Integrand size = 29, antiderivative size = 139 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {6 \cos (e+f x)}{5 (c+d) f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 \cos (e+f x)}{5 (c+d)^2 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 \cos (e+f x)}{5 (c+d)^3 f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
-2/5*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-8/ 15*a*cos(f*x+e)/(c+d)^2/f/(c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-16 /15*a*cos(f*x+e)/(c+d)^3/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
Time = 0.75 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)} \left (15 c^2+10 c d+3 d^2+4 d (5 c+d) \sin (e+f x)+8 d^2 \sin ^2(e+f x)\right )}{5 \sqrt {3} (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{5/2}} \]
(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[1 + Sin[e + f*x]]*(15*c^2 + 10*c*d + 3*d^2 + 4*d*(5*c + d)*Sin[e + f*x] + 8*d^2*Sin[e + f*x]^2))/(5*S qrt[3]*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f* x])^(5/2))
Time = 0.59 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {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.
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a}}{(c+d \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a}}{(c+d \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{5/2}}dx}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{(c+d \sin (e+f x))^{3/2}}dx}{3 (c+d)}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3250 |
| \(\displaystyle \frac {4 \left (-\frac {4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}\right )}{5 (c+d)}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}\) |
(-2*a*Cos[e + f*x])/(5*(c + d)*f*Sqrt[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))
3.6.70.3.1 Defintions of rubi rules used
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. \(374\) vs. \(2(124)=248\).
Time = 3.58 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.70
| method | result | size |
| default | \(\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (8 \left (\sin ^{4}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}+4 \left (\sin ^{5}\left (f x +e \right )\right ) d^{5}-4 \left (\sin ^{3}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}-7 d^{5} \left (\sin ^{4}\left (f x +e \right )\right )-21 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+2 \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}-3 \left (\sin ^{3}\left (f x +e \right )\right ) c^{2} d^{3}+11 c \,d^{4} \left (\sin ^{3}\left (f x +e \right )\right )+3 d^{5} \left (\sin ^{3}\left (f x +e \right )\right )+7 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}-6 c^{2} d^{3} \left (\sin ^{2}\left (f x +e \right )\right )-11 c \,d^{4} \left (\sin ^{2}\left (f x +e \right )\right )+25 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d +19 c^{3} \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+15 c^{5} \sin \left (f x +e \right )+35 \sin \left (f x +e \right ) c^{4} d +22 \sin \left (f x +e \right ) c^{3} d^{2}+9 \sin \left (f x +e \right ) c^{2} d^{3}-15 c^{5}-35 c^{4} d -22 c^{3} d^{2}\right )}{15 f {\left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )}^{3} \left (c +d \right )^{3}}\) | \(375\) |
2/15/f*sec(f*x+e)*(a*(sin(f*x+e)+1))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(8*sin(f *x+e)^4*cos(f*x+e)^2*d^5+4*sin(f*x+e)^5*d^5-4*sin(f*x+e)^3*cos(f*x+e)^2*c* d^4-7*d^5*sin(f*x+e)^4-21*sin(f*x+e)^2*cos(f*x+e)^2*c^2*d^3+2*sin(f*x+e)^2 *cos(f*x+e)^2*c*d^4-3*sin(f*x+e)^3*c^2*d^3+11*c*d^4*sin(f*x+e)^3+3*d^5*sin (f*x+e)^3+7*sin(f*x+e)*cos(f*x+e)^2*c^3*d^2-6*c^2*d^3*sin(f*x+e)^2-11*c*d^ 4*sin(f*x+e)^2+25*cos(f*x+e)^2*c^4*d+19*c^3*cos(f*x+e)^2*d^2+15*c^5*sin(f* x+e)+35*sin(f*x+e)*c^4*d+22*sin(f*x+e)*c^3*d^2+9*sin(f*x+e)*c^2*d^3-15*c^5 -35*c^4*d-22*c^3*d^2)/(cos(f*x+e)^2*d^2+c^2-d^2)^3/(c+d)^3
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (124) = 248\).
