Integrand size = 37, antiderivative size = 193 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {8 a b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}} \] Output:
8/5*a^2*(d*sin(f*x+e))^(1/2)/d/f/g^3/(g*cos(f*x+e))^(1/2)+8/5*a*b*(d*sin(f *x+e))^(3/2)/d^2/f/g^3/(g*cos(f*x+e))^(1/2)+2/5*(d*sin(f*x+e))^(1/2)*(a+b* sin(f*x+e))^2/d/f/g/(g*cos(f*x+e))^(5/2)+8/5*a*b*(g*cos(f*x+e))^(1/2)*Elli pticE(cos(e+1/4*Pi+f*x),2^(1/2))*(d*sin(f*x+e))^(1/2)/d/f/g^4/sin(2*f*x+2* e)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.73 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \left (15 a^2+10 a b \cos ^2(e+f x)^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {9}{4},\frac {7}{4},\sin ^2(e+f x)\right ) \sin (e+f x)+3 \left (-4 a^2+b^2\right ) \sin ^2(e+f x)\right ) \tan (e+f x)}{15 f g^2 (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \] Input:
Integrate[(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(7/2)*Sqrt[d*Sin[e + f* x]]),x]
Output:
(2*(15*a^2 + 10*a*b*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[3/4, 9/4, 7/4 , Sin[e + f*x]^2]*Sin[e + f*x] + 3*(-4*a^2 + b^2)*Sin[e + f*x]^2)*Tan[e + f*x])/(15*f*g^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])
Time = 1.05 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {3042, 3367, 3042, 3317, 3042, 3043, 3051, 3042, 3052, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{\sqrt {d \sin (e+f x)} (g \cos (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{\sqrt {d \sin (e+f x)} (g \cos (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3367 |
| \(\displaystyle \frac {4 a \int \frac {a+b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \int \frac {a+b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle \frac {4 a \left (a \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx+\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \left (a \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}}dx+\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle \frac {4 a \left (\frac {b \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}}dx}{d}+\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3051 |
| \(\displaystyle \frac {4 a \left (\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{d}+\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \left (\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{g^2}\right )}{d}+\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle \frac {4 a \left (\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}+\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \left (\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}+\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a \left (\frac {2 a \sqrt {d \sin (e+f x)}}{d f g \sqrt {g \cos (e+f x)}}+\frac {b \left (\frac {2 (d \sin (e+f x))^{3/2}}{d f g \sqrt {g \cos (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{f g^2 \sqrt {\sin (2 e+2 f x)}}\right )}{d}\right )}{5 g^2}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}\) |
Input:
Int[(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(7/2)*Sqrt[d*Sin[e + f*x]]),x ]
Output:
(2*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2)/(5*d*f*g*(g*Cos[e + f*x])^ (5/2)) + (4*a*((2*a*Sqrt[d*Sin[e + f*x]])/(d*f*g*Sqrt[g*Cos[e + f*x]]) + ( b*((2*(d*Sin[e + f*x])^(3/2))/(d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos [e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(f*g^2*Sqrt[ Sin[2*e + 2*f*x]])))/d))/(5*g^2)
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_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(b*Sin[e + f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1) /(a*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Sin[e + f*x ])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m , -1] && IntegersQ[2*m, 2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*(g*Cos [e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^m/(d*f*g*(2*m + 1))), x] + Simp[2*a*(m/(g^2*(2*m + 1))) Int[(g*Cos[e + f*x])^(p + 2)*( (a + b*Sin[e + f*x])^(m - 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
Time = 6.23 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(-\frac {2 \left (\left (-4 \cos \left (f x +e \right )-4\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, a b \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (2 \cos \left (f x +e \right )+2\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, a b \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+a^{2} \left (-4 \sin \left (f x +e \right )-\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )+b a \left (4 \cos \left (f x +e \right )-2-2 \sec \left (f x +e \right )^{2}\right )-b^{2} \sin \left (f x +e \right ) \tan \left (f x +e \right )^{2}\right )}{5 g^{3} f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\) | \(283\) |
