Integrand size = 25, antiderivative size = 258 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e} \]
-26/3465*a*b*(79*a^2+74*b^2)*(e*cos(d*x+c))^(5/2)/d/e-2/693*b*(167*a^2+54* b^2)*(e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))/d/e-34/99*a*b*(e*cos(d*x+c))^(5 /2)*(a+b*sin(d*x+c))^2/d/e-2/11*b*(e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3/ d/e+2/231*(77*a^4+132*a^2*b^2+12*b^4)*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos (1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/( e*cos(d*x+c))^(1/2)+2/231*(77*a^4+132*a^2*b^2+12*b^4)*e*sin(d*x+c)*(e*cos( d*x+c))^(1/2)/d
Time = 2.80 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.73 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\frac {(e \cos (c+d x))^{3/2} \left (240 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (-1848 b \left (12 a^3+7 a b^2\right )-2464 \left (9 a^3 b+4 a b^3\right ) \cos (2 (c+d x))+3080 a b^3 \cos (4 (c+d x))+30 \left (616 a^4+660 a^2 b^2+39 b^4\right ) \sin (c+d x)-45 b \left (264 a^2 b+31 b^3\right ) \sin (3 (c+d x))+315 b^4 \sin (5 (c+d x))\right )\right )}{27720 d \cos ^{\frac {3}{2}}(c+d x)} \]
((e*Cos[c + d*x])^(3/2)*(240*(77*a^4 + 132*a^2*b^2 + 12*b^4)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-1848*b*(12*a^3 + 7*a*b^2) - 2464*(9*a^ 3*b + 4*a*b^3)*Cos[2*(c + d*x)] + 3080*a*b^3*Cos[4*(c + d*x)] + 30*(616*a^ 4 + 660*a^2*b^2 + 39*b^4)*Sin[c + d*x] - 45*b*(264*a^2*b + 31*b^3)*Sin[3*( c + d*x)] + 315*b^4*Sin[5*(c + d*x)])))/(27720*d*Cos[c + d*x]^(3/2))
Time = 1.29 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 3171, 27, 3042, 3341, 27, 3042, 3341, 27, 3042, 3148, 3042, 3115, 3042, 3121, 3042, 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.
| \(\displaystyle \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \left (11 a^2+17 b \sin (c+d x) a+6 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \left (11 a^2+17 b \sin (c+d x) a+6 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \left (11 a^2+17 b \sin (c+d x) a+6 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (a \left (99 a^2+122 b^2\right )+b \left (167 a^2+54 b^2\right ) \sin (c+d x)\right )dx-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (a \left (99 a^2+122 b^2\right )+b \left (167 a^2+54 b^2\right ) \sin (c+d x)\right )dx-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (a \left (99 a^2+122 b^2\right )+b \left (167 a^2+54 b^2\right ) \sin (c+d x)\right )dx-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} (e \cos (c+d x))^{3/2} \left (9 \left (77 a^4+132 b^2 a^2+12 b^4\right )+13 a b \left (79 a^2+74 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int (e \cos (c+d x))^{3/2} \left (9 \left (77 a^4+132 b^2 a^2+12 b^4\right )+13 a b \left (79 a^2+74 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int (e \cos (c+d x))^{3/2} \left (9 \left (77 a^4+132 b^2 a^2+12 b^4\right )+13 a b \left (79 a^2+74 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \int (e \cos (c+d x))^{3/2}dx-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\) |
(-2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3)/(11*d*e) + ((-34*a*b* (e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(9*d*e) + ((-2*b*(167*a^2 + 54*b^2)*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(7*d*e) + ((-26*a*b* (79*a^2 + 74*b^2)*(e*Cos[c + d*x])^(5/2))/(5*d*e) + 9*(77*a^4 + 132*a^2*b^ 2 + 12*b^4)*((2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqr t[e*Cos[c + d*x]]) + (2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/7)/9) /11
3.6.66.3.1 Defintions of rubi rules used
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[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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Leaf count of result is larger than twice the leaf count of optimal. \(638\) vs. \(2(258)=516\).
Time = 12.33 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.48
| method | result | size |
| default | \(-\frac {2 e^{2} \left (20160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-50400 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+49280 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-47520 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+41040 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-123200 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+71280 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}-11160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-22176 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b +101024 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+4620 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}-27720 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+33264 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b -28336 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-2310 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}+1980 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+180 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+1155 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{4}+1980 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b^{2}+180 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{4}-16632 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b -1232 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+2772 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +1232 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}\right )}{3465 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(639\) |
| parts | \(\text {Expression too large to display}\) | \(726\) |
-2/3465/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^2*(20160* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12*b^4-50400*cos(1/2*d*x+1/2*c)*sin( 1/2*d*x+1/2*c)^10*b^4+49280*sin(1/2*d*x+1/2*c)^11*a*b^3-47520*cos(1/2*d*x+ 1/2*c)*sin(1/2*d*x+1/2*c)^8*a^2*b^2+41040*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1 /2*c)^8*b^4-123200*sin(1/2*d*x+1/2*c)^9*a*b^3+71280*cos(1/2*d*x+1/2*c)*sin (1/2*d*x+1/2*c)^6*a^2*b^2-11160*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*b^ 4-22176*sin(1/2*d*x+1/2*c)^7*a^3*b+101024*sin(1/2*d*x+1/2*c)^7*a*b^3+4620* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^4-27720*cos(1/2*d*x+1/2*c)*sin(1 /2*d*x+1/2*c)^4*a^2*b^2+33264*sin(1/2*d*x+1/2*c)^5*a^3*b-28336*sin(1/2*d*x +1/2*c)^5*a*b^3-2310*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^4+1980*cos( 1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2*b^2+180*cos(1/2*d*x+1/2*c)*sin(1/2 *d*x+1/2*c)^2*b^4+1155*(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))*a^4+1980*(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))*a^2*b^2+180*(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))*b^4-16632*sin(1/2*d*x+1/2* c)^3*a^3*b-1232*sin(1/2*d*x+1/2*c)^3*a*b^3+2772*sin(1/2*d*x+1/2*c)*a^3*b+1 232*sin(1/2*d*x+1/2*c)*a*b^3)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.83 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\frac {-15 i \, \sqrt {2} {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (1540 \, a b^{3} e \cos \left (d x + c\right )^{4} - 2772 \, {\left (a^{3} b + a b^{3}\right )} e \cos \left (d x + c\right )^{2} + 15 \, {\left (21 \, b^{4} e \cos \left (d x + c\right )^{4} - 3 \, {\left (66 \, a^{2} b^{2} + 13 \, b^{4}\right )} e \cos \left (d x + c\right )^{2} + {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{3465 \, d} \]
1/3465*(-15*I*sqrt(2)*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^(3/2)*weierstrassP Inverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*I*sqrt(2)*(77*a^4 + 132 *a^2*b^2 + 12*b^4)*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c)) + 2*(1540*a*b^3*e*cos(d*x + c)^4 - 2772*(a^3*b + a*b^3)*e*cos(d *x + c)^2 + 15*(21*b^4*e*cos(d*x + c)^4 - 3*(66*a^2*b^2 + 13*b^4)*e*cos(d* x + c)^2 + (77*a^4 + 132*a^2*b^2 + 12*b^4)*e)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]