Integrand size = 14, antiderivative size = 256 \[ \int (a+b \sin (c+d x))^{7/2} \, dx=-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{105 d}-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {32 a \left (11 a^2+13 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{105 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{105 d \sqrt {a+b \sin (c+d x)}} \]
-24/35*a*b*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/d-2/7*b*cos(d*x+c)*(a+b*sin(d *x+c))^(5/2)/d-2/105*b*(71*a^2+25*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/d -32/105*a*(11*a^2+13*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/ 4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2)) *(a+b*sin(d*x+c))^(1/2)/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+2/105*(71*a^4-46* a^2*b^2-25*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d *x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin (d*x+c))/(a+b))^(1/2)/d/(a+b*sin(d*x+c))^(1/2)
Time = 0.76 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.86 \[ \int (a+b \sin (c+d x))^{7/2} \, dx=\frac {-64 a \left (11 a^3+11 a^2 b+13 a b^2+13 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+4 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b \cos (c+d x) \left (488 a^3+262 a b^2-162 a b^2 \cos (2 (c+d x))+b \left (752 a^2+145 b^2\right ) \sin (c+d x)-15 b^3 \sin (3 (c+d x))\right )}{210 d \sqrt {a+b \sin (c+d x)}} \]
(-64*a*(11*a^3 + 11*a^2*b + 13*a*b^2 + 13*b^3)*EllipticE[(-2*c + Pi - 2*d* x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 4*(71*a^4 - 46*a ^2*b^2 - 25*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x]*(488*a^3 + 262*a*b^2 - 162*a*b^ 2*Cos[2*(c + d*x)] + b*(752*a^2 + 145*b^2)*Sin[c + d*x] - 15*b^3*Sin[3*(c + d*x)]))/(210*d*Sqrt[a + b*Sin[c + d*x]])
Time = 1.33 (sec) , antiderivative size = 267, 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.
| \(\displaystyle \int (a+b \sin (c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (c+d x))^{7/2}dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \sin (c+d x))^{3/2} \left (7 a^2+12 b \sin (c+d x) a+5 b^2\right )dx-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int (a+b \sin (c+d x))^{3/2} \left (7 a^2+12 b \sin (c+d x) a+5 b^2\right )dx-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int (a+b \sin (c+d x))^{3/2} \left (7 a^2+12 b \sin (c+d x) a+5 b^2\right )dx-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \sin (c+d x)} \left (a \left (35 a^2+61 b^2\right )+b \left (71 a^2+25 b^2\right ) \sin (c+d x)\right )dx-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \sin (c+d x)} \left (a \left (35 a^2+61 b^2\right )+b \left (71 a^2+25 b^2\right ) \sin (c+d x)\right )dx-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \sin (c+d x)} \left (a \left (35 a^2+61 b^2\right )+b \left (71 a^2+25 b^2\right ) \sin (c+d x)\right )dx-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {105 a^4+254 b^2 a^2+16 b \left (11 a^2+13 b^2\right ) \sin (c+d x) a+25 b^4}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 a^4+254 b^2 a^2+16 b \left (11 a^2+13 b^2\right ) \sin (c+d x) a+25 b^4}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 a^4+254 b^2 a^2+16 b \left (11 a^2+13 b^2\right ) \sin (c+d x) a+25 b^4}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (16 a \left (11 a^2+13 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (16 a \left (11 a^2+13 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {16 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {16 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (71 a^4-46 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}\right )-\frac {2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )-\frac {24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}\) |
(-2*b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*d) + ((-24*a*b*Cos[c + d *x]*(a + b*Sin[c + d*x])^(3/2))/(5*d) + ((-2*b*(71*a^2 + 25*b^2)*Cos[c + d *x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((32*a*(11*a^2 + 13*b^2)*EllipticE[( c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b *Sin[c + d*x])/(a + b)]) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqr t[a + b*Sin[c + d*x]]))/3)/5)/7
3.1.50.3.1 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[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. \(1039\) vs. \(2(298)=596\).
Time = 1.19 (sec) , antiderivative size = 1040, normalized size of antiderivative = 4.06
2/105*(15*b^5*sin(d*x+c)^5+105*a^5*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d *x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*si n(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+71*((a+b*sin(d*x+c))/(a-b))^(1 /2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellipt icF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b+78*((a+b*sin (d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/( a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))* a^3*b^2-46*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)* (-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),( (a-b)/(a+b))^(1/2))*a^2*b^3-183*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+ c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d *x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4-25*((a+b*sin(d*x+c))/(a-b)) ^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ell ipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^5-176*((a+b*s in(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b /(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2) )*a^5-32*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(- (1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a -b)/(a+b))^(1/2))*a^3*b^2+208*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c) -1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.92 \[ \int (a+b \sin (c+d x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (37 \, a^{4} - 346 \, a^{2} b^{2} - 75 \, b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, 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 i \, a}{3 \, b}\right ) + \sqrt {2} {\left (37 \, a^{4} - 346 \, a^{2} b^{2} - 75 \, b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, 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 i \, a}{3 \, b}\right ) + 48 \, \sqrt {2} {\left (11 i \, a^{3} b + 13 i \, a b^{3}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, 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 i \, a^{3} - 9 i \, 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 i \, a}{3 \, b}\right )\right ) + 48 \, \sqrt {2} {\left (-11 i \, a^{3} b - 13 i \, a b^{3}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, 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 i \, a^{3} + 9 i \, 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 i \, a}{3 \, b}\right )\right ) - 6 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{3} - 66 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (61 \, a^{2} b^{2} + 20 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{315 \, b d} \]
-1/315*(sqrt(2)*(37*a^4 - 346*a^2*b^2 - 75*b^4)*sqrt(I*b)*weierstrassPInve rse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*co s(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + sqrt(2)*(37*a^4 - 346*a^2*b^ 2 - 75*b^4)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27 *(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) + 48*sqrt(2)*(11*I*a^3*b + 13*I*a*b^3)*sqrt(I*b)*weierstrassZeta (-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassPIn verse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b* cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 48*sqrt(2)*(-11*I*a^3*b - 13*I*a*b^3)*sqrt(-I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(- 8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8 /27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 6*(15*b^4*cos(d*x + c)^3 - 66*a*b^3*cos(d*x + c)*sin(d*x + c) - 2*(61*a^2*b^2 + 20*b^4)*cos(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b*d )
Timed out. \[ \int (a+b \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (a+b \sin (c+d x))^{7/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
\[ \int (a+b \sin (c+d x))^{7/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (a+b \sin (c+d x))^{7/2} \, dx=\int {\left (a+b\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]