Time = 0.31 (sec) , antiderivative size = 556, normalized size of antiderivative = 4.00 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 \, {\left (8 \, d^{2} \cos \left (f x + e\right )^{3} - 4 \, {\left (5 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} - {\left (15 \, c^{2} + 10 \, c d + 11 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} + 4 \, {\left (5 \, c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{15 \, {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{4} - 3 \, {\left (c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right )} f \cos \left (f x + e\right )^{3} - {\left (3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f - {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right ) - {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f\right )} \sin \left (f x + e\right )\right )}} \]
2/15*(8*d^2*cos(f*x + e)^3 - 4*(5*c*d - d^2)*cos(f*x + e)^2 - 15*c^2 + 10* c*d - 7*d^2 - (15*c^2 + 10*c*d + 11*d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e )^2 - 15*c^2 + 10*c*d - 7*d^2 + 4*(5*c*d + d^2)*cos(f*x + e))*sin(f*x + e) )*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4 - 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c *d^5)*f*cos(f*x + e)^3 - (3*c^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 + 10*c^3*d ^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 2 0*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 + 3*c*d^ 5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))
Timed out. \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (124) = 248\).
Time = 0.36 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.91 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left ({\left (15 \, c^{3} + 10 \, c^{2} d + 3 \, c d^{2}\right )} \sqrt {a} - \frac {{\left (15 \, c^{3} - 60 \, c^{2} d - 25 \, c d^{2} - 6 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (45 \, c^{3} - 40 \, c^{2} d + 93 \, c d^{2} + 10 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, {\left (9 \, c^{3} - 22 \, c^{2} d + 13 \, c d^{2} - 12 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, {\left (9 \, c^{3} - 22 \, c^{2} d + 13 \, c d^{2} - 12 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (45 \, c^{3} - 40 \, c^{2} d + 93 \, c d^{2} + 10 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {{\left (15 \, c^{3} - 60 \, c^{2} d - 25 \, c d^{2} - 6 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {{\left (15 \, c^{3} + 10 \, c^{2} d + 3 \, c d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + \frac {3 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\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 {7}{2}} f} \]
-2/15*((15*c^3 + 10*c^2*d + 3*c*d^2)*sqrt(a) - (15*c^3 - 60*c^2*d - 25*c*d ^2 - 6*d^3)*sqrt(a)*sin(f*x + e)/(cos(f*x + e) + 1) + (45*c^3 - 40*c^2*d + 93*c*d^2 + 10*d^3)*sqrt(a)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5*(9*c^3 - 22*c^2*d + 13*c*d^2 - 12*d^3)*sqrt(a)*sin(f*x + e)^3/(cos(f*x + e) + 1) ^3 + 5*(9*c^3 - 22*c^2*d + 13*c*d^2 - 12*d^3)*sqrt(a)*sin(f*x + e)^4/(cos( f*x + e) + 1)^4 - (45*c^3 - 40*c^2*d + 93*c*d^2 + 10*d^3)*sqrt(a)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + (15*c^3 - 60*c^2*d - 25*c*d^2 - 6*d^3)*sqrt( a)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - (15*c^3 + 10*c^2*d + 3*c*d^2)*sqr t(a)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^3 + 3*c^2*d + 3*c*d^2 + d^3 + 3*(c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*(c^3 + 3*c^2*d + 3*c*d^2 + d ^3)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)* 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)^(7/2)*f)
Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (124) = 248\).
