Integrand size = 29, antiderivative size = 217 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {18 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {18 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {24 (c+13 d) \cos (e+f x)}{35 d (c+d)^3 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {48 (c+13 d) \cos (e+f x)}{35 d (c+d)^4 f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
2/7*a^2*(c-d)*cos(f*x+e)/d/(c+d)/f/(c+d*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)) ^(1/2)-2/35*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^2/f/(c+d*sin(f*x+e))^(5/2)/(a+ a*sin(f*x+e))^(1/2)-8/105*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^3/f/(c+d*sin(f*x +e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-16/105*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^4 /f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
Time = 2.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.90 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=-\frac {2 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} \left (175 c^3+147 c^2 d+253 c d^2+41 d^3-2 d \left (7 c^2+92 c d+13 d^2\right ) \cos (2 (e+f x))+\left (35 c^3+469 c^2 d+191 c d^2+117 d^3\right ) \sin (e+f x)-2 c d^2 \sin (3 (e+f x))-26 d^3 \sin (3 (e+f x))\right )}{35 (c+d)^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c+d \sin (e+f x))^{7/2}} \]
(-2*Sqrt[3]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^(3/2) *(175*c^3 + 147*c^2*d + 253*c*d^2 + 41*d^3 - 2*d*(7*c^2 + 92*c*d + 13*d^2) *Cos[2*(e + f*x)] + (35*c^3 + 469*c^2*d + 191*c*d^2 + 117*d^3)*Sin[e + f*x ] - 2*c*d^2*Sin[3*(e + f*x)] - 26*d^3*Sin[3*(e + f*x)]))/(35*(c + d)^4*f*( Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c + d*Sin[e + f*x])^(7/2))
Time = 0.92 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3241, 27, 2011, 3042, 3251, 3042, 3251, 3042, 3250}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2}}{(c+d \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3241 |
| \(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}-\frac {2 a \int -\frac {a (c+13 d)+a \sin (e+f x) (c+13 d)}{2 \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {a (c+13 d)+a \sin (e+f x) (c+13 d)}{\sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 2011 |
| \(\displaystyle \frac {a (c+13 d) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (c+13 d) \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{7/2}}dx}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {a (c+13 d) \left (\frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (c+13 d) \left (\frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {a (c+13 d) \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (c+13 d) \left (\frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}+\frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3250 |
| \(\displaystyle \frac {2 a^2 (c-d) \cos (e+f x)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac {a (c+13 d) \left (\frac {4 \left (-\frac {4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\right )}{7 d (c+d)}\) |
(2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) + (a*(c + 13*d)*((-2*a*Cos[e + f*x])/(5*(c + d)*f*S qrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) + (4*((-2*a*Cos[e + f* x])/(3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4 *a*Cos[e + f*x])/(3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(5*(c + d))))/(7*d*(c + d))
3.6.78.3.1 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[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x ] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x , a + b*x])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b *Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a* d))), x] + Simp[b^2/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b* c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], 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[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq rt[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]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) 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] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Leaf count of result is larger than twice the leaf count of optimal. \(763\) vs. \(2(205)=410\).
