3.3.13 \(\int \frac {A+B \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx\) [213]

Optimal. Leaf size=250 \[ -\frac {3 (121 A-21 B) \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \]

[Out]

-3/128*(121*A-21*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))/a^(7/2)/d*2
^(1/2)-1/6*(A-B)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(7/2)/cos(d*x+c)^(1/2)-1/48*(19*A-7*B)*sin(d*x+c)/a/d/(a+a*cos(
d*x+c))^(5/2)/cos(d*x+c)^(1/2)-1/192*(199*A-43*B)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(3/2)/cos(d*x+c)^(1/2)+1/1
92*(691*A-103*B)*sin(d*x+c)/a^3/d/cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2)

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Rubi [A]
time = 0.51, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3057, 3063, 12, 2861, 211} \begin {gather*} -\frac {3 (121 A-21 B) \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 )}{64 \sqrt {2} a^{7/2} d}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]

[Out]

(-3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*S
qrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)
*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sin[c + d*x])/(192*a^2
*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x
]]*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 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)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}+\frac {\int \frac {\frac {1}{2} a (13 A-B)-3 a (A-B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}+\frac {\int \frac {\frac {3}{4} a^2 (41 A-5 B)-a^2 (19 A-7 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\frac {1}{8} a^3 (691 A-103 B)-\frac {1}{4} a^3 (199 A-43 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {\int -\frac {9 a^4 (121 A-21 B)}{16 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{24 a^7}\\ &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}-\frac {(3 (121 A-21 B)) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {(3 (121 A-21 B)) \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 )}{64 a^2 d}\\ &=-\frac {3 (121 A-21 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 )}{64 \sqrt {2} a^{7/2} d}-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.77, size = 240, normalized size = 0.96 \begin {gather*} \frac {\cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {9 i (121 A-21 B) e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+\frac {(5284 A-532 B+9 (941 A-121 B) \cos (c+d x)+4 (937 A-133 B) \cos (2 (c+d x))+691 A \cos (3 (c+d x))-103 B \cos (3 (c+d x))) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{16 \sqrt {\cos (c+d x)}}\right )}{24 d (a (1+\cos (c+d x)))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]

[Out]

(Cos[(c + d*x)/2]^7*(((-9*I)*(121*A - 21*B)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))
]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] + ((52
84*A - 532*B + 9*(941*A - 121*B)*Cos[c + d*x] + 4*(937*A - 133*B)*Cos[2*(c + d*x)] + 691*A*Cos[3*(c + d*x)] -
103*B*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/(16*Sqrt[Cos[c + d*x]])))/(24*d*(a*(1 + Cos[c + d
*x]))^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(580\) vs. \(2(213)=426\).
time = 0.38, size = 581, normalized size = 2.32

method	result	size
default	\(-\frac {\sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (-1+\cos \left (d x +c \right )\right )^{2} \left (-1089 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+189 B \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-3267 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+567 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-3267 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right )+567 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right )+1382 A \left (\cos ^{4}\left (d x +c \right )\right )-1089 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-206 B \left (\cos ^{4}\left (d x +c \right )\right )+189 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}\, \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+2366 A \left (\cos ^{3}\left (d x +c \right )\right )-326 B \left (\cos ^{3}\left (d x +c \right )\right )-550 A \left (\cos ^{2}\left (d x +c \right )\right )+142 B \left (\cos ^{2}\left (d x +c \right )\right )-2430 A \cos \left (d x +c \right )+390 B \cos \left (d x +c \right )-768 A \right )}{384 d \,a^{4} \sin \left (d x +c \right )^{5} \left (1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\)	\(581\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/384/d*(a*(1+cos(d*x+c)))^(1/2)*(-1+cos(d*x+c))^2*(-1089*A*cos(d*x+c)^3*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*2^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+189*B*cos(d*x+c)^3*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)*2^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-3267*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c
)^2*2^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+567*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^2
*2^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-3267*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*2^(1/2)*arcsin
((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)+567*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*2^(1/2)*arcsin((-1+
cos(d*x+c))/sin(d*x+c))*cos(d*x+c)+1382*A*cos(d*x+c)^4-1089*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*2^(
1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-206*B*cos(d*x+c)^4+189*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*
2^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+2366*A*cos(d*x+c)^3-326*B*cos(d*x+c)^3-550*A*cos(d*x+c)^2+142*B*cos
(d*x+c)^2-2430*A*cos(d*x+c)+390*B*cos(d*x+c)-768*A)/a^4/sin(d*x+c)^5/(1+cos(d*x+c))/cos(d*x+c)^(1/2)

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Maxima [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))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]
time = 0.42, size = 298, normalized size = 1.19 \begin {gather*} -\frac {9 \, \sqrt {2} {\left ({\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \, {\left ({\left (691 \, A - 103 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (937 \, A - 133 \, B\right )} \cos \left (d x + c\right )^{2} + 39 \, {\left (41 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + 384 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

-1/384*(9*sqrt(2)*((121*A - 21*B)*cos(d*x + c)^5 + 4*(121*A - 21*B)*cos(d*x + c)^4 + 6*(121*A - 21*B)*cos(d*x
+ c)^3 + 4*(121*A - 21*B)*cos(d*x + c)^2 + (121*A - 21*B)*cos(d*x + c))*sqrt(a)*arctan(1/2*sqrt(2)*sqrt(a*cos(
d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c))) - 2*((691*A - 103*B
)*cos(d*x + c)^3 + 2*(937*A - 133*B)*cos(d*x + c)^2 + 39*(41*A - 5*B)*cos(d*x + c) + 384*A)*sqrt(a*cos(d*x + c
) + a)*sqrt(cos(d*x + c))*sin(d*x + c))/(a^4*d*cos(d*x + c)^5 + 4*a^4*d*cos(d*x + c)^4 + 6*a^4*d*cos(d*x + c)^
3 + 4*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c))

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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))/cos(d*x+c)**(3/2)/(a+a*cos(d*x+c))**(7/2),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))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),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 {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cos(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(7/2)),x)

[Out]

int((A + B*cos(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(7/2)), x)

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