Integrand size = 35, antiderivative size = 274 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {a^{5/2} (326 A+283 B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^3 (326 A+283 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (326 A+283 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (170 A+157 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (10 A+13 B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {a B \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \] Output:
1/128*a^(5/2)*(326*A+283*B)*arcsin(a^(1/2)*sin(d*x+c)/(a+a*cos(d*x+c))^(1/ 2))/d+1/128*a^3*(326*A+283*B)*cos(d*x+c)^(1/2)*sin(d*x+c)/d/(a+a*cos(d*x+c ))^(1/2)+1/192*a^3*(326*A+283*B)*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d* x+c))^(1/2)+1/240*a^3*(170*A+157*B)*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos (d*x+c))^(1/2)+1/40*a^2*(10*A+13*B)*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^(1/2 )*sin(d*x+c)/d+1/5*a*B*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/ d
Time = 2.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.58 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (15 \sqrt {2} (326 A+283 B) \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \sqrt {\cos (c+d x)} (5810 A+5521 B+(3620 A+3874 B) \cos (c+d x)+4 (230 A+331 B) \cos (2 (c+d x))+120 A \cos (3 (c+d x))+348 B \cos (3 (c+d x))+48 B \cos (4 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3840 d} \] Input:
Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x ]),x]
Output:
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(326*A + 283* B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(5810*A + 5521* B + (3620*A + 3874*B)*Cos[c + d*x] + 4*(230*A + 331*B)*Cos[2*(c + d*x)] + 120*A*Cos[3*(c + d*x)] + 348*B*Cos[3*(c + d*x)] + 48*B*Cos[4*(c + d*x)])*S in[(c + d*x)/2]))/(3840*d)
Time = 1.44 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3455, 27, 3042, 3455, 27, 3042, 3460, 3042, 3249, 3042, 3249, 3042, 3253, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2} (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2} (5 a (2 A+B)+a (10 A+13 B) \cos (c+d x))dx+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2} (5 a (2 A+B)+a (10 A+13 B) \cos (c+d x))dx+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (5 a (2 A+B)+a (10 A+13 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a} \left (5 (26 A+21 B) a^2+(170 A+157 B) \cos (c+d x) a^2\right )dx+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a} \left (5 (26 A+21 B) a^2+(170 A+157 B) \cos (c+d x) a^2\right )dx+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (5 (26 A+21 B) a^2+(170 A+157 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \left (\frac {3}{4} \int \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}dx+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \left (\frac {3}{4} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{6} a^2 (326 A+283 B) \left (\frac {3}{4} \left (\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{10} \left (\frac {a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}+\frac {1}{8} \left (\frac {a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {5}{6} a^2 (326 A+283 B) \left (\frac {3}{4} \left (\frac {\sqrt {a} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )\right )\right )+\frac {a B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
Input:
Int[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]
Output:
(a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + ( (a^2*(10*A + 13*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x ])/(4*d) + ((a^3*(170*A + 157*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqr t[a + a*Cos[c + d*x]]) + (5*a^2*(326*A + 283*B)*((a*Cos[c + d*x]^(3/2)*Sin [c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]) + (3*((Sqrt[a]*ArcSin[(Sqrt[a]*S in[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])))/4))/6)/8)/10
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), 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] && GtQ[n, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-2/f Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Co s[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && E qQ[a^2 - b^2, 0] && EqQ[d, a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*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] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, 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[n, -1]
Time = 17.79 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(\frac {\left (4890 A \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+4245 B \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\left (480 \cos \left (d x +c \right )^{3}+1840 \cos \left (d x +c \right )^{2}+3260 \cos \left (d x +c \right )+4890\right ) \sin \left (d x +c \right ) A \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (384 \cos \left (d x +c \right )^{4}+1392 \cos \left (d x +c \right )^{3}+2264 \cos \left (d x +c \right )^{2}+2830 \cos \left (d x +c \right )+4245\right ) \sin \left (d x +c \right ) B \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, a^{2}}{1920 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\) | \(249\) |
