Integrand size = 14, antiderivative size = 246 \[ \int (a+b \cos (c+d x))^{7/2} \, dx=\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (71 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {24 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \] Output:
32/105*a*(11*a^2+13*b^2)*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2* c),2^(1/2)*(b/(a+b))^(1/2))/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/105*(71*a^4 -46*a^2*b^2-25*b^4)*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x +1/2*c,2^(1/2)*(b/(a+b))^(1/2))/d/(a+b*cos(d*x+c))^(1/2)+2/105*b*(71*a^2+2 5*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d+24/35*a*b*(a+b*cos(d*x+c))^(3/2 )*sin(d*x+c)/d+2/7*b*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d
Time = 1.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.86 \[ \int (a+b \cos (c+d x))^{7/2} \, dx=\frac {64 a \left (11 a^3+11 a^2 b+13 a b^2+13 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-4 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+b \left (488 a^3+262 a b^2+b \left (752 a^2+145 b^2\right ) \cos (c+d x)+162 a b^2 \cos (2 (c+d x))+15 b^3 \cos (3 (c+d x))\right ) \sin (c+d x)}{210 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[(a + b*Cos[c + d*x])^(7/2),x]
Output:
(64*a*(11*a^3 + 11*a^2*b + 13*a*b^2 + 13*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(71*a^4 - 46*a^2*b^2 - 25 *b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(488*a^3 + 262*a*b^2 + b*(752*a^2 + 145*b^2)*Cos[c + d*x] + 162*a *b^2*Cos[2*(c + d*x)] + 15*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(210*d*Sqrt [a + b*Cos[c + d*x]])
Time = 1.35 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3042, 3135, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int (a+b \cos (c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (7 a^2+12 b \cos (c+d x) a+5 b^2\right )dx+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (7 a^2+12 b \cos (c+d x) a+5 b^2\right )dx+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (7 a^2+12 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+5 b^2\right )dx+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (a \left (35 a^2+61 b^2\right )+b \left (71 a^2+25 b^2\right ) \cos (c+d x)\right )dx+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (a \left (35 a^2+61 b^2\right )+b \left (71 a^2+25 b^2\right ) \cos (c+d x)\right )dx+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (a \left (35 a^2+61 b^2\right )+b \left (71 a^2+25 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {105 a^4+254 b^2 a^2+16 b \left (11 a^2+13 b^2\right ) \cos (c+d x) a+25 b^4}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 a^4+254 b^2 a^2+16 b \left (11 a^2+13 b^2\right ) \cos (c+d x) a+25 b^4}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 a^4+254 b^2 a^2+16 b \left (11 a^2+13 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+25 b^4}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (16 a \left (11 a^2+13 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (16 a \left (11 a^2+13 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {16 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {16 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}\right )\right )+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(7/2),x]
Output:
(2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((24*a*b*(a + b*Cos[ c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((32*a*(11*a^2 + 13*b^2)*Sqrt[a + b *Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/3 + (2*b*(71*a^2 + 25*b^2)*Sqrt[a + b*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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[1/n Int[(a + b* Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x] , x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
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] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(823\) vs. \(2(231)=462\).
Time = 9.91 (sec) , antiderivative size = 824, normalized size of antiderivative = 3.35
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b^{4}+648 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a \,b^{3}-600 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{4}+752 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b^{2}-1296 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a \,b^{3}+640 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{4}+244 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{3} b -1128 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b^{2}+860 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \,b^{3}-360 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{4}-71 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{4}+46 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b^{2}+25 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) b^{4}+176 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{4}-176 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3} b +208 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b^{2}-208 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{3}-244 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +376 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}-212 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}+80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}\right )}{105 \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}\, d}\) | \(824\) |
Input:
int((a+cos(d*x+c)*b)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/105*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*co s(1/2*d*x+1/2*c)^9*b^4+648*cos(1/2*d*x+1/2*c)^7*a*b^3-600*cos(1/2*d*x+1/2* c)^7*b^4+752*cos(1/2*d*x+1/2*c)^5*a^2*b^2-1296*cos(1/2*d*x+1/2*c)^5*a*b^3+ 640*cos(1/2*d*x+1/2*c)^5*b^4+244*cos(1/2*d*x+1/2*c)^3*a^3*b-1128*cos(1/2*d *x+1/2*c)^3*a^2*b^2+860*cos(1/2*d*x+1/2*c)^3*a*b^3-360*cos(1/2*d*x+1/2*c)^ 3*b^4-71*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b ))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4+46*(sin(1/2* d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticF (cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2+25*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1 /2*c),(-2*b/(a-b))^(1/2))*b^4+176*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1 /2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b) )^(1/2))*a^4-176*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a -b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+20 8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2 )*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-208*(sin(1/2*d* x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(c os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3-244*cos(1/2*d*x+1/2*c)*a^3*b+3 76*cos(1/2*d*x+1/2*c)*a^2*b^2-212*cos(1/2*d*x+1/2*c)*a*b^3+80*cos(1/2*d*x+ 1/2*c)*b^4)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.93 \[ \int (a+b \cos (c+d x))^{7/2} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (-37 i \, a^{4} + 346 i \, a^{2} b^{2} + 75 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {\frac {1}{2}} {\left (37 i \, a^{4} - 346 i \, a^{2} b^{2} - 75 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 48 \, \sqrt {\frac {1}{2}} {\left (-11 i \, a^{3} b - 13 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 48 \, \sqrt {\frac {1}{2}} {\left (11 i \, a^{3} b + 13 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{2} + 66 \, a b^{3} \cos \left (d x + c\right ) + 122 \, a^{2} b^{2} + 25 \, b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )\right )}}{315 \, b d} \] Input:
integrate((a+b*cos(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-2/315*(sqrt(1/2)*(-37*I*a^4 + 346*I*a^2*b^2 + 75*I*b^4)*sqrt(b)*weierstra ssPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b* cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(37*I*a^4 - 346*I* a^2*b^2 - 75*I*b^4)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, - 8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2 *a)/b) + 48*sqrt(1/2)*(-11*I*a^3*b - 13*I*a*b^3)*sqrt(b)*weierstrassZeta(4 /3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4 /3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 48*sqrt(1/2)*(11*I*a^3*b + 13*I*a*b^3)* sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b ^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b ^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(15*b^4*cos( d*x + c)^2 + 66*a*b^3*cos(d*x + c) + 122*a^2*b^2 + 25*b^4)*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b*d)
Timed out. \[ \int (a+b \cos (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{7/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^(7/2), x)
\[ \int (a+b \cos (c+d x))^{7/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(7/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^(7/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{7/2} \, dx=\int {\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2} \,d x \] Input:
int((a + b*cos(c + d*x))^(7/2),x)
Output:
int((a + b*cos(c + d*x))^(7/2), x)
\[ \int (a+b \cos (c+d x))^{7/2} \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{2} \] Input:
int((a+b*cos(d*x+c))^(7/2),x)
Output:
int(sqrt(cos(c + d*x)*b + a),x)*a**3 + 3*int(sqrt(cos(c + d*x)*b + a)*cos( c + d*x),x)*a**2*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*b**3 + 3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a*b**2