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

Optimal. Leaf size=194 \[ \frac{31 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{4 a^{3/2} d}-\frac{11 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right )}{\sqrt{2} a^{3/2} d}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x) \cos (c+d x)}{d (a-a \sec (c+d x))^{3/2}} \]

[Out]

(31*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(3/2)*d) - (11*A*ArcTan[(Sqrt[a]*Tan[c + d
*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Cos[c + d*x]*Sin[c + d*x])/(d*(a - a*Sec[c
+ d*x])^(3/2)) + (13*A*Sin[c + d*x])/(4*a*d*Sqrt[a - a*Sec[c + d*x]]) + (3*A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d
*Sqrt[a - a*Sec[c + d*x]])

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Rubi [A]  time = 0.529145, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4020, 4022, 3920, 3774, 203, 3795} \[ \frac{31 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{4 a^{3/2} d}-\frac{11 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right )}{\sqrt{2} a^{3/2} d}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x) \cos (c+d x)}{d (a-a \sec (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(31*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(3/2)*d) - (11*A*ArcTan[(Sqrt[a]*Tan[c + d
*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Cos[c + d*x]*Sin[c + d*x])/(d*(a - a*Sec[c
+ d*x])^(3/2)) + (13*A*Sin[c + d*x])/(4*a*d*Sqrt[a - a*Sec[c + d*x]]) + (3*A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d
*Sqrt[a - a*Sec[c + d*x]])

Rule 4020

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 4022

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]

Rule 3920

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 3774

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3795

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
]

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx &=-\frac{A \cos (c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{\int \frac{\cos ^2(c+d x) (6 a A+5 a A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{A \cos (c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{3 A \cos (c+d x) \sin (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}-\frac{\int \frac{\cos (c+d x) \left (-13 a^2 A-9 a^2 A \sec (c+d x)\right )}{\sqrt{a-a \sec (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{A \cos (c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \cos (c+d x) \sin (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}+\frac{\int \frac{\frac{31 a^3 A}{2}+\frac{13}{2} a^3 A \sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx}{4 a^4}\\ &=-\frac{A \cos (c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \cos (c+d x) \sin (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}+\frac{(31 A) \int \sqrt{a-a \sec (c+d x)} \, dx}{8 a^2}+\frac{(11 A) \int \frac{\sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx}{2 a}\\ &=-\frac{A \cos (c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \cos (c+d x) \sin (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}+\frac{(31 A) \operatorname{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 d}-\frac{(11 A) \operatorname{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 )}{a d}\\ &=\frac{31 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{4 a^{3/2} d}-\frac{11 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right )}{\sqrt{2} a^{3/2} d}-\frac{A \cos (c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \cos (c+d x) \sin (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.66512, size = 408, normalized size = 2.1 \[ A \left (\frac{\sin ^3\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2(c+d x) \left (\frac{3 \sin \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )}{2 d}+\frac{5 \sin \left (\frac{3 c}{2}\right ) \sin \left (\frac{3 d x}{2}\right )}{d}+\frac{\sin \left (\frac{5 c}{2}\right ) \sin \left (\frac{5 d x}{2}\right )}{2 d}-\frac{3 \cos \left (\frac{c}{2}\right ) \cos \left (\frac{d x}{2}\right )}{2 d}-\frac{5 \cos \left (\frac{3 c}{2}\right ) \cos \left (\frac{3 d x}{2}\right )}{d}-\frac{\cos \left (\frac{5 c}{2}\right ) \cos \left (\frac{5 d x}{2}\right )}{2 d}-\frac{2 \cot \left (\frac{c}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{2 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \csc ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\right )}{(a-a \sec (c+d x))^{3/2}}-\frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^3\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^{\frac{3}{2}}(c+d x) \left (31 \sinh ^{-1}\left (e^{i (c+d x)}\right )-44 \sqrt{2} \tanh ^{-1}\left (\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )+31 \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )}{2 \sqrt{2} d (a-a \sec (c+d x))^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

A*(-(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(31*ArcSinh[E^(I*(c + d*x))
] - 44*Sqrt[2]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 31*ArcTanh[Sqrt[1 + E^
((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sin[c/2 + (d*x)/2]^3)/(2*Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c +
d*x])^(3/2)) + (Sec[c + d*x]^2*((-3*Cos[c/2]*Cos[(d*x)/2])/(2*d) - (5*Cos[(3*c)/2]*Cos[(3*d*x)/2])/d - (Cos[(5
*c)/2]*Cos[(5*d*x)/2])/(2*d) - (2*Cot[c/2]*Csc[c/2 + (d*x)/2])/d + (2*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/
2])/d + (3*Sin[c/2]*Sin[(d*x)/2])/(2*d) + (5*Sin[(3*c)/2]*Sin[(3*d*x)/2])/d + (Sin[(5*c)/2]*Sin[(5*d*x)/2])/(2
*d))*Sin[c/2 + (d*x)/2]^3)/(a - a*Sec[c + d*x])^(3/2))

