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

Optimal. Leaf size=259 \[ \frac{(245 A+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A+19 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}} \]

[Out]

((11*A + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A
+ C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A + 799*C)*Sin[c + d*x])/(105*a*d*
Sqrt[a + a*Cos[c + d*x]]) - ((35*A + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((
7*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Cos[c
+ d*x]]*Sin[c + d*x])/(210*a^2*d)

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Rubi [A]  time = 0.791716, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3042, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac{(245 A+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(11 A+19 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11*A + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A
+ C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A + 799*C)*Sin[c + d*x])/(105*a*d*
Sqrt[a + a*Cos[c + d*x]]) - ((35*A + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((
7*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Cos[c
+ d*x]]*Sin[c + d*x])/(210*a^2*d)

Rule 3042

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

Rule 2983

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*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(
m + n + 1)), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*b*c*(
m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*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] && GtQ[n, 0] && (I
ntegerQ[n] || EqQ[m + 1/2, 0])

Rule 2968

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

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2751

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

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\cos ^3(c+d x) \left (-2 a (A+2 C)+\frac{1}{2} a (7 A+11 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{\cos ^2(c+d x) \left (\frac{3}{2} a^2 (7 A+11 C)-\frac{1}{4} a^2 (35 A+67 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{7 a^3}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(35 A+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{\cos (c+d x) \left (-\frac{1}{2} a^3 (35 A+67 C)+\frac{1}{8} a^3 (245 A+397 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(35 A+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{-\frac{1}{2} a^3 (35 A+67 C) \cos (c+d x)+\frac{1}{8} a^3 (245 A+397 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(35 A+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac{4 \int \frac{\frac{1}{16} a^4 (245 A+397 C)-\frac{1}{8} a^4 (455 A+799 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{105 a^5}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}-\frac{(35 A+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac{(11 A+19 C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}-\frac{(35 A+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}-\frac{(11 A+19 C) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{2 a d}\\ &=\frac{(11 A+19 C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a+a \cos (c+d x)}}-\frac{(35 A+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt{a+a \cos (c+d x)}}+\frac{(7 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt{a+a \cos (c+d x)}}+\frac{(245 A+397 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.998765, size = 157, normalized size = 0.61 \[ \frac{\frac{1}{2} \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^3\left (\frac{1}{2} (c+d x)\right ) (6 (140 A+277 C) \cos (c+d x)-4 (35 A+64 C) \cos (2 (c+d x))+1190 A+18 C \cos (3 (c+d x))-15 C \cos (4 (c+d x))+2161 C)-105 (11 A+19 C) \cos ^5\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{105 d \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )-1\right ) (a (\cos (c+d x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-105*(11*A + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (Cos[(c + d*x)/2]^3*(1190*A + 2161*C + 6*(1
40*A + 277*C)*Cos[c + d*x] - 4*(35*A + 64*C)*Cos[2*(c + d*x)] + 18*C*Cos[3*(c + d*x)] - 15*C*Cos[4*(c + d*x)])
*Sin[(c + d*x)/2])/2)/(105*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))

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Maple [A]  time = 0.071, size = 442, normalized size = 1.7 \begin{align*}{\frac{1}{420\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 960\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-1632\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+112\,\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( 5\,A+16\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-35\,\sqrt{2} \left ( 33\,A\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a-8\,A\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+57\,C\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a-16\,C\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1155\,\sqrt{2}\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) aA+1995\,\sqrt{2}\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) aC-945\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}-1785\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{a}^{-{\frac{5}{2}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(960*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*sin(1/2*d*x+1/2*c)^
8-1632*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*sin(1/2*d*x+1/2*c)^6+112*2^(1/2)*(a*sin(1/2*d*x+1/2*c)
^2)^(1/2)*a^(1/2)*(5*A+16*C)*sin(1/2*d*x+1/2*c)^4-35*2^(1/2)*(33*A*ln(4/cos(1/2*d*x+1/2*c)*(a^(1/2)*(a*sin(1/2
*d*x+1/2*c)^2)^(1/2)+a))*a-8*A*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+57*C*ln(4/cos(1/2*d*x+1/2*c)*(a^(1/2)*(a
*sin(1/2*d*x+1/2*c)^2)^(1/2)+a))*a-16*C*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2))*sin(1/2*d*x+1/2*c)^2+1155*2^(1
/2)*ln(4/cos(1/2*d*x+1/2*c)*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a))*a*A+1995*2^(1/2)*ln(4/cos(1/2*d*x+1/2*
c)*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a))*a*C-945*A*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)-1785*C
*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2))/cos(1/2*d*x+1/2*c)/a^(5/2)/sin(1/2*d*x+1/2*c)/(a*cos(1/2*d*x+
1/2*c)^2)^(1/2)/d

