Integrand size = 29, antiderivative size = 154 \[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=-\frac {3 \sqrt {3} (c+d)^2 \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{4 \sqrt {d} f}-\frac {9 (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \sqrt {3+3 \sin (e+f x)}}-\frac {3 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {3+3 \sin (e+f x)}} \]
-3/4*(c+d)^2*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d *sin(f*x+e))^(1/2))*a^(1/2)/f/d^(1/2)-1/2*a*cos(f*x+e)*(c+d*sin(f*x+e))^(3 /2)/f/(a+a*sin(f*x+e))^(1/2)-3/4*a*(c+d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2) /f/(a+a*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 1.26 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.36 \[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) \sqrt {3} \sqrt {1+\sin (e+f x)} \left (\frac {3 i (c+d)^2 e^{\frac {1}{2} i (e+f x)} \sqrt {2 c-i d e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )} \left ((-1)^{3/4} \sqrt {2} \arctan \left (\frac {\sqrt [4]{-1} \left (d-i c e^{i (e+f x)}\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )}}\right )-(1+i) \text {arctanh}\left (\frac {(-1)^{3/4} \left (c-i d e^{i (e+f x)}\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )}}\right )\right )}{\sqrt {d} \sqrt {-2 c e^{i (e+f x)}+i d \left (-1+e^{2 i (e+f x)}\right )} f}-\frac {(2-2 i) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} (5 c+3 d+2 d \sin (e+f x))}{f}\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )} \]
((1/16 + I/16)*Sqrt[3]*Sqrt[1 + Sin[e + f*x]]*(((3*I)*(c + d)^2*E^((I/2)*( e + f*x))*Sqrt[2*c - (I*d*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x))]*((- 1)^(3/4)*Sqrt[2]*ArcTan[((-1)^(1/4)*(d - I*c*E^(I*(e + f*x))))/(Sqrt[d]*Sq rt[-2*c*E^(I*(e + f*x)) + I*d*(-1 + E^((2*I)*(e + f*x)))])] - (1 + I)*ArcT anh[((-1)^(3/4)*(c - I*d*E^(I*(e + f*x))))/(Sqrt[d]*Sqrt[-2*c*E^(I*(e + f* x)) + I*d*(-1 + E^((2*I)*(e + f*x)))])]))/(Sqrt[d]*Sqrt[-2*c*E^(I*(e + f*x )) + I*d*(-1 + E^((2*I)*(e + f*x)))]*f) - ((2 - 2*I)*(Cos[(e + f*x)/2] - S in[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(5*c + 3*d + 2*d*Sin[e + f*x]))/ f))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])
Time = 0.60 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {3042, 3249, 3042, 3249, 3042, 3254, 218}
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.
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {3}{4} (c+d) \int \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}dx-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} (c+d) \int \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}dx-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {3}{4} (c+d) \left (\frac {1}{2} (c+d) \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} (c+d) \left (\frac {1}{2} (c+d) \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {3}{4} (c+d) \left (-\frac {a (c+d) \int \frac {1}{\frac {a^2 d \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}+a}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3}{4} (c+d) \left (-\frac {\sqrt {a} (c+d) \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\) |
-1/2*(a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*Sqrt[a + a*Sin[e + f*x ]]) + (3*(c + d)*(-((Sqrt[a]*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x]) /(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[d]*f)) - (a*C os[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])))/4
3.6.65.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))], x ] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Timed out.
\[\int \sqrt {a +a \sin \left (f x +e \right )}\, \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (130) = 260\).
Time = 0.59 (sec) , antiderivative size = 1069, normalized size of antiderivative = 6.94 \[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\text {Too large to display} \]
[1/32*(3*(c^2 + 2*c*d + d^2 + (c^2 + 2*c*d + d^2)*cos(f*x + e) + (c^2 + 2* c*d + d^2)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*co s(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 3 2*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4 *cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4 )*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3 *d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e )^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a *sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3 *d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*co s(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*( a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*c os(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(2*d*cos(f*x + e)^2 + (5*c + 3*d)*cos(f*x + e) + (2*d*cos(f*x + e) - 5*c - d)*sin(f*x + e ) + 5*c + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/16*(3*(c^2 + 2*c*d + d^2 + (c^2 + 2*c*d + d ^2)*cos(f*x + e) + (c^2 + 2*c*d + d^2)*sin(f*x + e))*sqrt(a/d)*arctan(1...
\[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int { \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int { \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx=\int \sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]