\(\int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx\) [177]

Optimal result

Integrand size = 34, antiderivative size = 236 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {59 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{4 a^{5/2} d}-\frac {167 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \cos (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {49 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {23 A \cos (c+d x) \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}} \]

[Out]

59/4*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)*sin(d*x+c)/d/(a-a*sec(d*x+
c))^(5/2)-15/8*A*cos(d*x+c)*sin(d*x+c)/a/d/(a-a*sec(d*x+c))^(3/2)-167/16*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)+49/8*A*sin(d*x+c)/a^2/d/(a-a*sec(d*x+c))^(1/2)+23/8*A*cos(d*x+c)*
sin(d*x+c)/a^2/d/(a-a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4105, 4107, 4005, 3859, 209, 3880} \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {59 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{4 a^{5/2} d}-\frac {167 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}+\frac {49 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {23 A \sin (c+d x) \cos (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {15 A \sin (c+d x) \cos (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac {A \sin (c+d x) \cos (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}} \]

[In]

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

[Out]

(59*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(5/2)*d) - (167*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]*Sin[c + d*x])/(2*d*(a - a*S
ec[c + d*x])^(5/2)) - (15*A*Cos[c + d*x]*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (49*A*Sin[c + d*x]
)/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (23*A*Cos[c + d*x]*Sin[c + d*x])/(8*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 (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}+\frac {\int \frac {\cos ^2(c+d x) (8 a A+7 a A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {A \cos (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^2(c+d x) \left (46 a^2 A+\frac {75}{2} a^2 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{8 a^4} \\ & = -\frac {A \cos (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {23 A \cos (c+d x) \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {\cos (c+d x) \left (-98 a^3 A-69 a^3 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{16 a^5} \\ & = -\frac {A \cos (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {49 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {23 A \cos (c+d x) \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {\int \frac {118 a^4 A+49 a^4 A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{16 a^6} \\ & = -\frac {A \cos (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {49 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {23 A \cos (c+d x) \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(59 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{8 a^3}+\frac {(167 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{16 a^2} \\ & = -\frac {A \cos (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {49 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {23 A \cos (c+d x) \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(59 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 )}{4 a^2 d}-\frac {(167 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 {59 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{4 a^{5/2} d}-\frac {167 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 (c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {15 A \cos (c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {49 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {23 A \cos (c+d x) \sin (c+d x)}{8 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 = 2.12 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {A \left (-30 (-1+\cos (c+d x)) \sin (c+d x)-4 \sin (2 (c+d x))-150 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2 \tan (c+d x)+\frac {167 (-1+\sec (c+d x))^2 \left (7 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) (1+2 \cos (c+d x)) \sqrt {1+\sec (c+d x)}\right ) \tan (c+d x)}{4 \sqrt {1+\sec (c+d x)}}\right )}{16 d (a-a \sec (c+d x))^{5/2}} \]

[In]

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

[Out]

(A*(-30*(-1 + Cos[c + d*x])*Sin[c + d*x] - 4*Sin[2*(c + d*x)] - 150*Hypergeometric2F1[1/2, 3, 3/2, 1 + Sec[c +
d*x]]*(-1 + Sec[c + d*x])^2*Tan[c + d*x] + (167*(-1 + Sec[c + d*x])^2*(7*ArcTanh[Sqrt[1 + Sec[c + d*x]]] - 4*
Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(1 + 2*Cos[c + d*x])*Sqrt[1 + Sec[c + d*x]])*Ta
n[c + d*x])/(4*Sqrt[1 + Sec[c + d*x]])))/(16*d*(a - a*Sec[c + d*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(201)=402\).

