Integrand size = 25, antiderivative size = 225 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 \cos (e+f x)}{(c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 (c-3 d) \cos (e+f x)}{(c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{(c-d) d (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d (c+d) f \sqrt {c+d \sin (e+f x)}} \]
-2/3*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(3/2)-2/3*a*(c-3*d)*cos(f*x+e)/ (c-d)/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)+2/3*a*(c-3*d)*(sin(1/2*e+1/4*Pi+1/2 *f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f* x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/(c-d)/d/(c+d)^2/f/((c+d *sin(f*x+e))/(c+d))^(1/2)-2/3*a*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/ 2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^ (1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d/(c+d)/f/(c+d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 7.06 (sec) , antiderivative size = 1870, normalized size of antiderivative = 8.31 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx =\text {Too large to display} \]
3*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*(c - 3*d)*Csc[e]*Sec[ e])/(3*(c - d)*d*(c + d)^2*f) + (2*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (2*Csc[e]*(3*c*Cos[e] - d*Cos[e] - c*Sin [f*x] + 3*d*Sin[f*x]))/(3*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x]))))/(Cos [e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (c*Sec[e]*(1 + Sin[e + f*x])*(-( (AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]] ]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqr t[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + C ot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e ]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sq rt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt [1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTa n[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*( c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/S qrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*(c - d )*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f *x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*...
Time = 1.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3233, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {2 \int -\frac {3 a (c-d)+a \sin (e+f x) (c-d)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a (c-d)+a \sin (e+f x) (c-d)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a (c-d)+a \sin (e+f x) (c-d)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {a (c-d) (3 c-d)-a (c-3 d) (c-d) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a (c-d) (3 c-d)-a (c-3 d) (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a (c-d) (3 c-d)-a (c-3 d) (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-3 d) (c-d) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-3 d) (c-d) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-3 d) (c-d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-3 d) (c-d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a (c-3 d) (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 d) (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 d) (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 a (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 d) (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a (c-3 d) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}\) |
(-2*a*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + ((-2*a*(c - 3*d)*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*a*(c - 3*d )*(c - d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*(c - d)^2*(c + d)*E llipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^2))
3.5.87.3.1 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[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[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*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] && LtQ[m, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(883\) vs. \(2(283)=566\).
Time = 5.83 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.93
| method | result | size |
| default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, a \left (\frac {\frac {2 d \left (\cos ^{2}\left (f x +e \right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}}{d}+\frac {\left (-c +d \right ) \left (\frac {2 \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{3 \left (c^{2}-d^{2}\right ) d \left (\sin \left (f x +e \right )+\frac {c}{d}\right )^{2}}+\frac {8 d \left (\cos ^{2}\left (f x +e \right )\right ) c}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 \left (3 c^{2}+d^{2}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (3 c^{4}-6 c^{2} d^{2}+3 d^{4}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {8 c d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(884\) |
| parts | \(\text {Expression too large to display}\) | \(1379\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*a*(1/d*(2*d*cos(f*x+e)^2/(c^2-d^2) /(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*sin(f *x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1 )*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x +e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x +e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)* d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d *sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e)) /(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+(-c+d)/d*(2/3/(c^2-d^2)/d*(-(-d*sin(f *x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^ 2)^2*c/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2* d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d)) ^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2) ^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c *d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c +d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e )^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d) )^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/ cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 920, normalized size of antiderivative = 4.09 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
1/9*((sqrt(2)*(2*a*c^2*d^2 + 3*a*c*d^3 - 3*a*d^4)*cos(f*x + e)^2 - 2*sqrt( 2)*(2*a*c^3*d + 3*a*c^2*d^2 - 3*a*c*d^3)*sin(f*x + e) - sqrt(2)*(2*a*c^4 + 3*a*c^3*d - a*c^2*d^2 + 3*a*c*d^3 - 3*a*d^4))*sqrt(I*d)*weierstrassPInver se(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos (f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (sqrt(2)*(2*a*c^2*d^2 + 3*a*c *d^3 - 3*a*d^4)*cos(f*x + e)^2 - 2*sqrt(2)*(2*a*c^3*d + 3*a*c^2*d^2 - 3*a* c*d^3)*sin(f*x + e) - sqrt(2)*(2*a*c^4 + 3*a*c^3*d - a*c^2*d^2 + 3*a*c*d^3 - 3*a*d^4))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 7*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) - 3*(sqrt(2)*(-I*a*c*d^3 + 3*I*a*d^4)*cos(f*x + e)^2 + 2*sqrt(2 )*(I*a*c^2*d^2 - 3*I*a*c*d^3)*sin(f*x + e) + sqrt(2)*(I*a*c^3*d - 3*I*a*c^ 2*d^2 + I*a*c*d^3 - 3*I*a*d^4))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3* d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3* I*d*sin(f*x + e) - 2*I*c)/d)) - 3*(sqrt(2)*(I*a*c*d^3 - 3*I*a*d^4)*cos(f*x + e)^2 + 2*sqrt(2)*(-I*a*c^2*d^2 + 3*I*a*c*d^3)*sin(f*x + e) + sqrt(2)*(- I*a*c^3*d + 3*I*a*c^2*d^2 - I*a*c*d^3 + 3*I*a*d^4))*sqrt(-I*d)*weierstrass Zeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstra ssPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3 *(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) + 6*((a*c*d^3 - 3*...
Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]