| parts | \(\frac {2 a^{2} \left (4 \sin \left (f x +e \right )+\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )}{5 f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}\, g^{3}}+\frac {2 b^{2} \sin \left (f x +e \right ) \tan \left (f x +e \right )^{2}}{5 f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}\, g^{3}}-\frac {4 a b \left (\left (1+\cos \left (f x +e \right )\right ) \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}\, \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\left (-2 \cos \left (f x +e \right )-2\right ) \operatorname {EllipticE}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sqrt {-2 \csc \left (f x +e \right )+2 \cot \left (f x +e \right )+2}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}+2 \cos \left (f x +e \right )-1-\sec \left (f x +e \right )^{2}\right )}{5 f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}\, g^{3}}\) | \(328\) |
Input:
int((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x,method= _RETURNVERBOSE)
Output:
-2/5/g^3/f/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*((-4*cos(f*x+e)-4)*(c sc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f *x+e)+cot(f*x+e))^(1/2)*a*b*EllipticE((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2* 2^(1/2))+(2*cos(f*x+e)+2)*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*(-2*csc(f*x+e)+2 *cot(f*x+e)+2)^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*a*b*EllipticF((csc(f*x +e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))+a^2*(-4*sin(f*x+e)-sec(f*x+e)*tan(f*x +e))+b*a*(4*cos(f*x+e)-2-2*sec(f*x+e)^2)-b^2*sin(f*x+e)*tan(f*x+e)^2)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=-\frac {2 \, {\left (2 i \, \sqrt {i \, d g} a b \cos \left (f x + e\right )^{3} E(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 i \, \sqrt {-i \, d g} a b \cos \left (f x + e\right )^{3} E(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 i \, \sqrt {i \, d g} a b \cos \left (f x + e\right )^{3} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + 2 i \, \sqrt {-i \, d g} a b \cos \left (f x + e\right )^{3} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - {\left ({\left (4 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (2 \, a b \cos \left (f x + e\right )^{2} + a b\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}\right )}}{5 \, d f g^{4} \cos \left (f x + e\right )^{3}} \] Input:
integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2/5*(2*I*sqrt(I*d*g)*a*b*cos(f*x + e)^3*elliptic_e(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1) - 2*I*sqrt(-I*d*g)*a*b*cos(f*x + e)^3*elliptic_e(arcs in(cos(f*x + e) - I*sin(f*x + e)), -1) - 2*I*sqrt(I*d*g)*a*b*cos(f*x + e)^ 3*elliptic_f(arcsin(cos(f*x + e) + I*sin(f*x + e)), -1) + 2*I*sqrt(-I*d*g) *a*b*cos(f*x + e)^3*elliptic_f(arcsin(cos(f*x + e) - I*sin(f*x + e)), -1) - ((4*a^2 - b^2)*cos(f*x + e)^2 + a^2 + b^2 + 2*(2*a*b*cos(f*x + e)^2 + a* b)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(d*sin(f*x + e)))/(d*f*g^4*cos(f *x + e)^3)
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(f*x+e))**2/(g*cos(f*x+e))**(7/2)/(d*sin(f*x+e))**(1/2), x)
Output:
Timed out
\[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {7}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)^2/((g*cos(f*x + e))^(7/2)*sqrt(d*sin(f*x + e))), x)
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int((a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(7/2)*(d*sin(e + f*x))^(1/2)) ,x)
Output:
int((a + b*sin(e + f*x))^2/((g*cos(e + f*x))^(7/2)*(d*sin(e + f*x))^(1/2)) , x)
\[ \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {\sqrt {g}\, \sqrt {d}\, \left (10 \cos \left (f x +e \right )^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}}{\cos \left (f x +e \right )^{4}}d x \right ) a b f +5 \cos \left (f x +e \right )^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}}{\cos \left (f x +e \right )^{4} \sin \left (f x +e \right )}d x \right ) a^{2} f -\cos \left (f x +e \right )^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}}{\cos \left (f x +e \right )^{2} \sin \left (f x +e \right )}d x \right ) b^{2} f +2 \sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}\, b^{2}\right )}{5 \cos \left (f x +e \right )^{3} d f \,g^{4}} \] Input:
int((a+b*sin(f*x+e))^2/(g*cos(f*x+e))^(7/2)/(d*sin(f*x+e))^(1/2),x)
Output:
(sqrt(g)*sqrt(d)*(10*cos(e + f*x)**3*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x)))/cos(e + f*x)**4,x)*a*b*f + 5*cos(e + f*x)**3*int((sqrt(sin(e + f*x) )*sqrt(cos(e + f*x)))/(cos(e + f*x)**4*sin(e + f*x)),x)*a**2*f - cos(e + f *x)**3*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x)))/(cos(e + f*x)**2*sin(e + f*x)),x)*b**2*f + 2*sqrt(sin(e + f*x))*sqrt(cos(e + f*x))*b**2))/(5*cos( e + f*x)**3*d*f*g**4)