Time = 1.22 (sec) , antiderivative size = 1050, normalized size of antiderivative = 7.55 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \]
4/15*sqrt(2)*(((5*(3*(c^10*d^6 - 2*c^9*d^7 - 3*c^8*d^8 + 8*c^7*d^9 + 2*c^6 *d^10 - 12*c^5*d^11 + 2*c^4*d^12 + 8*c^3*d^13 - 3*c^2*d^14 - 2*c*d^15 + d^ 16)*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5* c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^11 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^ 14 - 5*c^2*d^15 - c*d^16 + d^17) + 4*(3*c^10*d^6 - 14*c^9*d^7 + 15*c^8*d^8 + 24*c^7*d^9 - 58*c^6*d^10 + 12*c^5*d^11 + 54*c^4*d^12 - 40*c^3*d^13 - 9* c^2*d^14 + 18*c*d^15 - 5*d^16)/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^ 9 + 10*c^7*d^10 - 10*c^6*d^11 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5 *c^2*d^15 - c*d^16 + d^17))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 2*(45*c^10* d^6 - 250*c^9*d^7 + 601*c^8*d^8 - 664*c^7*d^9 - 166*c^6*d^10 + 1444*c^5*d^ 11 - 1510*c^4*d^12 + 104*c^3*d^13 + 889*c^2*d^14 - 634*c*d^15 + 141*d^16)/ (c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^11 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5*c^2*d^15 - c*d^16 + d^17))*tan (-1/8*pi + 1/4*f*x + 1/4*e)^2 + 20*(3*c^10*d^6 - 14*c^9*d^7 + 15*c^8*d^8 + 24*c^7*d^9 - 58*c^6*d^10 + 12*c^5*d^11 + 54*c^4*d^12 - 40*c^3*d^13 - 9*c^ 2*d^14 + 18*c*d^15 - 5*d^16)/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^11 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5*c ^2*d^15 - c*d^16 + d^17))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 15*(c^10*d^6 - 2*c^9*d^7 - 3*c^8*d^8 + 8*c^7*d^9 + 2*c^6*d^10 - 12*c^5*d^11 + 2*c^4*d^1 2 + 8*c^3*d^13 - 3*c^2*d^14 - 2*c*d^15 + d^16)/(c^11*d^6 - c^10*d^7 - 5...
Time = 17.15 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.60 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{15\,d\,f\,{\left (c+d\right )}^3}-\frac {32\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d\,f\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (240\,c^2+80\,d^2\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^3}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (c^2\,240{}\mathrm {i}+d^2\,80{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^3}+\frac {32\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d^2\,f\,{\left (c+d\right )}^3}-\frac {c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{3\,d^2\,f\,{\left (c+d\right )}^3}\right )}{{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3}{{\left (c+d\right )}^3}-\frac {3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (6\,c+d\right )}{d}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c\,6{}\mathrm {i}+d\,1{}\mathrm {i}\right )}{d}-\frac {3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3\,{\left (c+d\right )}^3}} \]
-((c + d*sin(e + f*x))^(1/2)*((exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^(1/ 2)*32i)/(15*d*f*(c + d)^3) - (32*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^( 1/2))/(15*d*f*(c + d)^3) + (exp(e*4i + f*x*4i)*(240*c^2 + 80*d^2)*(a + a*s in(e + f*x))^(1/2))/(15*d^3*f*(c + d)^3) - (exp(e*3i + f*x*3i)*(c^2*240i + d^2*80i)*(a + a*sin(e + f*x))^(1/2))/(15*d^3*f*(c + d)^3) + (32*c*exp(e*2 i + f*x*2i)*(a + a*sin(e + f*x))^(1/2))/(3*d^2*f*(c + d)^3) - (c*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*32i)/(3*d^2*f*(c + d)^3)))/(exp(e*7i + f*x*7i) + (c*1i + d*1i)^3/(c + d)^3 - (3*exp(e*5i + f*x*5i)*(2*c*d + 4*c ^2 + d^2))/d^2 - (exp(e*1i + f*x*1i)*(6*c + d))/d + (exp(e*3i + f*x*3i)*(1 2*c*d^2 + 12*c^2*d + 8*c^3 + 3*d^3))/d^3 + (exp(e*6i + f*x*6i)*(c*6i + d*1 i))/d - (3*exp(e*2i + f*x*2i)*(c*1i + d*1i)^3*(2*c*d + 4*c^2 + d^2))/(d^2* (c + d)^3) + (exp(e*4i + f*x*4i)*(c*1i + d*1i)^3*(12*c*d^2 + 12*c^2*d + 8* c^3 + 3*d^3))/(d^3*(c + d)^3))