\(\int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx\) [291]

Optimal result

Integrand size = 37, antiderivative size = 126 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=-\frac {\sqrt {a} (A d+B (c+2 d)) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{3/2} (c+d)^{3/2} f}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \]

[Out]

-(A*d+B*(c+2*d))*arctanh(cos(f*x+e)*a^(1/2)*d^(1/2)/(c+d)^(1/2)/(a+a*sin(f*x+e))^(1/2))*a^(1/2)/d^(3/2)/(c+d)^
(3/2)/f+a*(-A*d+B*c)*cos(f*x+e)/d/(c+d)/f/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {3059, 2852, 214} \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {a (B c-A d) \cos (e+f x)}{d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {\sqrt {a} (A d+B (c+2 d)) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{3/2} f (c+d)^{3/2}} \]

[In]

Int[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]

[Out]

-((Sqrt[a]*(A*d + B*(c + 2*d))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])
/(d^(3/2)*(c + d)^(3/2)*f)) + (a*(B*c - A*d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e
+ f*x]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2852

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[-2*(
b/f), Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*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]

Rule 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), 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,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {a (B c-A d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+-\frac {(-a A d-B (a c+2 a d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 d (a c+a d)} \\ & = \frac {a (B c-A d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {(a (A d+B (c+2 d))) \text {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d (c+d) f} \\ & = -\frac {\sqrt {a} (A d+B (c+2 d)) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{3/2} (c+d)^{3/2} f}+\frac {a (B c-A d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 7.02 (sec) , antiderivative size = 901, normalized size of antiderivative = 7.15 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {a (1+\sin (e+f x))} \left (\frac {(A d+B (c+2 d)) \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right ) \left ((-1+i) x \cos (e)+\frac {\text {RootSum}\left [-d+2 i c e^{i e} \text {$\#$1}^2+d e^{2 i e} \text {$\#$1}^4\&,\frac {(1+i) d \sqrt {e^{-i e}} f x-(2-2 i) d \sqrt {e^{-i e}} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right )-i \sqrt {d} \sqrt {c+d} f x \text {$\#$1}+2 \sqrt {d} \sqrt {c+d} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}+\frac {(1-i) c f x \text {$\#$1}^2}{\sqrt {e^{-i e}}}+\frac {(2+2 i) c \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {e^{-i e}}}-\sqrt {d} \sqrt {c+d} e^{i e} f x \text {$\#$1}^3-2 i \sqrt {d} \sqrt {c+d} e^{i e} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{d-i c e^{i e} \text {$\#$1}^2}\&\right ] (\cos (e)+i (-1+\sin (e))) \sqrt {\cos (e)-i \sin (e)}}{4 f}+(1+i) x \sin (e)\right )}{(c+d)^{3/2} (\cos (e)+i (-1+\sin (e))) \sqrt {\cos (e)-i \sin (e)}}+\frac {(A d+B (c+2 d)) \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right ) \left ((1-i) x \cos (e)-(1+i) x \sin (e)+\frac {\text {RootSum}\left [-d+2 i c e^{i e} \text {$\#$1}^2+d e^{2 i e} \text {$\#$1}^4\&,\frac {(1-i) d \sqrt {e^{-i e}} f x+(2+2 i) d \sqrt {e^{-i e}} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right )+\sqrt {d} \sqrt {c+d} f x \text {$\#$1}+2 i \sqrt {d} \sqrt {c+d} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}-\frac {(1+i) c f x \text {$\#$1}^2}{\sqrt {e^{-i e}}}+\frac {(2-2 i) c \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {e^{-i e}}}-i \sqrt {d} \sqrt {c+d} e^{i e} f x \text {$\#$1}^3+2 \sqrt {d} \sqrt {c+d} e^{i e} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{d-i c e^{i e} \text {$\#$1}^2}\&\right ] \sqrt {\cos (e)-i \sin (e)} (-1-i \cos (e)+\sin (e))}{4 f}\right )}{(c+d)^{3/2} (\cos (e)+i (-1+\sin (e))) \sqrt {\cos (e)-i \sin (e)}}-\frac {(2-2 i) \sqrt {d} (-B c+A d) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d) f (c+d \sin (e+f x))}\right )}{d^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2,x]

[Out]

