Integrand size = 25, antiderivative size = 138 \[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 a E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a (c-d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \] Output:
-2*a*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin (f*x+e))^(1/2)/d/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2*a*(c-d)*InverseJacobiA M(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^( 1/2)/d/f/(c+d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.61 (sec) , antiderivative size = 880, normalized size of antiderivative = 6.38 \[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + a*Sin[e + f*x])/Sqrt[c + d*Sin[e + f*x]],x]
Output:
a*((Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e ]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos [f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*( -1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]] ])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot [e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[ 1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot [e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTa n[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[ 1 + Cot[e]^2]*Sin[e]]))/(f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*Tan[e])/(d*f*(Cos[e/2 + (f* x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e] *(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[ e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(- 1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]] *(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan...
Time = 0.62 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
\(\displaystyle \int \frac {a \sin (e+f x)+a}{\sqrt {c+d \sin (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a \sin (e+f x)+a}{\sqrt {c+d \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {a \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (c-d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (c-d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {a \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (c-d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (c-d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2 a \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (c-d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {2 a \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2 a \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\) |
Input:
Int[(a + a*Sin[e + f*x])/Sqrt[c + d*Sin[e + f*x]],x]
Output:
(2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]] )/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c - d)*EllipticF[(e - P i/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt [c + d*Sin[e + f*x]])
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Time = 2.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {2 a \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {d \left (-1+\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (2 \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d -\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c -\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(205\) |
parts | \(\frac {2 a \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {d \left (-1+\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{d \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {2 a \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {d \left (-1+\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c +\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d -\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(330\) |
risch | \(\text {Expression too large to display}\) | \(1428\) |
Input:
int((a+sin(f*x+e)*a)/(c+d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2*a*(c-d)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-d*(-1+sin(f*x+e))/(c+d))^(1/2)* (-d*(1+sin(f*x+e))/(c-d))^(1/2)*(2*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2 ),((c-d)/(c+d))^(1/2))*d-EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/( c+d))^(1/2))*c-EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2 ))*d)/d^2/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.67 \[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \, {\left (3 i \, a \sqrt {\frac {1}{2} i \, d} d {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 i \, a \sqrt {-\frac {1}{2} i \, d} d {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + {\left (2 \, a c - 3 \, a d\right )} \sqrt {\frac {1}{2} i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (2 \, a c - 3 \, a d\right )} \sqrt {-\frac {1}{2} i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}}{3 \, d^{2} f} \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2/3*(3*I*a*sqrt(1/2*I*d)*d*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 7*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) - 3*I*a*sqrt(-1/2*I*d)*d*weierstrassZeta(-4/3*(4*c^2 - 3*d ^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3 *I*d*sin(f*x + e) + 2*I*c)/d)) + (2*a*c - 3*a*d)*sqrt(1/2*I*d)*weierstrass PInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3 *d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (2*a*c - 3*a*d)*sqrt(-1 /2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9* I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d))/(d^2 *f)
\[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=a \left (\int \frac {\sin {\left (e + f x \right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx + \int \frac {1}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(1/2),x)
Output:
a*(Integral(sin(e + f*x)/sqrt(c + d*sin(e + f*x)), x) + Integral(1/sqrt(c + d*sin(e + f*x)), x))
\[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)
\[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)
Time = 17.00 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.28 \[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {a\,\left (2\,c\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,d}{c+d}\right )-2\,\left (c+d\right )\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,d}{c+d}\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {c+d\,\sin \left (e+f\,x\right )}{c+d}}}{d\,f\,\cos \left (e+f\,x\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}-\frac {2\,a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |\frac {2\,d}{c+d}\right )\,\sqrt {\frac {c+d\,\sin \left (e+f\,x\right )}{c+d}}}{f\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \] Input:
int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(1/2),x)
Output:
(a*(2*c*ellipticF(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), (2*d)/(c + d )) - 2*(c + d)*ellipticE(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), (2*d) /(c + d)))*(cos(e + f*x)^2)^(1/2)*((c + d*sin(e + f*x))/(c + d))^(1/2))/(d *f*cos(e + f*x)*(c + d*sin(e + f*x))^(1/2)) - (2*a*ellipticF(pi/4 - e/2 - (f*x)/2, (2*d)/(c + d))*((c + d*sin(e + f*x))/(c + d))^(1/2))/(f*(c + d*si n(e + f*x))^(1/2))
\[ \int \frac {a+a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=a \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right ) d +c}d x +\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right ) d +c}d x \right ) \] Input:
int((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x)
Output:
a*(int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)*d + c),x) + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)*d + c),x))