\(\int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx\) [159]

Optimal result

Integrand size = 33, antiderivative size = 268 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {(47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}} \]

[Out]

-1/8*(47*A-38*B)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(3/2)/d-1/2*(A-B)*cos(d*x+c)^2*sin(d*x+c)
/d/(a+a*sec(d*x+c))^(3/2)+1/4*(17*A-13*B)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2))/a^(3/2
)/d*2^(1/2)+7/8*(3*A-2*B)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2)-1/12*(13*A-12*B)*cos(d*x+c)*sin(d*x+c)/a/d/(a+
a*sec(d*x+c))^(1/2)+1/6*(5*A-3*B)*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 268, 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, 4107, 4005, 3859, 209, 3880} \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {(47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a \sec (c+d x)+a}}+\frac {(5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{6 a d \sqrt {a \sec (c+d x)+a}}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac {(13 A-12 B) \sin (c+d x) \cos (c+d x)}{12 a d \sqrt {a \sec (c+d x)+a}} \]

[In]

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

[Out]

-1/8*((47*A - 38*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((17*A - 13*B)*ArcT
an[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Cos[c + d*x]^2
*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (7*(3*A - 2*B)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]
) - ((13*A - 12*B)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 3*B)*Cos[c + d*x]^2*
Sin[c + d*x])/(6*a*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-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^3(c+d x) \left (a (5 A-3 B)-\frac {7}{2} a (A-B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\cos ^2(c+d x) \left (-a^2 (13 A-12 B)+\frac {5}{2} a^2 (5 A-3 B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{6 a^3} \\ & = -\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\cos (c+d x) \left (\frac {21}{2} a^3 (3 A-2 B)-\frac {3}{2} a^3 (13 A-12 B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{12 a^4} \\ & = -\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {-\frac {3}{4} a^4 (47 A-38 B)+\frac {21}{4} a^4 (3 A-2 B) \sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{12 a^5} \\ & = -\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}-\frac {(47 A-38 B) \int \sqrt {a+a \sec (c+d x)} \, dx}{16 a^2}+\frac {(17 A-13 B) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a} \\ & = -\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}+\frac {(47 A-38 B) \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 d}-\frac {(17 A-13 B) \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 )}{2 a d} \\ & = -\frac {(47 A-38 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A-13 B) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {7 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A-12 B) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A-3 B) \cos ^2(c+d x) \sin (c+d x)}{6 a 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 = 6.13 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {B \cos (c+d x) \sin (c+d x)}{2 d (a (1+\sec (c+d x)))^{3/2}}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a (1+\sec (c+d x)))^{3/2}}-\frac {B (1+\sec (c+d x))^{3/2} \left (\frac {40 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1+\sec (c+d x)}}-\frac {13 \left (7 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )-\cos (c+d x) \sqrt {1-\sec (c+d x)}+2 \cos ^2(c+d x) \sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{16 (a (1+\sec (c+d x)))^{3/2}}-\frac {A (1+\sec (c+d x))^{3/2} \left (\frac {336 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {17 \left (27 \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-24 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )-21 \cos (c+d x) \sqrt {1-\sec (c+d x)}+2 \cos ^2(c+d x) \sqrt {1-\sec (c+d x)}-8 \cos ^3(c+d x) \sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{96 (a (1+\sec (c+d x)))^{3/2}} \]

[In]

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

[Out]

-1/2*(B*Cos[c + d*x]*Sin[c + d*x])/(d*(a*(1 + Sec[c + d*x]))^(3/2)) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a*
(1 + Sec[c + d*x]))^(3/2)) - (B*(1 + Sec[c + d*x])^(3/2)*((40*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]
*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) - (13*(7*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*ArcTanh[Sqrt[1
- Sec[c + d*x]]/Sqrt[2]] - Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]])*Tan[
c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(16*(a*(1 + Sec[c + d*x]))^(3/2)) - (A*(1 + Sec[
c + d*x])^(3/2)*((336*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]
) + (17*(27*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 24*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - 21*Cos[c +
d*x]*Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]] - 8*Cos[c + d*x]^3*Sqrt[1 - Sec[c + d*x]
])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(96*(a*(1 + Sec[c + d*x]))^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1024\) vs. \(2(233)=466\).

