\(\int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx\) [178]

Optimal result

Integrand size = 34, antiderivative size = 280 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {203 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}} \]

[Out]

203/8*A*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/a^(5/2)/d-1/2*A*cos(d*x+c)^2*sin(d*x+c)/d/(a-a*sec(d
*x+c))^(5/2)-19/8*A*cos(d*x+c)^2*sin(d*x+c)/a/d/(a-a*sec(d*x+c))^(3/2)-287/16*A*arctan(1/2*a^(1/2)*tan(d*x+c)*
2^(1/2)/(a-a*sec(d*x+c))^(1/2))/a^(5/2)/d*2^(1/2)+21/2*A*sin(d*x+c)/a^2/d/(a-a*sec(d*x+c))^(1/2)+119/24*A*cos(
d*x+c)*sin(d*x+c)/a^2/d/(a-a*sec(d*x+c))^(1/2)+77/24*A*cos(d*x+c)^2*sin(d*x+c)/a^2/d/(a-a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4105, 4107, 4005, 3859, 209, 3880} \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {203 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \sin (c+d x) \cos ^2(c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \sin (c+d x) \cos (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {19 A \sin (c+d x) \cos ^2(c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}} \]

[In]

Int[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]

[Out]

(203*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*a^(5/2)*d) - (287*A*ArcTan[(Sqrt[a]*Tan[c +
d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a -
a*Sec[c + d*x])^(5/2)) - (19*A*Cos[c + d*x]^2*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (21*A*Sin[c +
d*x])/(2*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (119*A*Cos[c + d*x]*Sin[c + d*x])/(24*a^2*d*Sqrt[a - a*Sec[c + d*x
]]) + (77*A*Cos[c + d*x]^2*Sin[c + d*x])/(24*a^2*d*Sqrt[a - a*Sec[c + d*x]])

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 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3880

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]

Rule 4005

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

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]

Rule 4107

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

Rubi steps \begin{align*} \text {integral}& = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}+\frac {\int \frac {\cos ^3(c+d x) (10 a A+9 a A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^3(c+d x) \left (77 a^2 A+\frac {133}{2} a^2 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{8 a^4} \\ & = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x) \left (-238 a^3 A-\frac {385}{2} a^3 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{24 a^5} \\ & = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {\int \frac {\cos (c+d x) \left (504 a^4 A+357 a^4 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{48 a^6} \\ & = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {-609 a^5 A-252 a^5 A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{48 a^7} \\ & = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(203 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{16 a^3}+\frac {(287 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{16 a^2} \\ & = -\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(203 A) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^2 d}-\frac {(287 A) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^2 d} \\ & = \frac {203 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 2.78 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {A \sec ^2(c+d x) \left (-912 (-1+\cos (c+d x)) \cos ^4(c+d x)-192 \cos ^5(c+d x)+6384 (-1+\cos (c+d x))^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1+\sec (c+d x)\right )+\frac {287 (-1+\cos (c+d x))^2 \left (27 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )-24 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) \left (21+2 \cos (c+d x)+8 \cos ^2(c+d x)\right ) \sqrt {1+\sec (c+d x)}\right )}{\sqrt {1+\sec (c+d x)}}\right ) \tan (c+d x)}{384 d (a-a \sec (c+d x))^{5/2}} \]

[In]

Integrate[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]

[Out]

(A*Sec[c + d*x]^2*(-912*(-1 + Cos[c + d*x])*Cos[c + d*x]^4 - 192*Cos[c + d*x]^5 + 6384*(-1 + Cos[c + d*x])^2*H
ypergeometric2F1[1/2, 4, 3/2, 1 + Sec[c + d*x]] + (287*(-1 + Cos[c + d*x])^2*(27*ArcTanh[Sqrt[1 + Sec[c + d*x]
]] - 24*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(21 + 2*Cos[c + d*x] + 8*Cos[c + d*x]^2
)*Sqrt[1 + Sec[c + d*x]]))/Sqrt[1 + Sec[c + d*x]])*Tan[c + d*x])/(384*d*(a - a*Sec[c + d*x])^(5/2))

Maple [A] (verified)

Time = 30.62 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.58

method	result	size
default	\(-\frac {A \sqrt {2}\, \left (8 \cos \left (d x +c \right )^{5} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+30 \cos \left (d x +c \right )^{4} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+113 \cos \left (d x +c \right )^{3} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-294 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+609 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {2}\, \cos \left (d x +c \right )^{2}+861 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{2}-133 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \cos \left (d x +c \right )-1218 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-1722 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+252 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+609 \sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+861 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )\right ) \csc \left (d x +c \right )}{48 a^{2} d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )-1\right ) \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}}\)	\(441\)