| parts | \(\frac {A \left (489 \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\left (48 \cos \left (d x +c \right )^{3}+184 \cos \left (d x +c \right )^{2}+326 \cos \left (d x +c \right )+489\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, a^{2}}{192 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}+\frac {B \left (4245 \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\left (384 \cos \left (d x +c \right )^{4}+1392 \cos \left (d x +c \right )^{3}+2264 \cos \left (d x +c \right )^{2}+2830 \cos \left (d x +c \right )+4245\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, a^{2}}{1920 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\) | \(312\) |
Input:
int(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x,method=_RET URNVERBOSE)
Output:
1/1920/d*(4890*A*arctan(tan(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+4245 *B*arctan(tan(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+(480*cos(d*x+c)^3+ 1840*cos(d*x+c)^2+3260*cos(d*x+c)+4890)*sin(d*x+c)*A*(cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)+(384*cos(d*x+c)^4+1392*cos(d*x+c)^3+2264*cos(d*x+c)^2+2830*co s(d*x+c)+4245)*sin(d*x+c)*B*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^ (1/2)*(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/(cos(d*x+c)+1)/(cos(d*x+c)/(cos(d*x+c )+1))^(1/2)*2^(1/2)*a^2
Time = 0.17 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.78 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {{\left (384 \, B a^{2} \cos \left (d x + c\right )^{4} + 48 \, {\left (10 \, A + 29 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (230 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 10 \, {\left (326 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right ) + 15 \, {\left (326 \, A + 283 \, B\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 15 \, {\left ({\left (326 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right ) + {\left (326 \, A + 283 \, B\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )}\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x, algo rithm="fricas")
Output:
1/1920*((384*B*a^2*cos(d*x + c)^4 + 48*(10*A + 29*B)*a^2*cos(d*x + c)^3 + 8*(230*A + 283*B)*a^2*cos(d*x + c)^2 + 10*(326*A + 283*B)*a^2*cos(d*x + c) + 15*(326*A + 283*B)*a^2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin (d*x + c) + 15*((326*A + 283*B)*a^2*cos(d*x + c) + (326*A + 283*B)*a^2)*sq rt(a)*arctan(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))))/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(3/2)*(a+a*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 10042 vs. \(2 (236) = 472\).
Time = 0.80 (sec) , antiderivative size = 10042, normalized size of antiderivative = 36.65 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x, algo rithm="maxima")
Output:
1/7680*((10*(cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c)))^2 + sin( 2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c)))^2 + 2*cos(2/5*arctan2(sin (5*d*x + 5*c), cos(5*d*x + 5*c))) + 1)^(3/4)*((135*a^2*sin(4/5*arctan2(sin (5*d*x + 5*c), cos(5*d*x + 5*c))) + 88*a^2*sin(3/5*arctan2(sin(5*d*x + 5*c ), cos(5*d*x + 5*c))) + 135*a^2*sin(1/5*arctan2(sin(5*d*x + 5*c), cos(5*d* x + 5*c))))*cos(3/2*arctan2(sin(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))), cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 1)) - (135* a^2*cos(4/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 88*a^2*cos(3/5* arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) - 135*a^2*cos(1/5*arctan2(sin (5*d*x + 5*c), cos(5*d*x + 5*c))) - 88*a^2)*sin(3/2*arctan2(sin(2/5*arctan 2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))), cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 1)))*sqrt(a) + 6*(cos(2/5*arctan2(sin(5*d*x + 5*c), c os(5*d*x + 5*c)))^2 + sin(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) ^2 + 2*cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 1)^(1/4)*(8* (a^2*cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c)))^2*sin(5*d*x + 5* c) + a^2*sin(5*d*x + 5*c)*sin(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5* c)))^2 + 2*a^2*cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c)))*sin(5* d*x + 5*c) + a^2*sin(5*d*x + 5*c))*cos(5/2*arctan2(sin(2/5*arctan2(sin(5*d *x + 5*c), cos(5*d*x + 5*c))), cos(2/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 1)) - 5*(35*a^2*sin(4/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x...
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x, algo rithm="giac")
Output:
integrate((B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^(3/ 2), x)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^{3/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int(cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2),x)
Output:
int(cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a +2 \left (\int \sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b +2 \left (\int \sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) b \right ) \] Input:
int(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x)
Output:
sqrt(a)*a**2*(int(sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x),x )*a + int(sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*b + int(sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a + 2*in t(sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*b + 2*int(s qrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a + int(sqrt(c os(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*b)