Time = 29.89 (sec) , antiderivative size = 1025, normalized size of antiderivative = 3.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1025\)

[In]

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

[Out]

1/24/a^2/d*(102*A*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+
c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*cos(d*x+c)^2+8*A*cos(d*x+c)^4*sin(d*x+c)-78*B*2^(1/2)*(-cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*cos(d*x+
c)^2+204*A*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(
d*x+c)+csc(d*x+c)^2-1)^(1/2))*cos(d*x+c)-141*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+
c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-6*A*sin(d*x+c)*cos(d*x+c)^3-156*B*2^(1/2)*(-cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*co
s(d*x+c)+114*B*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1
))^(1/2))*cos(d*x+c)^2+12*B*cos(d*x+c)^3*sin(d*x+c)+102*A*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot
(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-282*A*arctanh(sin(d*x+c)/(cos(d*x
+c)+1)/(-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)+37*A*cos(d*x+c)^2*sin
(d*x+c)-78*B*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*(-c
os(d*x+c)/(cos(d*x+c)+1))^(1/2)+228*B*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-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)-18*B*sin(d*x+c)*cos(d*x+c)^2-141*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+63*A*cos(d*x+c)*sin(d*x+c)+114*B*arc
tanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-42*B*cos
(d*x+c)*sin(d*x+c))*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^2

Fricas [A] (verification not implemented)

none

Time = 3.34 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.52 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {6 \, \sqrt {2} {\left ({\left (17 \, A - 13 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (17 \, A - 13 \, B\right )} \cos \left (d x + c\right ) + 17 \, A - 13 \, B\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 3 \, {\left ({\left (47 \, A - 38 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (47 \, A - 38 \, B\right )} \cos \left (d x + c\right ) + 47 \, A - 38 \, B\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (8 \, A \cos \left (d x + c\right )^{4} - 6 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (37 \, A - 18 \, B\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{48 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, -\frac {6 \, \sqrt {2} {\left ({\left (17 \, A - 13 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (17 \, A - 13 \, B\right )} \cos \left (d x + c\right ) + 17 \, A - 13 \, B\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 ) - 3 \, {\left ({\left (47 \, A - 38 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (47 \, A - 38 \, B\right )} \cos \left (d x + c\right ) + 47 \, A - 38 \, B\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 ) - {\left (8 \, A \cos \left (d x + c\right )^{4} - 6 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{3} + {\left (37 \, A - 18 \, B\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]

[In]

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

[Out]

[1/48*(6*sqrt(2)*((17*A - 13*B)*cos(d*x + c)^2 + 2*(17*A - 13*B)*cos(d*x + c) + 17*A - 13*B)*sqrt(-a)*log(-(2*
sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*
cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 3*((47*A - 38*B)*cos(d*x + c)^2 + 2*(47*A - 38*B)*c
os(d*x + c) + 47*A - 38*B)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c
))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(8*A*cos(d*x + c)^4 - 6*(A - 2*B)*c
os(d*x + c)^3 + (37*A - 18*B)*cos(d*x + c)^2 + 21*(3*A - 2*B)*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x
+ c))*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d), -1/24*(6*sqrt(2)*((17*A - 13*B)*cos
(d*x + c)^2 + 2*(17*A - 13*B)*cos(d*x + c) + 17*A - 13*B)*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))) - 3*((47*A - 38*B)*cos(d*x + c)^2 + 2*(47*A - 38*B)*cos(d*x +
c) + 47*A - 38*B)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))
- (8*A*cos(d*x + c)^4 - 6*(A - 2*B)*cos(d*x + c)^3 + (37*A - 18*B)*cos(d*x + c)^2 + 21*(3*A - 2*B)*cos(d*x + c
))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)
]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (233) = 466\).

Time = 2.44 (sec) , antiderivative size = 816, normalized size of antiderivative = 3.04 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/48*(6*sqrt(2)*(17*A - 13*B)*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/(s
qrt(-a)*a*sgn(cos(d*x + c))) - 3*(47*A - 38*B)*log(abs(-1947111321950560360698936123457536*(sqrt(-a)*tan(1/2*d
*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - 3894222643901120721397872246915072*sqrt(2)*abs(a) + 584
1333965851681082096808370372608*a)/abs(-1947111321950560360698936123457536*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sq
rt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + 3894222643901120721397872246915072*sqrt(2)*abs(a) + 5841333965851681082
096808370372608*a))/(sqrt(-a)*abs(a)*sgn(cos(d*x + c))) - 12*(sqrt(2)*A*a*sgn(cos(d*x + c)) - sqrt(2)*B*a*sgn(
cos(d*x + c)))*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*tan(1/2*d*x + 1/2*c)/a^3 - 4*sqrt(2)*(339*(sqrt(-a)*tan(1/2
*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A - 174*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1
/2*d*x + 1/2*c)^2 + a))^10*B - 3165*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*A*
a + 1842*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*B*a + 9198*(sqrt(-a)*tan(1/2*
d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*A*a^2 - 5292*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*ta
n(1/2*d*x + 1/2*c)^2 + a))^6*B*a^2 - 4938*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)
)^4*A*a^3 + 2820*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*B*a^3 + 975*(sqrt(-a)
*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*A*a^4 - 582*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sq
rt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*B*a^4 - 73*A*a^5 + 42*B*a^5)/(((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*t
an(1/2*d*x + 1/2*c)^2 + a))^4 - 6*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a +
a^2)^3*sqrt(-a)*sgn(cos(d*x + c))))/d

Mupad [F(-1)]

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

[In]

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

[Out]

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