*x+1/2*c)^4-
47*A*cos(1/2*d*x+1/2*c)^2+37*B*cos(1/2*d*x+1/2*c)^2+3*A-3*B)/cos(1/2*d*x+1/2*c)^5/(-2*sin(1/2*d*x+1/2*c)^4+sin
(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.87 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {5 \, {\left (\sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (33 i \, A - 13 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (33 i \, A - 13 i \, B\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-33 i \, A + 13 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-33 i \, A + 13 i \, B\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (17 i \, A - 7 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (17 i \, A - 7 i \, B\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-17 i \, A + 7 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-17 i \, A + 7 i \, B\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (20 \, A \cos \left (d x + c\right )^{4} + 3 \, {\left (79 \, A - 29 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (188 \, A - 73 \, B\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (33 \, A - 13 \, B\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

integrate((A+B*sec(d*x+c))/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/60*(5*(sqrt(2)*(33*I*A - 13*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(33*I*A - 13*I*B)*cos(d*x + c)^2 + 3*sqrt(2)*(3
3*I*A - 13*I*B)*cos(d*x + c) + sqrt(2)*(33*I*A - 13*I*B))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x
+ c)) + 5*(sqrt(2)*(-33*I*A + 13*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-33*I*A + 13*I*B)*cos(d*x + c)^2 + 3*sqrt(2)
*(-33*I*A + 13*I*B)*cos(d*x + c) + sqrt(2)*(-33*I*A + 13*I*B))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin
(d*x + c)) + 21*(sqrt(2)*(17*I*A - 7*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(17*I*A - 7*I*B)*cos(d*x + c)^2 + 3*sqrt(
2)*(17*I*A - 7*I*B)*cos(d*x + c) + sqrt(2)*(17*I*A - 7*I*B))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0,
cos(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(-17*I*A + 7*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-17*I*A + 7*I*B)*
cos(d*x + c)^2 + 3*sqrt(2)*(-17*I*A + 7*I*B)*cos(d*x + c) + sqrt(2)*(-17*I*A + 7*I*B))*weierstrassZeta(-4, 0,
weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(20*A*cos(d*x + c)^4 + 3*(79*A - 29*B)*cos(d*x
+ c)^3 + 2*(188*A - 73*B)*cos(d*x + c)^2 + 5*(33*A - 13*B)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^3
*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sec(d*x+c))/sec(d*x+c)**(3/2)/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((A+B*sec(d*x+c))/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((A+B*sec(d*x+c))/sec(d*x+c)^(3/2)/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)/((a*sec(d*x + c) + a)^3*sec(d*x + c)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((A + B/cos(c + d*x))/((a + a/cos(c + d*x))^3*(1/cos(c + d*x))^(3/2)),x)

[Out]

int((A + B/cos(c + d*x))/((a + a/cos(c + d*x))^3*(1/cos(c + d*x))^(3/2)), x)