Time = 3.81 (sec) , antiderivative size = 764, normalized size of antiderivative = 3.52
| method | result | size |
| default | \(\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (-4 \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{2} d^{5}-140 c^{7}-35 \left (\cos ^{2}\left (f x +e \right )\right ) c^{7}+148 c^{5} d^{2}+388 c^{4} d^{3}-252 c^{6} d +497 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{2} d^{5}+564 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) c^{5} d^{2}+8 \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{4}\left (f x +e \right )\right ) c \,d^{6}-76 d^{7} \left (\sin ^{5}\left (f x +e \right )\right )+76 d^{7} \left (\sin ^{4}\left (f x +e \right )\right )+750 c^{4} \left (\cos ^{4}\left (f x +e \right )\right ) d^{3}+104 \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{4}\left (f x +e \right )\right ) d^{7}-195 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{4}\left (f x +e \right )\right ) d^{7}+106 \left (\cos ^{4}\left (f x +e \right )\right ) c^{5} d^{2}-323 \left (\cos ^{2}\left (f x +e \right )\right ) c^{5} d^{2}+52 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{5}\left (f x +e \right )\right ) d^{7}-120 c \,d^{6} \left (\sin ^{5}\left (f x +e \right )\right )+60 c \,d^{6} \left (\sin ^{4}\left (f x +e \right )\right )+8 c^{3} d^{4} \left (\sin ^{3}\left (f x +e \right )\right )+212 c^{2} d^{5} \left (\sin ^{3}\left (f x +e \right )\right )+60 c \,d^{6} \left (\sin ^{3}\left (f x +e \right )\right )-68 c^{3} d^{4} \left (\sin ^{2}\left (f x +e \right )\right )-212 c^{2} d^{5} \left (\sin ^{2}\left (f x +e \right )\right )+119 \left (\cos ^{2}\left (f x +e \right )\right ) c^{6} d -1153 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d^{3}+140 \sin \left (f x +e \right ) c^{7}-29 \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{3} d^{4}-375 \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{2} d^{5}-41 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{4}\left (f x +e \right )\right ) c \,d^{6}-120 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{3} d^{4}-76 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{2} d^{5}+112 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{6} d +644 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d^{3}-4 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c^{4} d^{3}+56 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{5}\left (f x +e \right )\right ) c \,d^{6}-221 \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{3} d^{4}-388 c^{4} d^{3} \sin \left (f x +e \right )+60 c^{3} d^{4} \sin \left (f x +e \right )+252 \sin \left (f x +e \right ) c^{6} d -148 \sin \left (f x +e \right ) c^{5} d^{2}\right ) a}{105 f {\left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )}^{4} \left (c +d \right )^{4}}\) | \(764\) |
2/105/f*sec(f*x+e)*(a*(sin(f*x+e)+1))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(-140*c ^7-221*cos(f*x+e)^2*sin(f*x+e)^2*c^3*d^4+497*cos(f*x+e)^2*sin(f*x+e)^2*c^2 *d^5+564*cos(f*x+e)^2*sin(f*x+e)*c^5*d^2+8*cos(f*x+e)^4*sin(f*x+e)^4*c*d^6 -4*cos(f*x+e)^4*sin(f*x+e)^3*c^2*d^5+56*cos(f*x+e)^2*sin(f*x+e)^5*c*d^6-29 *cos(f*x+e)^4*sin(f*x+e)^2*c^3*d^4-375*cos(f*x+e)^4*sin(f*x+e)^2*c^2*d^5-4 1*cos(f*x+e)^2*sin(f*x+e)^4*c*d^6-120*cos(f*x+e)^2*sin(f*x+e)^3*c^3*d^4-76 *cos(f*x+e)^2*sin(f*x+e)^3*c^2*d^5+148*c^5*d^2+388*c^4*d^3-252*c^6*d-323*c os(f*x+e)^2*c^5*d^2+106*cos(f*x+e)^4*c^5*d^2+104*cos(f*x+e)^4*sin(f*x+e)^4 *d^7+52*cos(f*x+e)^2*sin(f*x+e)^5*d^7-195*cos(f*x+e)^2*sin(f*x+e)^4*d^7-35 *cos(f*x+e)^2*c^7+140*sin(f*x+e)*c^7+750*c^4*cos(f*x+e)^4*d^3-120*c*d^6*si n(f*x+e)^5+60*c*d^6*sin(f*x+e)^4+8*c^3*d^4*sin(f*x+e)^3+212*c^2*d^5*sin(f* x+e)^3+60*c*d^6*sin(f*x+e)^3-68*c^3*d^4*sin(f*x+e)^2-212*c^2*d^5*sin(f*x+e )^2-388*c^4*d^3*sin(f*x+e)+60*c^3*d^4*sin(f*x+e)+112*sin(f*x+e)*cos(f*x+e) ^2*c^6*d+644*sin(f*x+e)*cos(f*x+e)^2*c^4*d^3-4*sin(f*x+e)*cos(f*x+e)^4*c^4 *d^3+119*cos(f*x+e)^2*c^6*d-1153*cos(f*x+e)^2*c^4*d^3+252*sin(f*x+e)*c^6*d -148*sin(f*x+e)*c^5*d^2-76*d^7*sin(f*x+e)^5+76*d^7*sin(f*x+e)^4)*a/(cos(f* x+e)^2*d^2+c^2-d^2)^4/(c+d)^4
Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (205) = 410\).
