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

Optimal. Leaf size=254 \[ \frac{2 (21 A-3 B+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A-93 B+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 (9 B-C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}} \]

[Out]

-((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(
147*A - 111*B + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Cos[c + d*x]^2*
Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(9*B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Co
s[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt
[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)

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Rubi [A]  time = 0.837927, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3045, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac{2 (21 A-3 B+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A-93 B+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 (9 B-C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(
147*A - 111*B + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Cos[c + d*x]^2*
Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(9*B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Co
s[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt
[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)

Rule 3045

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

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+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx &=\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \int \frac{\cos ^3(c+d x) \left (\frac{1}{2} a (9 A+8 C)+\frac{1}{2} a (9 B-C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{9 a}\\ &=\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{4 \int \frac{\cos ^2(c+d x) \left (\frac{3}{2} a^2 (9 B-C)+\frac{3}{4} a^2 (21 A-3 B+19 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{63 a^2}\\ &=\frac{2 (21 A-3 B+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{8 \int \frac{\cos (c+d x) \left (\frac{3}{2} a^3 (21 A-3 B+19 C)-\frac{3}{8} a^3 (21 A-93 B+29 C) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=\frac{2 (21 A-3 B+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{8 \int \frac{\frac{3}{2} a^3 (21 A-3 B+19 C) \cos (c+d x)-\frac{3}{8} a^3 (21 A-93 B+29 C) \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=\frac{2 (21 A-3 B+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (21 A-93 B+29 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 a d}+\frac{16 \int \frac{-\frac{3}{16} a^4 (21 A-93 B+29 C)+\frac{3}{8} a^4 (147 A-111 B+143 C) \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx}{945 a^4}\\ &=\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (21 A-3 B+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (21 A-93 B+29 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 a d}+(-A+B-C) \int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (21 A-3 B+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (21 A-93 B+29 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 a d}+\frac{(2 (A-B+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 )}{d}\\ &=-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \cos (c+d x)}}\right )}{\sqrt{a} d}+\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (21 A-3 B+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (9 B-C) \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (21 A-93 B+29 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{315 a d}\\ \end{align*}

Mathematica [A]  time = 0.78595, size = 144, normalized size = 0.57 \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \left (2 \sin \left (\frac{1}{2} (c+d x)\right ) (-2 (84 A-507 B+131 C) \cos (c+d x)+4 (63 A-9 B+92 C) \cos (2 (c+d x))+2436 A+90 B \cos (3 (c+d x))-1068 B-10 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2389 C)-2520 (A-B+C) \tanh ^{-1}\left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1260 d \sqrt{a (\cos (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]*(-2520*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(2436*A - 1068*B + 2389*C - 2*(84*A - 507*B
+ 131*C)*Cos[c + d*x] + 4*(63*A - 9*B + 92*C)*Cos[2*(c + d*x)] + 90*B*Cos[3*(c + d*x)] - 10*C*Cos[3*(c + d*x)
] + 35*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(1260*d*Sqrt[a*(1 + Cos[c + d*x])])

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Maple [A]  time = 0.15, size = 392, normalized size = 1.5 \begin{align*}{\frac{1}{315\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 1120\,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}-720\,\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( B+3\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+504\,\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( A+2\,B+4\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-420\,\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \left ( A+2\,B+2\,C \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-315\,\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+315\,\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 ) aB-315\,\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+630\,A\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a}+630\,C\sqrt{2}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{a} \right ){a}^{-{\frac{3}{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+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x)

[Out]

1/315*cos(1/2*d*x+1/2*c)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(1120*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)^8-720*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*(B+3*C)*sin(1/2*d*x+1/2*c)^6+504*2^(1
/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2)*(A+2*B+4*C)*sin(1/2*d*x+1/2*c)^4-420*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^
2)^(1/2)*a^(1/2)*(A+2*B+2*C)*sin(1/2*d*x+1/2*c)^2-315*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+315*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*B-31
5*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+630*A*2^(1/2)*(a*sin(1/2*d*x
+1/2*c)^2)^(1/2)*a^(1/2)+630*C*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^(1/2))/a^(3/2)/sin(1/2*d*x+1/2*c)/(a*c
os(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+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.99137, size = 582, normalized size = 2.29 \begin{align*} \frac{4 \,{\left (35 \, C \cos \left (d x + c\right )^{4} + 5 \,{\left (9 \, B - C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (21 \, A - 3 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (21 \, A - 93 \, B + 29 \, C\right )} \cos \left (d x + c\right ) + 273 \, A - 129 \, B + 257 \, C\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac{315 \, \sqrt{2}{\left ({\left (A - B + C\right )} a \cos \left (d x + c\right ) +{\left (A - B + C\right )} a\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} + \frac{2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt{a}}}{630 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/630*(4*(35*C*cos(d*x + c)^4 + 5*(9*B - C)*cos(d*x + c)^3 + 3*(21*A - 3*B + 19*C)*cos(d*x + c)^2 - (21*A - 93
*B + 29*C)*cos(d*x + c) + 273*A - 129*B + 257*C)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c) + 315*sqrt(2)*((A - B +
C)*a*cos(d*x + c) + (A - B + C)*a)*log(-(cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqr
t(a) - 2*cos(d*x + c) - 3)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))/sqrt(a))/(a*d*cos(d*x + c) + a*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+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 2.0079, size = 363, normalized size = 1.43 \begin{align*} \frac{\frac{315 \,{\left (\sqrt{2} A - \sqrt{2} B + \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 )}{\sqrt{a}} + \frac{2 \,{\left (315 \, \sqrt{2} A a^{4} + 315 \, \sqrt{2} C a^{4} +{\left (1050 \, \sqrt{2} A a^{4} - 420 \, \sqrt{2} B a^{4} + 840 \, \sqrt{2} C a^{4} +{\left (1512 \, \sqrt{2} A a^{4} - 756 \, \sqrt{2} B a^{4} + 1638 \, \sqrt{2} C a^{4} +{\left (1134 \, \sqrt{2} A a^{4} - 612 \, \sqrt{2} B a^{4} + 936 \, \sqrt{2} C a^{4} +{\left (357 \, \sqrt{2} A a^{4} - 276 \, \sqrt{2} B a^{4} + 383 \, \sqrt{2} C a^{4}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\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{9}{2}}}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/315*(315*(sqrt(2)*A - sqrt(2)*B + 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)))/sqrt(a) + 2*(315*sqrt(2)*A*a^4 + 315*sqrt(2)*C*a^4 + (1050*sqrt(2)*A*a^4 - 420*sqrt(2)*B*a^4 + 8
40*sqrt(2)*C*a^4 + (1512*sqrt(2)*A*a^4 - 756*sqrt(2)*B*a^4 + 1638*sqrt(2)*C*a^4 + (1134*sqrt(2)*A*a^4 - 612*sq
rt(2)*B*a^4 + 936*sqrt(2)*C*a^4 + (357*sqrt(2)*A*a^4 - 276*sqrt(2)*B*a^4 + 383*sqrt(2)*C*a^4)*tan(1/2*d*x + 1/
2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)/(a*tan(1/
2*d*x + 1/2*c)^2 + a)^(9/2))/d