3.2.74 \(\int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx\) [174]

Optimal. Leaf size=236 \[ \frac {85 A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{3/2} d}-\frac {15 A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} a^{3/2} d}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}} \]

[Out]

85/8*A*arctan(a^(1/2)*tan(d*x+c)/(a-a*sec(d*x+c))^(1/2))/a^(3/2)/d-A*cos(d*x+c)^2*sin(d*x+c)/d/(a-a*sec(d*x+c)
)^(3/2)-15/2*A*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)+35/8*A*sin(d*x+
c)/a/d/(a-a*sec(d*x+c))^(1/2)+25/12*A*cos(d*x+c)*sin(d*x+c)/a/d/(a-a*sec(d*x+c))^(1/2)+4/3*A*cos(d*x+c)^2*sin(
d*x+c)/a/d/(a-a*sec(d*x+c))^(1/2)

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Rubi [A]
time = 0.46, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4105, 4107, 4005, 3859, 209, 3880} \begin {gather*} \frac {85 A \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{3/2} d}-\frac {15 A \text {ArcTan}\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 {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {25 A \sin (c+d x) \cos (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(85*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*a^(3/2)*d) - (15*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]^2*Sin[c + d*x])/(d*(a - a*Sec[
c + d*x])^(3/2)) + (35*A*Sin[c + d*x])/(8*a*d*Sqrt[a - a*Sec[c + d*x]]) + (25*A*Cos[c + d*x]*Sin[c + d*x])/(12
*a*d*Sqrt[a - a*Sec[c + d*x]]) + (4*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*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*} \int \frac {\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^3(c+d x) (8 a A+7 a A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x) \left (-25 a^2 A-20 a^2 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{6 a^3}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}+\frac {\int \frac {\cos (c+d x) \left (\frac {105 a^3 A}{2}+\frac {75}{2} a^3 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{12 a^4}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {-\frac {255 a^4 A}{4}-\frac {105}{4} a^4 A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{12 a^5}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}+\frac {(85 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{16 a^2}+\frac {(15 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{2 a}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}+\frac {(85 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 d}-\frac {(15 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 )}{a d}\\ &=\frac {85 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{3/2} d}-\frac {15 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 ^2(c+d x) \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {35 A \sin (c+d x)}{8 a d \sqrt {a-a \sec (c+d x)}}+\frac {25 A \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a-a \sec (c+d x)}}+\frac {4 A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a-a \sec (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.87, size = 314, normalized size = 1.33 \begin {gather*} \frac {A \left (-\frac {5 \sqrt {2} 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)}} \left (-17 i d x+17 \sinh ^{-1}\left (e^{i (c+d x)}\right )+24 \sqrt {2} \log \left (1-e^{i (c+d x)}\right )+17 \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )-24 \sqrt {2} \log \left (1+e^{i (c+d x)}+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right )}{d}+\frac {\left (-61 \cos \left (\frac {1}{2} (c+d x)\right )-120 \cos \left (\frac {3}{2} (c+d x)\right )+72 \cos \left (\frac {5}{2} (c+d x)\right )+11 \cos \left (\frac {7}{2} (c+d x)\right )+2 \cos \left (\frac {9}{2} (c+d x)\right )\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)}}{6 d}\right ) \sec ^{\frac {3}{2}}(c+d x) \sin ^3\left (\frac {1}{2} (c+d x)\right )}{8 (a-a \sec (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(A*((-5*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*((-17*I)*d*x + 1
7*ArcSinh[E^(I*(c + d*x))] + 24*Sqrt[2]*Log[1 - E^(I*(c + d*x))] + 17*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] -
24*Sqrt[2]*Log[1 + E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*E^((I/2)*(c + d*x))) + ((-61
*Cos[(c + d*x)/2] - 120*Cos[(3*(c + d*x))/2] + 72*Cos[(5*(c + d*x))/2] + 11*Cos[(7*(c + d*x))/2] + 2*Cos[(9*(c
+ d*x))/2])*Csc[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]])/(6*d))*Sec[c + d*x]^(3/2)*Sin[(c + d*x)/2]^3)/(8*(a - a*Se
c[c + d*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1103\) vs. \(2(205)=410\).
time = 8.85, size = 1104, normalized size = 4.68