Time = 0.37 (sec) , antiderivative size = 937, normalized size of antiderivative = 4.32 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {2 \, {\left (8 \, {\left (a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} - 140 \, a c^{3} + 308 \, a c^{2} d - 244 \, a c d^{2} + 76 \, a d^{3} + 4 \, {\left (7 \, a c^{2} d + 92 \, a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (35 \, a c^{3} + 441 \, a c^{2} d - 167 \, a c d^{2} + 195 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (175 \, a c^{3} + 161 \, a c^{2} d + 437 \, a c d^{2} + 67 \, a d^{3}\right )} \cos \left (f x + e\right ) + {\left (140 \, a c^{3} - 308 \, a c^{2} d + 244 \, a c d^{2} - 76 \, a d^{3} + 8 \, {\left (a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - 4 \, {\left (7 \, a c^{2} d + 90 \, a c d^{2} - 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, a c^{3} + 469 \, a c^{2} d + 193 \, a c d^{2} + 143 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{105 \, {\left ({\left (c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{5} + {\left (4 \, c^{5} d^{3} + 17 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 22 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, c^{6} d^{2} + 12 \, c^{5} d^{3} + 19 \, c^{4} d^{4} + 16 \, c^{3} d^{5} + 9 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (2 \, c^{7} d + 11 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 43 \, c^{4} d^{4} + 42 \, c^{3} d^{5} + 25 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{8} + 4 \, c^{7} d + 12 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 38 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right ) + {\left (c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f + {\left ({\left (c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{4} - 4 \, {\left (c^{5} d^{3} + 4 \, c^{4} d^{4} + 6 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, c^{6} d^{2} + 14 \, c^{5} d^{3} + 27 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 17 \, c^{2} d^{6} + 6 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{2} + 4 \, {\left (c^{7} d + 4 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 8 \, c^{4} d^{4} + 7 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]
2/105*(8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^4 - 140*a*c^3 + 308*a*c^2*d - 2 44*a*c*d^2 + 76*a*d^3 + 4*(7*a*c^2*d + 92*a*c*d^2 + 13*a*d^3)*cos(f*x + e) ^3 - (35*a*c^3 + 441*a*c^2*d - 167*a*c*d^2 + 195*a*d^3)*cos(f*x + e)^2 - ( 175*a*c^3 + 161*a*c^2*d + 437*a*c*d^2 + 67*a*d^3)*cos(f*x + e) + (140*a*c^ 3 - 308*a*c^2*d + 244*a*c*d^2 - 76*a*d^3 + 8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - 4*(7*a*c^2*d + 90*a*c*d^2 - 13*a*d^3)*cos(f*x + e)^2 - (35*a*c^3 + 469*a*c^2*d + 193*a*c*d^2 + 143*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt( a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^4*d^4 + 4*c^3*d^5 + 6*c^2 *d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^5 + (4*c^5*d^3 + 17*c^4*d^4 + 28*c^3* d^5 + 22*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^4 - 2*(3*c^6*d^2 + 12*c^5 *d^3 + 19*c^4*d^4 + 16*c^3*d^5 + 9*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e) ^3 - 2*(2*c^7*d + 11*c^6*d^2 + 28*c^5*d^3 + 43*c^4*d^4 + 42*c^3*d^5 + 25*c ^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^2 + (c^8 + 4*c^7*d + 12*c^6*d^2 + 2 8*c^5*d^3 + 38*c^4*d^4 + 28*c^3*d^5 + 12*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f* x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^ 5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f + ((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4* c*d^7 + d^8)*f*cos(f*x + e)^4 - 4*(c^5*d^3 + 4*c^4*d^4 + 6*c^3*d^5 + 4*c^2 *d^6 + c*d^7)*f*cos(f*x + e)^3 - 2*(3*c^6*d^2 + 14*c^5*d^3 + 27*c^4*d^4 + 28*c^3*d^5 + 17*c^2*d^6 + 6*c*d^7 + d^8)*f*cos(f*x + e)^2 + 4*(c^7*d + 4*c ^6*d^2 + 7*c^5*d^3 + 8*c^4*d^4 + 7*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f...
