Integrand size = 37, antiderivative size = 159 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\left (2 a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (4 e-\pi )+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{3 f g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
4/3*a*b*(d*sin(f*x+e))^(3/2)/d^2/f/g/(g*cos(f*x+e))^(3/2)+2/3*(a^2+b^2)*(d *sin(f*x+e))^(1/2)/d/f/g/(g*cos(f*x+e))^(3/2)-1/3*(2*a^2-b^2)*(sin(e+1/4*P i+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*sin (2*f*x+2*e)^(1/2)/f/g^2/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \left (15 a^2 \cos ^2(e+f x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{4},\frac {5}{4},\sin ^2(e+f x)\right )+b \sin (e+f x) \left (10 a+3 b \cos ^2(e+f x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {9}{4},\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right ) \tan (e+f x)}{15 f g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
(2*(15*a^2*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[1/4, 7/4, 5/4, Sin[e + f*x]^2] + b*Sin[e + f*x]*(10*a + 3*b*(Cos[e + f*x]^2)^(3/4)*Hypergeometri c2F1[5/4, 7/4, 9/4, Sin[e + f*x]^2]*Sin[e + f*x]))*Tan[e + f*x])/(15*f*g^2 *Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])
Time = 0.75 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {3042, 3390, 3042, 3043, 3681, 362, 266, 768, 858, 807, 230}
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.
\(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{\sqrt {d \sin (e+f x)} (g \cos (e+f x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{\sqrt {d \sin (e+f x)} (g \cos (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 3390 |
\(\displaystyle \int \frac {a^2+b^2 \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}}dx+\frac {2 a b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{5/2}}dx}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a^2+b^2 \sin (e+f x)^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}}dx+\frac {2 a b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{5/2}}dx}{d}\) |
\(\Big \downarrow \) 3043 |
\(\displaystyle \int \frac {a^2+b^2 \sin (e+f x)^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}}dx+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3681 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \int \frac {a^2+b^2 \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} \left (1-\sin ^2(e+f x)\right )^{7/4}}d\sin (e+f x)}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 362 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \left (\frac {1}{3} \left (2 a^2-b^2\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \left (1-\sin ^2(e+f x)\right )^{3/4}}d\sin (e+f x)+\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}\right )}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \left (\frac {2 \left (2 a^2-b^2\right ) \int \frac {1}{\left (1-\sin ^2(e+f x)\right )^{3/4}}d\sqrt {d \sin (e+f x)}}{3 d}+\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}\right )}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \left (\frac {2 \left (2 a^2-b^2\right ) (d \sin (e+f x))^{3/2} \left (1-d^2 \csc ^4(e+f x)\right )^{3/4} \int \frac {\csc ^3(e+f x)}{\left (1-d^2 \csc ^4(e+f x)\right )^{3/4}}d\sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}+\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}\right )}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \left (\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}-\frac {2 \left (2 a^2-b^2\right ) (d \sin (e+f x))^{3/2} \left (1-d^2 \csc ^4(e+f x)\right )^{3/4} \int \frac {\csc (e+f x)}{\left (1-d^4 \sin ^2(e+f x)\right )^{3/4}}d\csc (e+f x)}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}\right )}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \left (\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}-\frac {\left (2 a^2-b^2\right ) (d \sin (e+f x))^{3/2} \left (1-d^2 \csc ^4(e+f x)\right )^{3/4} \int \frac {1}{\left (1-d^3 \sin (e+f x)\right )^{3/4}}d(d \sin (e+f x))}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}\right )}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 230 |
\(\displaystyle \frac {\cos ^2(e+f x)^{3/4} \left (\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d \left (1-\sin ^2(e+f x)\right )^{3/4}}-\frac {2 \left (2 a^2-b^2\right ) (d \sin (e+f x))^{3/2} \left (1-d^2 \csc ^4(e+f x)\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (d^2 \sin (e+f x)\right ),2\right )}{3 d^2 \left (1-\sin ^2(e+f x)\right )^{3/4}}\right )}{f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\) |
(4*a*b*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2)) + ((Cos[ e + f*x]^2)^(3/4)*((2*(a^2 + b^2)*Sqrt[d*Sin[e + f*x]])/(3*d*(1 - Sin[e + f*x]^2)^(3/4)) - (2*(2*a^2 - b^2)*(1 - d^2*Csc[e + f*x]^4)^(3/4)*EllipticF [ArcSin[d^2*Sin[e + f*x]]/2, 2]*(d*Sin[e + f*x])^(3/2))/(3*d^2*(1 - Sin[e + f*x]^2)^(3/4))))/(f*g*(g*Cos[e + f*x])^(3/2))
3.13.77.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] ))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^( m_.), x_Symbol] :> Simp[(a*Sin[e + f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/ (a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] & & NeQ[m, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[2*a*(b/d) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e + f* x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(c_.))^(m_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*c^(2*IntPart[(m - 1)/2] + 1)*((c*Co s[e + f*x])^(2*FracPart[(m - 1)/2])/(f*(Cos[e + f*x]^2)^FracPart[(m - 1)/2] )) Subst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x] , x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !In tegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(159)=318\).
Time = 2.62 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.67
method | result | size |
default | \(\frac {\sqrt {2}\, \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2} \cos \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2} \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 \sqrt {2}\, a b \cos \left (f x +e \right )+\sqrt {2}\, a^{2} \tan \left (f x +e \right )+\sqrt {2}\, b^{2} \tan \left (f x +e \right )+2 \sqrt {2}\, a b \sec \left (f x +e \right )\right )}{3 g^{2} f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\) | \(425\) |
parts | \(\frac {a^{2} \sqrt {2}\, \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \tan \left (f x +e \right )\right )}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}+\frac {b^{2} \sqrt {2}\, \left (-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \tan \left (f x +e \right )\right )}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}+\frac {4 a b \sin \left (f x +e \right ) \tan \left (f x +e \right )}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}\) | \(462\) |
1/3/g^2/f*2^(1/2)/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(2*(-cot(f*x+e )+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f* x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^2*co s(f*x+e)-(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2) *(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2) ,1/2*2^(1/2))*b^2*cos(f*x+e)+2*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(-csc(f*x+ e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+ e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^2-(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*( -csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF(( -cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*b^2-2*2^(1/2)*a*b*cos(f*x+e)+ 2^(1/2)*a^2*tan(f*x+e)+2^(1/2)*b^2*tan(f*x+e)+2*2^(1/2)*a*b*sec(f*x+e))
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=-\frac {{\left (2 \, a^{2} - b^{2}\right )} \sqrt {i \, d g} \cos \left (f x + e\right )^{2} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + {\left (2 \, a^{2} - b^{2}\right )} \sqrt {-i \, d g} \cos \left (f x + e\right )^{2} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 \, {\left (2 \, a b \sin \left (f x + e\right ) + a^{2} + b^{2}\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}}{3 \, d f g^{3} \cos \left (f x + e\right )^{2}} \]
-1/3*((2*a^2 - b^2)*sqrt(I*d*g)*cos(f*x + e)^2*elliptic_f(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1) + (2*a^2 - b^2)*sqrt(-I*d*g)*cos(f*x + e)^2*ell iptic_f(arcsin(cos(f*x + e) - I*sin(f*x + e)), -1) - 2*(2*a*b*sin(f*x + e) + a^2 + b^2)*sqrt(g*cos(f*x + e))*sqrt(d*sin(f*x + e)))/(d*f*g^3*cos(f*x + e)^2)
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]