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Maple [B]  time = 0.351, size = 883, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*A/d*2^(1/2)*(-1+cos(d*x+c))^4*(-278*cos(d*x+c)^3*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-930*arctan
(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+132*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*cos(d*x+c)^2*2^(1/
2)+180*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*cos(d*x+c)^2*2^(1/2)+180*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*cos(
d*x+c)*2^(1/2)+132*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*cos(d*x+c)^3*2^(1/2)-220*(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(3/2)*cos(d*x+c)^3*2^(1/2)+660*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3*2^(1/2)+60*(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(7/2)*cos(d*x+c)^3*2^(1/2)-132*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*2^(1/2)+930*arcta
n(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-195*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)-660*2^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-132*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*cos(d*
x+c)*2^(1/2)+30*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^5*2^(1/2)-220*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(
3/2)*cos(d*x+c)^2*2^(1/2)+195*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^4*2^(1/2)+660*arctan(1/(-2*cos(d
*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2*2^(1/2)+220*cos(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(3/2)+
288*cos(d*x+c)^2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-40*cos(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)-660*cos(d*x+c)*2^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+220*2^(1/2)*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(3/2)-930*cos(d*x+c)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+60*(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(7/2)*2^(1/2)+930*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3)/(-2*c
os(d*x+c)/(cos(d*x+c)+1))^(3/2)/(a*(-1+cos(d*x+c))/cos(d*x+c))^(3/2)/sin(d*x+c)^7

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.55599, size = 1415, normalized size = 7.29 \begin{align*} \left [-\frac{22 \, \sqrt{2}{\left (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 ) + 31 \,{\left (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 ) + 2 \,{\left (2 \, A \cos \left (d x + c\right )^{4} + 9 \, A \cos \left (d x + c\right )^{3} - 6 \, A \cos \left (d x + c\right )^{2} - 13 \, A \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{8 \,{\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac{22 \, \sqrt{2}{\left (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 ) - 31 \,{\left (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 ) -{\left (2 \, A \cos \left (d x + c\right )^{4} + 9 \, A \cos \left (d x + c\right )^{3} - 6 \, A \cos \left (d x + c\right )^{2} - 13 \, A \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{4 \,{\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(22*sqrt(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)))*si
n(d*x + c) + 31*(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))/sin(d*x + c))*sin(d*x + c) + 2*(2*A*cos(d*x + c
)^4 + 9*A*cos(d*x + c)^3 - 6*A*cos(d*x + c)^2 - 13*A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d*x + c)))/((
a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c)), 1/4*(22*sqrt(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) - 31*(A*cos(d*x + c) - A)*s
qrt(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*A
*cos(d*x + c)^4 + 9*A*cos(d*x + c)^3 - 6*A*cos(d*x + c)^2 - 13*A*cos(d*x + c))*sqrt((a*cos(d*x + c) - a)/cos(d
*x + c)))/((a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} A \left (\int \frac{\cos ^{2}{\left (c + d x \right )}}{- a \sqrt{- a \sec{\left (c + d x \right )} + a} \sec{\left (c + d x \right )} + a \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 \sqrt{- a \sec{\left (c + d x \right )} + a} \sec{\left (c + d x \right )} + a \sqrt{- a \sec{\left (c + d x \right )} + a}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.17359, size = 393, normalized size = 2.03 \begin{align*} -\frac{A{\left (\frac{22 \, \sqrt{2} \arctan \left (\frac{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{31 \, \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{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{\sqrt{2}{\left (7 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{\frac{3}{2}} + 18 \, \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a} a\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{2} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{2 \, \sqrt{2} \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a}}{a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*A*(22*sqrt(2)*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/(a^(3/2)*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)
*sgn(tan(1/2*d*x + 1/2*c))) - 31*arctan(1/2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/(a^(3/2)*sgn(t
an(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c))) - sqrt(2)*(7*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/2) + 18*s
qrt(a*tan(1/2*d*x + 1/2*c)^2 - a)*a)/((a*tan(1/2*d*x + 1/2*c)^2 + a)^2*a*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(t
an(1/2*d*x + 1/2*c))) - 2*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/(a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(
tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c)^2))/d