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (205) = 410\).
Time = 0.41 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.46 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, {\left ({\left (175 \, c^{4} + 133 \, c^{3} d + 69 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {3}{2}} - \frac {3 \, {\left (35 \, c^{4} - 385 \, c^{3} d - 189 \, c^{2} d^{2} - 67 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {18 \, {\left (35 \, c^{4} - 28 \, c^{3} d + 166 \, c^{2} d^{2} + 44 \, c d^{3} + 7 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, {\left (35 \, c^{4} - 220 \, c^{3} d + 102 \, c^{2} d^{2} - 244 \, c d^{3} - 25 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {42 \, {\left (20 \, c^{4} - 61 \, c^{3} d + 117 \, c^{2} d^{2} - 55 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {42 \, {\left (20 \, c^{4} - 61 \, c^{3} d + 117 \, c^{2} d^{2} - 55 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {14 \, {\left (35 \, c^{4} - 220 \, c^{3} d + 102 \, c^{2} d^{2} - 244 \, c d^{3} - 25 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {18 \, {\left (35 \, c^{4} - 28 \, c^{3} d + 166 \, c^{2} d^{2} + 44 \, c d^{3} + 7 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {3 \, {\left (35 \, c^{4} - 385 \, c^{3} d - 189 \, c^{2} d^{2} - 67 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {{\left (175 \, c^{4} + 133 \, c^{3} d + 69 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{3}}{105 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} + \frac {3 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {9}{2}} f} \]
-2/105*((175*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2) - 3*(35*c^4 - 385*c^3*d - 189*c^2*d^2 - 67*c*d^3 - 10*d^4)*a^(3/2)*sin(f*x + e)/(cos(f *x + e) + 1) + 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a^( 3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 14*(35*c^4 - 220*c^3*d + 102*c^ 2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^3 + 35*d^4)*a^(3/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^ 3 + 35*d^4)*a^(3/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 14*(35*c^4 - 220 *c^3*d + 102*c^2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a^( 3/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*(35*c^4 - 385*c^3*d - 189*c^2 *d^2 - 67*c*d^3 - 10*d^4)*a^(3/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - (1 75*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2)*sin(f*x + e)^9/(cos(f* x + e) + 1)^9)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 + 3*(c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^ 4)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*(c^4 + 4*c^3*d + 6*c^2*d^2 + 4* c*d^3 + d^4)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (c^4 + 4*c^3*d + 6*c^2* d^2 + 4*c*d^3 + d^4)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)*(c + 2*d*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)^(9/2)*f)
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
