Integrand size = 35, antiderivative size = 245 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (63 A b^2+8 a^2 C+45 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 a C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d} \] Output:
2/15*b*(9*a^2*(5*A+3*C)+b^2*(9*A+7*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2 ))/d+2/21*a*(7*a^2*(3*A+C)+3*b^2*(7*A+5*C))*InverseJacobiAM(1/2*d*x+1/2*c, 2^(1/2))/d+2/63*a*(63*A*b^2+8*C*a^2+45*C*b^2)*cos(d*x+c)^(1/2)*sin(d*x+c)/ d+2/315*b*(24*a^2*C+7*b^2*(9*A+7*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+4/21*a* C*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+2/9*C*cos(d*x+c)^(1/2)* (a+b*cos(d*x+c))^3*sin(d*x+c)/d
Time = 3.79 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {84 \left (9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+60 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (7 b \left (36 A b^2+108 a^2 C+43 b^2 C\right ) \cos (c+d x)+5 \left (252 a A b^2+84 a^3 C+234 a b^2 C+54 a b^2 C \cos (2 (c+d x))+7 b^3 C \cos (3 (c+d x))\right )\right ) \sin (c+d x)}{630 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x ]],x]
Output:
(84*(9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 60 *a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[ Cos[c + d*x]]*(7*b*(36*A*b^2 + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252 *a*A*b^2 + 84*a^3*C + 234*a*b^2*C + 54*a*b^2*C*Cos[2*(c + d*x)] + 7*b^3*C* Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)
Time = 1.57 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 3529, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
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.
\(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3529 |
\(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+a (9 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+a (9 A+C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (6 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (9 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3528 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left ((63 A+13 C) a^2+2 b (63 A+43 C) \cos (c+d x) a+\left (24 C a^2+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left ((63 A+13 C) a^2+2 b (63 A+43 C) \cos (c+d x) a+\left (24 C a^2+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((63 A+13 C) a^2+2 b (63 A+43 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (24 C a^2+7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3512 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 (63 A+13 C) a^3+15 \left (8 C a^2+63 A b^2+45 b^2 C\right ) \cos ^2(c+d x) a+21 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (63 A+13 C) a^3+15 \left (8 C a^2+63 A b^2+45 b^2 C\right ) \cos ^2(c+d x) a+21 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (63 A+13 C) a^3+15 \left (8 C a^2+63 A b^2+45 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+21 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 a \left (7 (3 A+C) a^2+3 b^2 (7 A+5 C)\right )+7 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 a \left (7 (3 A+C) a^2+3 b^2 (7 A+5 C)\right )+7 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 a \left (7 (3 A+C) a^2+3 b^2 (7 A+5 C)\right )+7 b \left (9 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {10 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+3 \left (\frac {10 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {14 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {12 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\) |
Input:
Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]
Output:
(2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((12* a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2*b* (24*a^2*C + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3 *((14*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/d + (10*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/ d) + (10*a*(63*A*b^2 + 8*a^2*C + 45*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x] )/d)/5)/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
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] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(717\) vs. \(2(228)=456\).
Time = 41.30 (sec) , antiderivative size = 718, normalized size of antiderivative = 2.93
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-1120 C \,b^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (2160 C a \,b^{2}+2240 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-504 A \,b^{3}-1512 a^{2} b C -3240 C a \,b^{2}-2072 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1260 a A \,b^{2}+504 A \,b^{3}+420 a^{3} C +1512 a^{2} b C +2520 C a \,b^{2}+952 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-630 a A \,b^{2}-126 A \,b^{3}-210 a^{3} C -378 a^{2} b C -720 C a \,b^{2}-168 C \,b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+315 A \,a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+315 a A \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-945 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b -189 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}+105 a^{3} C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+225 C a \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-567 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b -147 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(718\) |
parts | \(\text {Expression too large to display}\) | \(983\) |
Input:
int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x,method=_RETUR NVERBOSE)
Output:
-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*C*b^ 3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(2160*C*a*b^2+2240*C*b^3)*sin(1 /2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*b^3-1512*C*a^2*b-3240*C*a*b^2-2 072*C*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1260*A*a*b^2+504*A*b^3 +420*C*a^3+1512*C*a^2*b+2520*C*a*b^2+952*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1 /2*d*x+1/2*c)+(-630*A*a*b^2-126*A*b^3-210*C*a^3-378*C*a^2*b-720*C*a*b^2-16 8*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+315*A*a^3*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),2^(1/2))+315*a*A*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^ 2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-945*A*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^ (1/2))*a^2*b-189*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1) ^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^3+105*a^3*C*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),2^(1/2))+225*C*a*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^ 2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-567*C*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^ (1/2))*a^2*b-147*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1) ^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^3)/(-2*sin(1/2*d*x+1/2*c)^4 +sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, {\left (35 \, C b^{3} \cos \left (d x + c\right )^{3} + 135 \, C a b^{2} \cos \left (d x + c\right )^{2} + 105 \, C a^{3} + 45 \, {\left (7 \, A + 5 \, C\right )} a b^{2} + 7 \, {\left (27 \, C a^{2} b + {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} + 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} - 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b - i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b + i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x, algori thm="fricas")
Output:
1/315*(2*(35*C*b^3*cos(d*x + c)^3 + 135*C*a*b^2*cos(d*x + c)^2 + 105*C*a^3 + 45*(7*A + 5*C)*a*b^2 + 7*(27*C*a^2*b + (9*A + 7*C)*b^3)*cos(d*x + c))*s qrt(cos(d*x + c))*sin(d*x + c) - 15*sqrt(2)*(7*I*(3*A + C)*a^3 + 3*I*(7*A + 5*C)*a*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-7*I*(3*A + C)*a^3 - 3*I*(7*A + 5*C)*a*b^2)*weierstrassPInvers e(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*sqrt(2)*(-9*I*(5*A + 3*C)*a^2 *b - I*(9*A + 7*C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*sqrt(2)*(9*I*(5*A + 3*C)*a^2*b + I*(9 *A + 7*C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(1/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/sqrt(cos(d*x + c)) , x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/sqrt(cos(d*x + c)) , x)
Time = 1.09 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {C\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,A\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,A\,a^2\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {3\,A\,a\,b^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3)/cos(c + d*x)^(1/2),x)
Output:
(C*a^3*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2 , 2))/3))/d + (2*A*a^3*ellipticF(c/2 + (d*x)/2, 2))/d + (6*A*a^2*b*ellipti cE(c/2 + (d*x)/2, 2))/d + (3*A*a*b^2*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/ 3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*b^3*cos(c + d*x)^(7/2)*si n(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x) ^2)^(1/2)) - (2*C*b^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/ 4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (6*C*a^2*b*cos( c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/( 7*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a*b^2*cos(c + d*x)^(9/2)*sin(c + d*x)*h ypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2))
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{4}+3 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{3} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{3} c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{2} b^{2}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b^{3} c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{2} c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{3} \] Input:
int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(1/2),x)
Output:
int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a**4 + 3*int(sqrt(cos(c + d*x)),x)* a**3*b + int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**3*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**2*b**2 + int(sqrt(cos(c + d*x))*cos(c + d*x)**4 ,x)*b**3*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a*b**2*c + 3*int( sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**2*b*c + int(sqrt(cos(c + d*x))*co s(c + d*x)**2,x)*a*b**3