Time = 29.76 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.73

method	result	size
default	\(-\frac {A \sqrt {2}\, \left (4 \cos \left (d x +c \right )^{4} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+22 \cos \left (d x +c \right )^{3} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-57 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+118 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {2}\, \cos \left (d x +c \right )^{2}+167 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{2}-26 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \cos \left (d x +c \right )-236 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-334 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+49 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+118 \sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+167 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )\right ) \csc \left (d x +c \right )}{16 a^{2} d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )-1\right ) \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}}\)	\(408\)

[In]

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

[Out]

-1/16*A/a^2/d*2^(1/2)*(4*cos(d*x+c)^4*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+22*cos(d*x+c)^3*2^(1/2)*(-cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)-57*cos(d*x+c)^2*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+118*arctan((-cos(d*x+
c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)*cos(d*x+c)^2+167*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(
d*x+c)^2-26*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2^(1/2)*cos(d*x+c)-236*2^(1/2)*cos(d*x+c)*arctan((-cos(d*x+c)/(
cos(d*x+c)+1))^(1/2))-334*cos(d*x+c)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+49*2^(1/2)*(-cos(d
*x+c)/(cos(d*x+c)+1))^(1/2)+118*2^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+167*arctan(1/2*2^(1/2)/(-co
s(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)*c
sc(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.69 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\left [-\frac {167 \, \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 ) + 236 \, {\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 (4 \, A \cos \left (d x + c\right )^{5} + 22 \, A \cos \left (d x + c\right )^{4} - 57 \, A \cos \left (d x + c\right )^{3} - 26 \, A \cos \left (d x + c\right )^{2} + 49 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{32 \, {\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 {167 \, \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 ) - 236 \, {\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 (4 \, A \cos \left (d x + c\right )^{5} + 22 \, A \cos \left (d x + c\right )^{4} - 57 \, A \cos \left (d x + c\right )^{3} - 26 \, A \cos \left (d x + c\right )^{2} + 49 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{16 \, {\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)^2*(A+A*sec(d*x+c))/(a-a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[-1/32*(167*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) + 236*(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))/si
n(d*x + c))*sin(d*x + c) + 4*(4*A*cos(d*x + c)^5 + 22*A*cos(d*x + c)^4 - 57*A*cos(d*x + c)^3 - 26*A*cos(d*x +
c)^2 + 49*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)), 1/16*(167*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) - 236*(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*(4*A*cos(d*x + c)^5 + 22*A*cos(d*x + c)^4 - 57*A*cos(d*x + c)^3 - 26*A*cos(d*x + c)
^2 + 49*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]

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

[In]

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

[Out]

A*(Integral(cos(c + d*x)**2/(a**2*sqrt(-a*sec(c + d*x) + a)*sec(c + d*x)**2 - 2*a**2*sqrt(-a*sec(c + d*x) + a)
*sec(c + d*x) + a**2*sqrt(-a*sec(c + d*x) + a)), x) + Integral(cos(c + d*x)**2*sec(c + d*x)/(a**2*sqrt(-a*sec(
c + d*x) + a)*sec(c + d*x)**2 - 2*a**2*sqrt(-a*sec(c + d*x) + a)*sec(c + d*x) + a**2*sqrt(-a*sec(c + d*x) + a)
), x))

Maxima [F]

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

[In]

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

Giac [A] (verification not implemented)

none

Time = 1.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {167 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {236 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {\sqrt {2} {\left (69 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {7}{2}} A + 315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {5}{2}} A a + 444 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A a^{2} + 196 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a^{3}\right )}}{{\left ({\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} + 3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a + 2 \, a^{2}\right )}^{2} a^{2}}}{16 \, d} \]

[In]

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

[Out]

1/16*(167*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(5/2) - 236*A*arctan(1/2*sqrt(2)*sqrt
(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(5/2) - sqrt(2)*(69*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(7/2)*A + 315*(a*
tan(1/2*d*x + 1/2*c)^2 - a)^(5/2)*A*a + 444*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2)*A*a^2 + 196*sqrt(a*tan(1/2*d*
x + 1/2*c)^2 - a)*A*a^3)/(((a*tan(1/2*d*x + 1/2*c)^2 - a)^2 + 3*(a*tan(1/2*d*x + 1/2*c)^2 - a)*a + 2*a^2)^2*a^
2))/d

Mupad [F(-1)]

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

[In]

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

[Out]

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