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 1.7848, size = 637, normalized size = 2.46 \begin{align*} \frac{105 \, \sqrt{2}{\left ({\left (11 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (11 \, A + 19 \, C\right )} \cos \left (d x + c\right ) + 11 \, A + 19 \, C\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \,{\left (60 \, C \cos \left (d x + c\right )^{4} - 36 \, C \cos \left (d x + c\right )^{3} + 28 \,{\left (5 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{2} - 12 \,{\left (35 \, A + 67 \, C\right )} \cos \left (d x + c\right ) - 665 \, A - 1201 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{840 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(105*sqrt(2)*((11*A + 19*C)*cos(d*x + c)^2 + 2*(11*A + 19*C)*cos(d*x + c) + 11*A + 19*C)*sqrt(a)*log(-(a
*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(a)*sin(d*x + c) - 2*a*cos(d*x + c) - 3*a)/(cos(d*x +
c)^2 + 2*cos(d*x + c) + 1)) + 4*(60*C*cos(d*x + c)^4 - 36*C*cos(d*x + c)^3 + 28*(5*A + 7*C)*cos(d*x + c)^2 -
12*(35*A + 67*C)*cos(d*x + c) - 665*A - 1201*C)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 +
2*a^2*d*cos(d*x + c) + a^2*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.03828, size = 343, normalized size = 1.32 \begin{align*} -\frac{\frac{105 \,{\left (11 \, \sqrt{2} A + 19 \, \sqrt{2} C\right )} \log \left ({\left | -\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} \right |}\right )}{a^{\frac{3}{2}}} + \frac{{\left ({\left ({\left ({\left (\frac{105 \,{\left (\sqrt{2} A a^{5} + \sqrt{2} C a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{3}} + \frac{4 \,{\left (455 \, \sqrt{2} A a^{5} + 877 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{14 \,{\left (305 \, \sqrt{2} A a^{5} + 517 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{140 \,{\left (25 \, \sqrt{2} A a^{5} + 47 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{105 \,{\left (9 \, \sqrt{2} A a^{5} + 17 \, \sqrt{2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{7}{2}}}}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(105*(11*sqrt(2)*A + 19*sqrt(2)*C)*log(abs(-sqrt(a)*tan(1/2*d*x + 1/2*c) + sqrt(a*tan(1/2*d*x + 1/2*c)^
2 + a)))/a^(3/2) + ((((105*(sqrt(2)*A*a^5 + sqrt(2)*C*a^5)*tan(1/2*d*x + 1/2*c)^2/a^3 + 4*(455*sqrt(2)*A*a^5 +
877*sqrt(2)*C*a^5)/a^3)*tan(1/2*d*x + 1/2*c)^2 + 14*(305*sqrt(2)*A*a^5 + 517*sqrt(2)*C*a^5)/a^3)*tan(1/2*d*x
+ 1/2*c)^2 + 140*(25*sqrt(2)*A*a^5 + 47*sqrt(2)*C*a^5)/a^3)*tan(1/2*d*x + 1/2*c)^2 + 105*(9*sqrt(2)*A*a^5 + 17
*sqrt(2)*C*a^5)/a^3)*tan(1/2*d*x + 1/2*c)/(a*tan(1/2*d*x + 1/2*c)^2 + a)^(7/2))/d