Time = 21.53 (sec) , antiderivative size = 807, normalized size of antiderivative = 3.72 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx=\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {32\,a\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (c+13\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{105\,d^2\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (9\,c^3-5\,c^2\,d+9\,c\,d^2-d^3\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,9{}\mathrm {i}-c^2\,d\,5{}\mathrm {i}+c\,d^2\,9{}\mathrm {i}-d^3\,1{}\mathrm {i}\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (5\,c^3+65\,c^2\,d+c\,d^2+13\,d^3\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,5{}\mathrm {i}+c^2\,d\,65{}\mathrm {i}+c\,d^2\,1{}\mathrm {i}+d^3\,13{}\mathrm {i}\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}+\frac {32\,a\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,13{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{105\,d^2\,f\,{\left (c+d\right )}^4}+\frac {32\,a\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,13{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^4}+\frac {32\,a\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (c+13\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^4}\right )}{{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{{\left (c+d\right )}^4}-\frac {4\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+6\,c^2\,d+6\,c\,d^2+d^3\right )}{d^3}-\frac {4\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (6\,c^2+2\,c\,d+d^2\right )}{d^2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (8\,c+d\right )}{d}+\frac {2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (8\,c^4+16\,c^3\,d+24\,c^2\,d^2+12\,c\,d^3+3\,d^4\right )}{d^4}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (8\,c^3+6\,c^2\,d+6\,c\,d^2+d^3\right )\,4{}\mathrm {i}}{d^3\,{\left (c+d\right )}^4}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (6\,c^2+2\,c\,d+d^2\right )\,4{}\mathrm {i}}{d^2\,{\left (c+d\right )}^4}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (8\,c+d\right )\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{d\,{\left (c+d\right )}^4}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (8\,c^4+16\,c^3\,d+24\,c^2\,d^2+12\,c\,d^3+3\,d^4\right )\,2{}\mathrm {i}}{d^4\,{\left (c+d\right )}^4}} \]
((c + d*sin(e + f*x))^(1/2)*((32*a*exp(e*8i + f*x*8i)*(c + 13*d)*(a + a*si n(e + f*x))^(1/2))/(105*d^2*f*(c + d)^4) - (16*a*exp(e*4i + f*x*4i)*(a + a *sin(e + f*x))^(1/2)*(9*c*d^2 - 5*c^2*d + 9*c^3 - d^3))/(3*d^4*f*(c + d)^4 ) - (16*a*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*9i - c^2*d* 5i + c^3*9i - d^3*1i))/(3*d^4*f*(c + d)^4) - (16*a*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2 + 65*c^2*d + 5*c^3 + 13*d^3))/(15*d^4*f*(c + d)^4) - (16*a*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*1i + c ^2*d*65i + c^3*5i + d^3*13i))/(15*d^4*f*(c + d)^4) + (32*a*exp(e*1i + f*x* 1i)*(c*1i + d*13i)*(a + a*sin(e + f*x))^(1/2))/(105*d^2*f*(c + d)^4) + (32 *a*c*exp(e*7i + f*x*7i)*(c*1i + d*13i)*(a + a*sin(e + f*x))^(1/2))/(15*d^3 *f*(c + d)^4) + (32*a*c*exp(e*2i + f*x*2i)*(c + 13*d)*(a + a*sin(e + f*x)) ^(1/2))/(15*d^3*f*(c + d)^4)))/(exp(e*9i + f*x*9i) + ((c*1i + d*1i)^4*1i)/ (c + d)^4 - (4*exp(e*3i + f*x*3i)*(6*c*d^2 + 6*c^2*d + 8*c^3 + d^3))/d^3 - (4*exp(e*7i + f*x*7i)*(2*c*d + 6*c^2 + d^2))/d^2 + (exp(e*1i + f*x*1i)*(8 *c + d))/d + (2*exp(e*5i + f*x*5i)*(12*c*d^3 + 16*c^3*d + 8*c^4 + 3*d^4 + 24*c^2*d^2))/d^4 - (exp(e*6i + f*x*6i)*(c*1i + d*1i)^4*(6*c*d^2 + 6*c^2*d + 8*c^3 + d^3)*4i)/(d^3*(c + d)^4) - (exp(e*2i + f*x*2i)*(c*1i + d*1i)^4*( 2*c*d + 6*c^2 + d^2)*4i)/(d^2*(c + d)^4) + (exp(e*8i + f*x*8i)*(8*c + d)*( c*1i + d*1i)^4*1i)/(d*(c + d)^4) + (exp(e*4i + f*x*4i)*(c*1i + d*1i)^4*(12 *c*d^3 + 16*c^3*d + 8*c^4 + 3*d^4 + 24*c^2*d^2)*2i)/(d^4*(c + d)^4))