((1/4 + I/4)*Sqrt[a*(1 + Sin[e + f*x])]*(((A*d + B*(c + 2*d))*(Cos[e/2] + I*Sin[e/2])*((-1 + I)*x*Cos[e] + (Ro
otSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 + I)*d*Sqrt[E^((-I)*e)]*f*x - (2 - 2*I)*d*Sqrt[E^
((-I)*e)]*Log[E^((I/2)*f*x) - #1] - I*Sqrt[d]*Sqrt[c + d]*f*x*#1 + 2*Sqrt[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #
1]*#1 + ((1 - I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 + 2*I)*c*Log[E^((I/2)*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] -
Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 - (2*I)*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Log[E^((I/2)*f*x) - #1]*#1^3)/(d - I*
c*E^(I*e)*#1^2) & ]*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]])/(4*f) + (1 + I)*x*Sin[e]))/((c + d)^(3
/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) + ((A*d + B*(c + 2*d))*(Cos[e/2] + I*Sin[e/2])*((1 - I
)*x*Cos[e] - (1 + I)*x*Sin[e] + (RootSum[-d + (2*I)*c*E^(I*e)*#1^2 + d*E^((2*I)*e)*#1^4 & , ((1 - I)*d*Sqrt[E^
((-I)*e)]*f*x + (2 + 2*I)*d*Sqrt[E^((-I)*e)]*Log[E^((I/2)*f*x) - #1] + Sqrt[d]*Sqrt[c + d]*f*x*#1 + (2*I)*Sqrt
[d]*Sqrt[c + d]*Log[E^((I/2)*f*x) - #1]*#1 - ((1 + I)*c*f*x*#1^2)/Sqrt[E^((-I)*e)] + ((2 - 2*I)*c*Log[E^((I/2)
*f*x) - #1]*#1^2)/Sqrt[E^((-I)*e)] - I*Sqrt[d]*Sqrt[c + d]*E^(I*e)*f*x*#1^3 + 2*Sqrt[d]*Sqrt[c + d]*E^(I*e)*Lo
g[E^((I/2)*f*x) - #1]*#1^3)/(d - I*c*E^(I*e)*#1^2) & ]*Sqrt[Cos[e] - I*Sin[e]]*(-1 - I*Cos[e] + Sin[e]))/(4*f)
))/((c + d)^(3/2)*(Cos[e] + I*(-1 + Sin[e]))*Sqrt[Cos[e] - I*Sin[e]]) - ((2 - 2*I)*Sqrt[d]*(-(B*c) + A*d)*(Cos
[(e + f*x)/2] - Sin[(e + f*x)/2]))/((c + d)*f*(c + d*Sin[e + f*x]))))/(d^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*
x)/2]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(273\) vs. \(2(110)=220\).

Time = 0.84 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.17

method	result	size
default	\(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin \left (f x +e \right ) \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a d \left (d A +B c +2 d B \right )+A \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a c d +B \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a \,c^{2}+2 B \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a c d +A \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, d -B \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, c \right )}{d \left (c +d \right ) \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(274\)

[In]

int((A+B*sin(f*x+e))*(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

-(1+sin(f*x+e))*(-a*(sin(f*x+e)-1))^(1/2)*(sin(f*x+e)*arctanh((a-a*sin(f*x+e))^(1/2)*d/(a*c*d+a*d^2)^(1/2))*a*
d*(A*d+B*c+2*B*d)+A*arctanh((a-a*sin(f*x+e))^(1/2)*d/(a*c*d+a*d^2)^(1/2))*a*c*d+B*arctanh((a-a*sin(f*x+e))^(1/
2)*d/(a*c*d+a*d^2)^(1/2))*a*c^2+2*B*arctanh((a-a*sin(f*x+e))^(1/2)*d/(a*c*d+a*d^2)^(1/2))*a*c*d+A*(a-a*sin(f*x
+e))^(1/2)*(a*(c+d)*d)^(1/2)*d-B*(a-a*sin(f*x+e))^(1/2)*(a*(c+d)*d)^(1/2)*c)/d/(c+d)/(c+d*sin(f*x+e))/(a*(c+d)
*d)^(1/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (110) = 220\).

Time = 0.85 (sec) , antiderivative size = 1012, normalized size of antiderivative = 8.03 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))*(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

[-1/4*((B*c^2 + (A + 3*B)*c*d + (A + 2*B)*d^2 - (B*c*d + (A + 2*B)*d^2)*cos(f*x + e)^2 + (B*c^2 + (A + 2*B)*c*
d)*cos(f*x + e) + (B*c^2 + (A + 3*B)*c*d + (A + 2*B)*d^2 + (B*c*d + (A + 2*B)*d^2)*cos(f*x + e))*sin(f*x + e))
*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2
+ 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d
+ 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (
a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d
^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2
+ d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(B*c -
A*d + (B*c - A*d)*cos(f*x + e) - (B*c - A*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c*d^2 + d^3)*f*cos(f*x
+ e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d +
2*c*d^2 + d^3)*f)*sin(f*x + e)), 1/2*((B*c^2 + (A + 3*B)*c*d + (A + 2*B)*d^2 - (B*c*d + (A + 2*B)*d^2)*cos(f*
x + e)^2 + (B*c^2 + (A + 2*B)*c*d)*cos(f*x + e) + (B*c^2 + (A + 3*B)*c*d + (A + 2*B)*d^2 + (B*c*d + (A + 2*B)*
d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c
- 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(B*c - A*d + (B*c - A*d)*cos(f*x + e) - (B*c - A*d)*sin(f*x
+ e))*sqrt(a*sin(f*x + e) + a))/((c*d^2 + d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*
c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sin(f*x+e))*(a+a*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]

[In]

integrate((A+B*sin(f*x+e))*(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*sqrt(a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^2, x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {\sqrt {2} {\left (B c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + A d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, B d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c d - d^{2}}}\right )}{{\left (c d + d^{2}\right )} \sqrt {-c d - d^{2}}} - \frac {2 \, {\left (B c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )} {\left (c d + d^{2}\right )}}\right )}}{2 \, f} \]

[In]

integrate((A+B*sin(f*x+e))*(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

-1/2*sqrt(2)*sqrt(a)*(sqrt(2)*(B*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + A*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e
)) + 2*B*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*arctan(sqrt(2)*d*sin(-1/4*pi + 1/2*f*x + 1/2*e)/sqrt(-c*d - d^
2))/((c*d + d^2)*sqrt(-c*d - d^2)) - 2*(B*c*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)
- A*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e))/((2*d*sin(-1/4*pi + 1/2*f*x + 1/2*e
)^2 - c - d)*(c*d + d^2)))/f

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(1/2))/(c + d*sin(e + f*x))^2,x)

[Out]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(1/2))/(c + d*sin(e + f*x))^2, x)