[In]

int(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/48*A/a^2/d*2^(1/2)*(8*cos(d*x+c)^5*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+30*cos(d*x+c)^4*2^(1/2)*(-cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)+113*cos(d*x+c)^3*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-294*cos(d*x+c)^2*2^(
1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+609*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)*cos(d*x+c)^2+86
1*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-133*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2
^(1/2)*cos(d*x+c)-1218*2^(1/2)*cos(d*x+c)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-1722*cos(d*x+c)*arctan(1/
2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+252*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+609*2^(1/2)*arcta
n((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+861*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)))/(-cos(d*x+c)
/(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)-1)/(-a*(sec(d*x+c)-1))^(1/2)*csc(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 656, normalized size of antiderivative = 2.34 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\left [-\frac {861 \, \sqrt {2} {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} + {\left (3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 1218 \, {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 4 \, {\left (8 \, A \cos \left (d x + c\right )^{6} + 30 \, A \cos \left (d x + c\right )^{5} + 113 \, A \cos \left (d x + c\right )^{4} - 294 \, A \cos \left (d x + c\right )^{3} - 133 \, A \cos \left (d x + c\right )^{2} + 252 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{96 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}, \frac {861 \, \sqrt {2} {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 1218 \, {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 2 \, {\left (8 \, A \cos \left (d x + c\right )^{6} + 30 \, A \cos \left (d x + c\right )^{5} + 113 \, A \cos \left (d x + c\right )^{4} - 294 \, A \cos \left (d x + c\right )^{3} - 133 \, A \cos \left (d x + c\right )^{2} + 252 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{48 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}\right ] \]

[In]

integrate(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[-1/96*(861*sqrt(2)*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c) + A)*sqrt(-a)*log((2*sqrt(2)*(cos(d*x + c)^2 + cos(d*
x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)) + (3*a*cos(d*x + c) + a)*sin(d*x + c))/((cos(d*x + c)
- 1)*sin(d*x + c)))*sin(d*x + c) + 1218*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c) + A)*sqrt(-a)*log((2*(cos(d*x +
c)^2 + cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)) - (2*a*cos(d*x + c) + a)*sin(d*x + c))/s
in(d*x + c))*sin(d*x + c) + 4*(8*A*cos(d*x + c)^6 + 30*A*cos(d*x + c)^5 + 113*A*cos(d*x + c)^4 - 294*A*cos(d*x
+ c)^3 - 133*A*cos(d*x + c)^2 + 252*A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/((a^3*d*cos(d*x
+ c)^2 - 2*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c)), 1/48*(861*sqrt(2)*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c) +
A)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) - a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*
x + c) - 1218*(A*cos(d*x + c)^2 - 2*A*cos(d*x + c) + A)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) - a)/cos(d*x + c))
*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) - 2*(8*A*cos(d*x + c)^6 + 30*A*cos(d*x + c)^5 + 113*A*cos(d
*x + c)^4 - 294*A*cos(d*x + c)^3 - 133*A*cos(d*x + c)^2 + 252*A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*
x + c)))/((a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((A*sec(d*x + c) + A)*cos(d*x + c)^3/(-a*sec(d*x + c) + a)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 1.23 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {861 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {1218 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {2 \, \sqrt {2} {\left (129 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {5}{2}} A + 560 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A a + 636 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{3} a^{2}} - \frac {3 \, \sqrt {2} {\left (33 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A + 31 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a\right )}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{48 \, d} \]

[In]

integrate(cos(d*x+c)^3*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/48*(861*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(5/2) - 1218*A*arctan(1/2*sqrt(2)*sqr
t(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(5/2) - 2*sqrt(2)*(129*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(5/2)*A + 560
*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A*a + 636*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)*A*a^2)/((a*tan(1/2*d*x + 1/
2*c)^2 + a)^3*a^2) - 3*sqrt(2)*(33*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A + 31*sqrt(a*tan(1/2*d*x + 1/2*c)^2 -
a)*A*a)/(a^4*tan(1/2*d*x + 1/2*c)^4))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int((cos(c + d*x)^3*(A + A/cos(c + d*x)))/(a - a/cos(c + d*x))^(5/2),x)

[Out]

int((cos(c + d*x)^3*(A + A/cos(c + d*x)))/(a - a/cos(c + d*x))^(5/2), x)