3.6.29 \(\int \frac {(A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx\) [529]

Optimal. Leaf size=317 \[ \frac {(19 A-15 B) \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{2 \sqrt {2} a^{3/2} d}-\frac {(1201 A-1029 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}} \]

[Out]

-1/2*(A-B)*sec(d*x+c)^(7/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(3/2)+1/210*(397*A-273*B)*sec(d*x+c)^(3/2)*sin(d*x+c
)/a/d/(a+a*cos(d*x+c))^(1/2)-1/70*(67*A-63*B)*sec(d*x+c)^(5/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)+1/14*(11*
A-7*B)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(1/2)+1/4*(19*A-15*B)*arctan(1/2*sin(d*x+c)*a^(1/2)*2^
(1/2)/cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^(3/2)/d*2^(1/2)-1/210*(1201
*A-1029*B)*sin(d*x+c)*sec(d*x+c)^(1/2)/a/d/(a+a*cos(d*x+c))^(1/2)

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Rubi [A]
time = 0.71, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3040, 3057, 3063, 12, 2861, 211} \begin {gather*} \frac {(19 A-15 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(11 A-7 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{14 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A-B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(67 A-63 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{70 a d \sqrt {a \cos (c+d x)+a}}+\frac {(397 A-273 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{210 a d \sqrt {a \cos (c+d x)+a}}-\frac {(1201 A-1029 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{210 a d \sqrt {a \cos (c+d x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]

[Out]

((19*A - 15*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c
+ d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*
a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c +
d*x]]) - ((67*A - 63*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c +
d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14
*a*d*Sqrt[a + a*Cos[c + d*x]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

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

Rule 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c +
d*Sin[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0]

Rule 3040

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[(a + b*Sin[e + f*x])^m*((
c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && IntegerQ[n])

Rule 3057

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

Rule 3063

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

Rubi steps

\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx\\ &=-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a (11 A-7 B)-4 a (A-B) \cos (c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{4} a^2 (67 A-63 B)+\frac {3}{2} a^2 (11 A-7 B) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{7 a^3}\\ &=-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a^3 (397 A-273 B)-\frac {1}{2} a^3 (67 A-63 B) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=\frac {(397 A-273 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a^4 (1201 A-1029 B)+\frac {1}{8} a^4 (397 A-273 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{105 a^5}\\ &=-\frac {(1201 A-1029 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {105 a^5 (19 A-15 B)}{32 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{105 a^6}\\ &=-\frac {(1201 A-1029 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\left ((19 A-15 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac {(1201 A-1029 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}-\frac {\left ((19 A-15 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{2 d}\\ &=\frac {(19 A-15 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{2 \sqrt {2} a^{3/2} d}-\frac {(1201 A-1029 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}+\frac {(397 A-273 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a d \sqrt {a+a \cos (c+d x)}}-\frac {(67 A-63 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A-B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(11 A-7 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.12, size = 262, normalized size = 0.83 \begin {gather*} \frac {\cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (\frac {i (19 A-15 B) 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)}} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )}{d}-\frac {(2339 A-2751 B+24 (213 A-217 B) \cos (c+d x)+60 (67 A-63 B) \cos (2 (c+d x))+1608 A \cos (3 (c+d x))-1512 B \cos (3 (c+d x))+1201 A \cos (4 (c+d x))-1029 B \cos (4 (c+d x))) \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{840 d}\right )}{(a (1+\cos (c+d x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]

[Out]

(Cos[(c + d*x)/2]^3*((I*(19*A - 15*B)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d
*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/(d*E^((I/2)*(c + d*x))) - ((2339
*A - 2751*B + 24*(213*A - 217*B)*Cos[c + d*x] + 60*(67*A - 63*B)*Cos[2*(c + d*x)] + 1608*A*Cos[3*(c + d*x)] -
1512*B*Cos[3*(c + d*x)] + 1201*A*Cos[4*(c + d*x)] - 1029*B*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]*Sec[c + d*x]^(7/
2)*Tan[(c + d*x)/2])/(840*d)))/(a*(1 + Cos[c + d*x]))^(3/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs. \(2(270)=540\).
time = 0.52, size = 731, normalized size = 2.31

