Integrand size = 25, antiderivative size = 285 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=-\frac {2 \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 f}-\frac {6 (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}-\frac {6 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}+\frac {2 \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d f \sqrt {c+d \sin (e+f x)}} \]
-2/35*a*(5*c+7*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f-2/7*a*cos(f*x+e)*(c+ d*sin(f*x+e))^(5/2)/f-2/105*a*(15*c^2+56*c*d+25*d^2)*cos(f*x+e)*(c+d*sin(f *x+e))^(1/2)/f-2/105*a*(15*c^3+161*c^2*d+145*c*d^2+63*d^3)*(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)/d/f/((c+d*sin(f*x +e))/(c+d))^(1/2)+2/105*a*(c^2-d^2)*(15*c^2+56*c*d+25*d^2)*(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/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 = 6.71 (sec) , antiderivative size = 3531, normalized size of antiderivative = 12.39 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=\text {Result too large to show} \]
3*((c^3*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((C sc[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*Sqrt[1 + Cot[e]^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*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot [e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[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*S qrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqr t[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 - A rcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*S qrt[1 + Cot[e]^2]*Sin[e]]))/(7*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]) ^2) + (23*c^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]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^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*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - Ar cTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[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])...
Time = 1.51 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 3232, 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.
\(\displaystyle \int (a \sin (e+f x)+a) (c+d \sin (e+f x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x)+a) (c+d \sin (e+f x))^{5/2}dx\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {2}{7} \int \frac {1}{2} (c+d \sin (e+f x))^{3/2} (a (7 c+5 d)+a (5 c+7 d) \sin (e+f x))dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int (c+d \sin (e+f x))^{3/2} (a (7 c+5 d)+a (5 c+7 d) \sin (e+f x))dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \int (c+d \sin (e+f x))^{3/2} (a (7 c+5 d)+a (5 c+7 d) \sin (e+f x))dx-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (a \left (35 c^2+40 d c+21 d^2\right )+a \left (15 c^2+56 d c+25 d^2\right ) \sin (e+f x)\right )dx-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {c+d \sin (e+f x)} \left (a \left (35 c^2+40 d c+21 d^2\right )+a \left (15 c^2+56 d c+25 d^2\right ) \sin (e+f x)\right )dx-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {c+d \sin (e+f x)} \left (a \left (35 c^2+40 d c+21 d^2\right )+a \left (15 c^2+56 d c+25 d^2\right ) \sin (e+f x)\right )dx-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {a \left (105 c^3+135 d c^2+119 d^2 c+25 d^3\right )+a \left (15 c^3+161 d c^2+145 d^2 c+63 d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (105 c^3+135 d c^2+119 d^2 c+25 d^3\right )+a \left (15 c^3+161 d c^2+145 d^2 c+63 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (105 c^3+135 d c^2+119 d^2 c+25 d^3\right )+a \left (15 c^3+161 d c^2+145 d^2 c+63 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 a \left (15 c^3+161 c^2 d+145 c d^2+63 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a \left (c^2-d^2\right ) \left (15 c^2+56 c d+25 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 a \left (15 c^2+56 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 a (5 c+7 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\) |
(-2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + ((-2*a*(5*c + 7*d)* Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + ((-2*a*(15*c^2 + 56*c*d + 25*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((2*a*(15*c^3 + 16 1*c^2*d + 145*c*d^2 + 63*d^3)*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^2 - d^2)*(15*c^2 + 56*c*d + 25*d^2)*EllipticF[(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 ]]))/3)/5)/7
3.5.82.3.1 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[(-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. \(1314\) vs. \(2(332)=664\).
Time = 6.26 (sec) , antiderivative size = 1315, normalized size of antiderivative = 4.61
method | result | size |
default | \(\text {Expression too large to display}\) | \(1315\) |
parts | \(\text {Expression too large to display}\) | \(1835\) |
2/105*a*(-90*sin(f*x+e)*c^2*d^3+60*sin(f*x+e)^4*c*d^4+90*sin(f*x+e)^3*c^2* d^3+45*sin(f*x+e)^2*c^3*d^2-25*sin(f*x+e)*d^5+15*sin(f*x+e)^5*d^5-77*c^2*d ^3-25*c*d^4-45*c^3*d^2+120*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1) *d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e) )/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d+176*((c+d*sin(f*x+e))/(c-d))^(1/ 2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*Ellipti cF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^2-32*((c+d*si n(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/ (c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)) *c^2*d^3-176*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2 )*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2) ,((c-d)/(c+d))^(1/2))*c*d^4-161*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+ e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f *x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-130*((c+d*sin(f*x+e))/(c-d) )^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*El lipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^2+98*((c +d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e )+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^( 1/2))*c^2*d^3+145*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d)) ^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.03 \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (30 \, a c^{4} + 7 \, a c^{3} d - 115 \, a c^{2} d^{2} - 231 \, a c d^{3} - 75 \, a d^{4}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (30 \, a c^{4} + 7 \, a c^{3} d - 115 \, a c^{2} d^{2} - 231 \, a c d^{3} - 75 \, a d^{4}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (15 i \, a c^{3} d + 161 i \, a c^{2} d^{2} + 145 i \, a c d^{3} + 63 i \, a d^{4}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-15 i \, a c^{3} d - 161 i \, a c^{2} d^{2} - 145 i \, a c d^{3} - 63 i \, a d^{4}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - 6 \, {\left (15 \, a d^{4} \cos \left (f x + e\right )^{3} - 3 \, {\left (15 \, a c d^{3} + 7 \, a d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (45 \, a c^{2} d^{2} + 77 \, a c d^{3} + 40 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, d^{2} f} \]
-1/315*(sqrt(2)*(30*a*c^4 + 7*a*c^3*d - 115*a*c^2*d^2 - 231*a*c*d^3 - 75*a *d^4)*sqrt(I*d)*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 ) + sqrt(2)*(30*a*c^4 + 7*a*c^3*d - 115*a*c^2*d^2 - 231*a*c*d^3 - 75*a*d^4 )*sqrt(-I*d)*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)*(15*I*a*c^3*d + 161*I*a*c^2*d^2 + 145*I*a*c*d^3 + 63*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)*(-15*I*a*c^3*d - 161*I*a*c^2*d^2 - 145*I*a*c*d^3 - 63*I*a*d^4)*s qrt(-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)) - 6*(15*a*d^4*cos(f*x + e)^3 - 3*(15*a*c*d^3 + 7*a*d^4)*cos(f*x + e)*sin(f *x + e) - (45*a*c^2*d^2 + 77*a*c*d^3 + 40*a*d^4)*cos(f*x + e))*sqrt(d*sin( f*x + e) + c))/(d^2*f)
\[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=a \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx\right ) \]
a*(Integral(c**2*sqrt(c + d*sin(e + f*x)), x) + Integral(c**2*sqrt(c + d*s in(e + f*x))*sin(e + f*x), x) + Integral(d**2*sqrt(c + d*sin(e + f*x))*sin (e + f*x)**2, x) + Integral(d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3, x) + Integral(2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral( 2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2, x))
\[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (3+3 \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=\int \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]