Integrand size = 23, antiderivative size = 159 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\frac {2 a \left (5 a^2+9 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (21 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a b^2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d} \] Output:
2/5*a*(5*a^2+9*b^2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21*b*(21*a^2 +5*b^2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/21*b*(21*a^2+5*b^2)*cos (d*x+c)^(1/2)*sin(d*x+c)/d+32/35*a*b^2*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/7*b ^2*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))*sin(d*x+c)/d
Time = 1.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.69 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\frac {42 \left (5 a^3+9 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (21 a^2 b+5 b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+b \sqrt {\cos (c+d x)} \left (210 a^2+65 b^2+126 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))\right ) \sin (c+d x)}{105 d} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3,x]
Output:
(42*(5*a^3 + 9*a*b^2)*EllipticE[(c + d*x)/2, 2] + 10*(21*a^2*b + 5*b^3)*El lipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(210*a^2 + 65*b^2 + 126*a*b *Cos[c + d*x] + 15*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)
Time = 0.80 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 3272, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (16 a b^2 \cos ^2(c+d x)+b \left (21 a^2+5 b^2\right ) \cos (c+d x)+a \left (7 a^2+3 b^2\right )\right )dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\cos (c+d x)} \left (16 a b^2 \cos ^2(c+d x)+b \left (21 a^2+5 b^2\right ) \cos (c+d x)+a \left (7 a^2+3 b^2\right )\right )dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (16 a b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (21 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (7 a^2+3 b^2\right )\right )dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (7 a \left (5 a^2+9 b^2\right )+5 b \left (21 a^2+5 b^2\right ) \cos (c+d x)\right )dx+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\cos (c+d x)} \left (7 a \left (5 a^2+9 b^2\right )+5 b \left (21 a^2+5 b^2\right ) \cos (c+d x)\right )dx+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (7 a \left (5 a^2+9 b^2\right )+5 b \left (21 a^2+5 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (5 b \left (21 a^2+5 b^2\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+7 a \left (5 a^2+9 b^2\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+9 b^2\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b \left (21 a^2+5 b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+9 b^2\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b \left (21 a^2+5 b^2\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+9 b^2\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b \left (21 a^2+5 b^2\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (5 b \left (21 a^2+5 b^2\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {14 a \left (5 a^2+9 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {14 a \left (5 a^2+9 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+5 b \left (21 a^2+5 b^2\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {32 a b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3,x]
Output:
(2*b^2*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d) + ((32* a*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + ((14*a*(5*a^2 + 9*b^2)*Elli pticE[(c + d*x)/2, 2])/d + 5*b*(21*a^2 + 5*b^2)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(420\) vs. \(2(146)=292\).
Time = 15.12 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.65
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} b^{3}+\left (-504 b^{2} a -360 b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (420 a^{2} b +504 b^{2} a +280 b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-210 a^{2} b -126 b^{2} a -80 b^{3}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+105 a^{2} b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 b^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-105 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-189 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}\right )}{105 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(421\) |
| parts | \(\text {Expression too large to display}\) | \(725\) |
Input:
int(cos(d*x+c)^(1/2)*(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
Output:
-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*cos(1/ 2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*b^3+(-504*a*b^2-360*b^3)*sin(1/2*d*x+1/2 *c)^6*cos(1/2*d*x+1/2*c)+(420*a^2*b+504*a*b^2+280*b^3)*sin(1/2*d*x+1/2*c)^ 4*cos(1/2*d*x+1/2*c)+(-210*a^2*b-126*a*b^2-80*b^3)*sin(1/2*d*x+1/2*c)^2*co s(1/2*d*x+1/2*c)+105*a^2*b*(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))+25*b^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))-105*(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^3-189*(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*b^2)/(-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)^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.29 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\frac {2 \, {\left (15 \, b^{3} \cos \left (d x + c\right )^{2} + 63 \, a b^{2} \cos \left (d x + c\right ) + 105 \, a^{2} b + 25 \, b^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, \sqrt {2} {\left (21 i \, a^{2} b + 5 i \, b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-21 i \, a^{2} b - 5 i \, b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-5 i \, a^{3} - 9 i \, a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (5 i \, a^{3} + 9 i \, a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{105 \, d} \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3,x, algorithm="fricas")
Output:
1/105*(2*(15*b^3*cos(d*x + c)^2 + 63*a*b^2*cos(d*x + c) + 105*a^2*b + 25*b ^3)*sqrt(cos(d*x + c))*sin(d*x + c) - 5*sqrt(2)*(21*I*a^2*b + 5*I*b^3)*wei erstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*sqrt(2)*(-21*I* a^2*b - 5*I*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*sqrt(2)*(-5*I*a^3 - 9*I*a*b^2)*weierstrassZeta(-4, 0, weierstrassPIn verse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*sqrt(2)*(5*I*a^3 + 9*I*a *b^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*s in(d*x + c))))/d
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**3,x)
Output:
Timed out
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3,x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^3*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3,x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^3*sqrt(cos(d*x + c)), x)
Time = 36.44 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\frac {2\,\left (a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+a^2\,b\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {2\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^3,x)
Output:
(2*(a^3*ellipticE(c/2 + (d*x)/2, 2) + a^2*b*ellipticF(c/2 + (d*x)/2, 2) + a^2*b*cos(c + d*x)^(1/2)*sin(c + d*x)))/d - (2*b^3*cos(c + d*x)^(9/2)*sin( c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2 )^(1/2)) - (6*a*b^2*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))
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{2} \] Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3,x)
Output:
int(sqrt(cos(c + d*x)),x)*a**3 + 3*int(sqrt(cos(c + d*x))*cos(c + d*x),x)* a**2*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*b**3 + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a*b**2