method	result	size
default	\(\text {Expression too large to display}\)	\(1104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168*A/d*(-1+cos(d*x+c))^5*(-2520*2^(1/2)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-735*(-2*cos(d*x+c)/(
1+cos(d*x+c)))^(1/2)*2^(1/2)-3570*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-720*2^(1/2)*cos(d*x
+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(7/2)+1008*2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(5/2)+720*
2^(1/2)*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(7/2)-1680*2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)
))^(3/2)+5040*2^(1/2)*cos(d*x+c)^3*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+1680*(-2*cos(d*x+c)/(1+cos(d
*x+c)))^(3/2)*2^(1/2)*cos(d*x+c)+1130*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)*cos(d*x+c)^3+952*(-2*cos(d*
x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)*cos(d*x+c)^2-875*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)*cos(d*x+c)-50
40*2^(1/2)*cos(d*x+c)*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+56*2^(1/2)*cos(d*x+c)^7*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)+7140*cos(d*x+c)^3*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+360*2^(1/2)*(-2
*cos(d*x+c)/(1+cos(d*x+c)))^(7/2)+1225*2^(1/2)*cos(d*x+c)^5*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-1008*2^(1/2)*
cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(5/2)-2103*2^(1/2)*cos(d*x+c)^4*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)
-504*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(5/2)+350*2^(1/2)*cos(d*x+c)^6*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2
)-168*2^(1/2)*cos(d*x+c)^4*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)-672*2^(1/2)*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(9/2)-360*2^(1/2)*cos(d*x+c)^4*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(7/2)-1008*2^(1/2)*cos(d*x+c)^2*(-2*co
s(d*x+c)/(1+cos(d*x+c)))^(9/2)-672*2^(1/2)*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)+504*2^(1/2)*cos(d*x
+c)^4*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(5/2)-840*2^(1/2)*cos(d*x+c)^4*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(3/2)+2520*
2^(1/2)*cos(d*x+c)^4*arctan(1/(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-168*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))
^(9/2)+3570*cos(d*x+c)^4*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))+840*(-2*cos(d*x+c)/(1+cos(d*
x+c)))^(3/2)*2^(1/2)-7140*cos(d*x+c)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)))/(-2*cos(d*x+c)/
(1+cos(d*x+c)))^(3/2)/(a*(-1+cos(d*x+c))/cos(d*x+c))^(3/2)/sin(d*x+c)^9*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]
time = 1.52, size = 572, normalized size = 2.42 \begin {gather*} \left [-\frac {180 \, \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 ) + 255 \, {\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 (8 \, A \cos \left (d x + c\right )^{5} + 26 \, A \cos \left (d x + c\right )^{4} + 73 \, A \cos \left (d x + c\right )^{3} - 50 \, A \cos \left (d x + c\right )^{2} - 105 \, 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^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {180 \, \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 ) - 255 \, {\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 (8 \, A \cos \left (d x + c\right )^{5} + 26 \, A \cos \left (d x + c\right )^{4} + 73 \, A \cos \left (d x + c\right )^{3} - 50 \, A \cos \left (d x + c\right )^{2} - 105 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{24 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(180*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)))*
sin(d*x + c) + 255*(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*(8*A*cos(d*x
+ c)^5 + 26*A*cos(d*x + c)^4 + 73*A*cos(d*x + c)^3 - 50*A*cos(d*x + c)^2 - 105*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)), 1/24*(180*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) - 255*(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) - (8*A*cos(d*x + c)^5 + 26*A*cos(d*x + c)^4 + 73*A*cos(d*x + c)^3 - 50*A*cos(d*x + c)
^2 - 105*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} A \left (\int \frac {\cos ^{3}{\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 ^{3}{\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(Integral(cos(c + d*x)**3/(-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)**3*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 = 1.16, size = 208, normalized size = 0.88 \begin {gather*} \frac {\frac {180 \, \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 {3}{2}}} - \frac {255 \, 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 {3}{2}}} - \frac {12 \, \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {\sqrt {2} {\left (63 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {5}{2}} A + 272 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A a + 324 \, \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}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(180*sqrt(2)*A*arctan(sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(3/2) - 255*A*arctan(1/2*sqrt(2)*sqrt
(a*tan(1/2*d*x + 1/2*c)^2 - a)/sqrt(a))/a^(3/2) - 12*sqrt(2)*sqrt(a*tan(1/2*d*x + 1/2*c)^2 - a)*A/(a^2*tan(1/2
*d*x + 1/2*c)^2) - sqrt(2)*(63*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(5/2)*A + 272*(a*tan(1/2*d*x + 1/2*c)^2 - a)^(3/
2)*A*a + 324*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))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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