method	result	size
default	\(\frac {\left (1995 A \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )-1575 B \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+7980 A \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-6300 B \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+11970 A \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-9450 B \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+7980 A \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-6300 B \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )+1995 A \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right )-1575 B \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right )-1201 A \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {2}+1029 B \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {2}+397 A \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}-273 B \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}+1000 A \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}-840 B \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}-232 A \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+168 B \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+96 A \cos \left (d x +c \right ) \sqrt {2}-84 B \cos \left (d x +c \right ) \sqrt {2}-60 A \sqrt {2}\right ) \cos \left (d x +c \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sin ^{5}\left (d x +c \right )\right ) \sqrt {2}}{420 d \left (-1+\cos \left (d x +c \right )\right )^{3} \left (1+\cos \left (d x +c \right )\right )^{4} a^{2}}\)	\(731\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/420/d*(1995*A*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^4*sin(d*x+c)-1
575*B*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^4*sin(d*x+c)+7980*A*(cos
(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)-6300*B*(cos(d*x+c)/(1
+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)+11970*A*(cos(d*x+c)/(1+cos(d*x+
c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-9450*B*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)
*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)+7980*A*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-
1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)*sin(d*x+c)-6300*B*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c)
)/sin(d*x+c))*cos(d*x+c)*sin(d*x+c)+1995*A*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c)
)*sin(d*x+c)-1575*B*(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*sin(d*x+c)-1201*A*cos
(d*x+c)^5*2^(1/2)+1029*B*cos(d*x+c)^5*2^(1/2)+397*A*cos(d*x+c)^4*2^(1/2)-273*B*cos(d*x+c)^4*2^(1/2)+1000*A*cos
(d*x+c)^3*2^(1/2)-840*B*cos(d*x+c)^3*2^(1/2)-232*A*cos(d*x+c)^2*2^(1/2)+168*B*cos(d*x+c)^2*2^(1/2)+96*A*cos(d*
x+c)*2^(1/2)-84*B*cos(d*x+c)*2^(1/2)-60*A*2^(1/2))*cos(d*x+c)*(1/cos(d*x+c))^(9/2)*(a*(1+cos(d*x+c)))^(1/2)*si
n(d*x+c)^5/(-1+cos(d*x+c))^3/(1+cos(d*x+c))^4*2^(1/2)/a^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((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)*sec(d*x + c)^(9/2)/(a*cos(d*x + c) + a)^(3/2), x)

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Fricas [A]
time = 0.40, size = 237, normalized size = 0.75 \begin {gather*} -\frac {105 \, \sqrt {2} {\left ({\left (19 \, A - 15 \, B\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (19 \, A - 15 \, B\right )} \cos \left (d x + c\right )^{4} + {\left (19 \, A - 15 \, B\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) + \frac {2 \, {\left ({\left (1201 \, A - 1029 \, B\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (67 \, A - 63 \, B\right )} \cos \left (d x + c\right )^{3} - 28 \, {\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{2} + 12 \, {\left (3 \, A - 7 \, B\right )} \cos \left (d x + c\right ) - 60 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{420 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/420*(105*sqrt(2)*((19*A - 15*B)*cos(d*x + c)^5 + 2*(19*A - 15*B)*cos(d*x + c)^4 + (19*A - 15*B)*cos(d*x + c
)^3)*sqrt(a)*arctan(sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) + 2*((1201*A -
1029*B)*cos(d*x + c)^4 + 12*(67*A - 63*B)*cos(d*x + c)^3 - 28*(7*A - 3*B)*cos(d*x + c)^2 + 12*(3*A - 7*B)*cos
(d*x + c) - 60*A)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^2*d*cos(d*x + c)^5 + 2*a^2*d*co
s(d*x + c)^4 + a^2*d*cos(d*x + c)^3)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)**(9/2)/(a+a*cos(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)/(a+a*cos(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+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(((A + B*cos(c + d*x))*(1/cos(c + d*x))^(9/2))/(a + a*cos(c + d*x))^(3/2),x)

[Out]

int(((A + B*cos(c + d*x))*(1/cos(c + d*x))^(9/2))/(a + a*cos(c + d*x))^(3/2), x)

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