Integrand size = 37, antiderivative size = 428 \[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {\sqrt {d} \sqrt {g} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b f}-\frac {\sqrt {d} \sqrt {g} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b f}+\frac {\sqrt {d} \sqrt {g} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\left (\sqrt {g}+\sqrt {g} \cot (e+f x)\right ) \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b f}-\frac {2 \sqrt {2} a d \sqrt {g} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a d \sqrt {g} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b \sqrt {-a+b} \sqrt {a+b} f \sqrt {d \sin (e+f x)}} \] Output:
-1/2*d^(1/2)*g^(1/2)*arctan(-1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2 )/(d*sin(f*x+e))^(1/2))*2^(1/2)/b/f-1/2*d^(1/2)*g^(1/2)*arctan(1+2^(1/2)*d ^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f*x+e))^(1/2))*2^(1/2)/b/f+1/2* d^(1/2)*g^(1/2)*arctanh(2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/(g^(1/2)+g^(1 /2)*cot(f*x+e))/(d*sin(f*x+e))^(1/2))*2^(1/2)/b/f-2*2^(1/2)*a*d*g^(1/2)*El lipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/( a+b)^(1/2),I)*sin(f*x+e)^(1/2)/b/(-a+b)^(1/2)/(a+b)^(1/2)/f/(d*sin(f*x+e)) ^(1/2)+2*2^(1/2)*a*d*g^(1/2)*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+si n(f*x+e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*sin(f*x+e)^(1/2)/b/(-a+b)^(1/2 )/(a+b)^(1/2)/f/(d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 12.41 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 \left (b \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )-a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} \left (a+b \sqrt {\sin ^2(e+f x)}\right )}{3 \left (a^2-b^2\right ) d f g \sin ^2(e+f x)^{3/4} (a+b \sin (e+f x))} \] Input:
Integrate[(Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]) ,x]
Output:
(2*(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a ^2 + b^2)] - a*AppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x ]^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*(a + b*S qrt[Sin[e + f*x]^2]))/(3*(a^2 - b^2)*d*f*g*(Sin[e + f*x]^2)^(3/4)*(a + b*S in[e + f*x]))
Time = 1.46 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {3042, 3388, 3042, 3055, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 3385, 3042, 3384, 993, 1542}
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.
| \(\displaystyle \int \frac {\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3388 |
| \(\displaystyle \frac {d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{b}-\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}\) |
| \(\Big \downarrow \) 3055 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \int \frac {g \cot (e+f x)}{d \left (\cot ^2(e+f x) g^2+g^2\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{b f}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\int \frac {\cot (e+f x) g+g}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}+\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} d \sqrt {g}}+\frac {\int \frac {\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {a d \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{b}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 3385 |
| \(\displaystyle -\frac {a d \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{b \sqrt {d \sin (e+f x)}}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a d \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{b \sqrt {d \sin (e+f x)}}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 3384 |
| \(\displaystyle \frac {4 \sqrt {2} a d g \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{b f \sqrt {d \sin (e+f x)}}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {4 \sqrt {2} a d g \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{b f \sqrt {d \sin (e+f x)}}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {4 \sqrt {2} a d g \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{b f \sqrt {d \sin (e+f x)}}-\frac {2 d^2 g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{b f}\) |
Input:
Int[(Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(a + b*Sin[e + f*x]),x]
Output:
(-2*d^2*g*((-(ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*S qrt[d*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g])) + ArcTan[1 + (Sqrt[2]*Sqr t[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[d ]*Sqrt[g]))/(2*d) - (-1/2*Log[g + g*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g ]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) + Log[g + g*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g]*Sqrt[g*Cos[e + f*x]])/Sq rt[d*Sin[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*Sqrt[g]))/(2*d)))/(b*f) + (4*Sqrt[2 ]*a*d*g*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(Sqrt[-a + b]*Sqrt[a + b]*S qrt[g]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]] /(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(2*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[g ]))*Sqrt[Sin[e + f*x]])/(b*f*Sqrt[d*Sin[e + f*x]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> With[{k = Denominator[m]}, Simp[(-k)*a*(b/f) Subst[Int[x ^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[ e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-4*Sqrt[2]*(g/f) S ubst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g *Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]] *((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Sin[e + f* x]]/Sqrt[d*Sin[e + f*x]] Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2 , 0]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int[(g *Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a*(d/b) Int[(g*C os[e + f*x])^p*((d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x])), x], x] /; F reeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (335 ) = 670\).
Time = 4.26 (sec) , antiderivative size = 855, normalized size of antiderivative = 2.00
| method | result | size |
| default | \(\frac {\left (-\frac {1}{2}+\frac {i}{2}\right ) \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \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 )}\, \left (a -b \right ) \left (2 i \sqrt {-a^{2}+b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \sqrt {-a^{2}+b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right )-i \sqrt {-a^{2}+b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) a +i \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) b -i \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) a -i \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) b +2 \sqrt {-a^{2}+b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-\sqrt {-a^{2}+b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right )-\sqrt {-a^{2}+b^{2}}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) a +\operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) b -\operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) a -\operatorname {EllipticPi}\left (\sqrt {\csc \left (f x +e \right )-\cot \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) b \right ) \left (\csc \left (f x +e \right )+\sec \left (f x +e \right ) \csc \left (f x +e \right )\right ) a}{f b \sqrt {-a^{2}+b^{2}}\, \left (-b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (b +\sqrt {-a^{2}+b^{2}}-a \right )}\) | \(855\) |
Input:
int((g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x,method=_R ETURNVERBOSE)
Output:
(-1/2+1/2*I)/f*(d*sin(f*x+e))^(1/2)*(g*cos(f*x+e))^(1/2)*(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)*(2*I*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^ (1/2),1/2-1/2*I,1/2*2^(1/2))-I*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot (f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))-I*(-a^2+b^2)^(1/2) *EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2* 2^(1/2))+I*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/ 2)+a),1/2*2^(1/2))*a+I*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(- a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b-I*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1 /2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a-I*EllipticPi((csc(f*x+e)-cot( f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b+2*(-a^2+b^2)^(1/2 )*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-(-a^2+ b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/ 2)+a),1/2*2^(1/2))-(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^( 1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))+EllipticPi((csc(f*x+e)-cot(f*x +e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a+EllipticPi((csc(f*x+ e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b-EllipticPi ((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a- EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2 ^(1/2))*b)*(csc(f*x+e)+sec(f*x+e)*csc(f*x+e))*a/b/(-a^2+b^2)^(1/2)/(-b+...
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, al gorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {d \sin {\left (e + f x \right )}} \sqrt {g \cos {\left (e + f x \right )}}}{a + b \sin {\left (e + f x \right )}}\, dx \] Input:
integrate((g*cos(f*x+e))**(1/2)*(d*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e)),x)
Output:
Integral(sqrt(d*sin(e + f*x))*sqrt(g*cos(e + f*x))/(a + b*sin(e + f*x)), x )
\[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate((g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, al gorithm="maxima")
Output:
integrate(sqrt(g*cos(f*x + e))*sqrt(d*sin(f*x + e))/(b*sin(f*x + e) + a), x)
\[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a} \,d x } \] Input:
integrate((g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, al gorithm="giac")
Output:
integrate(sqrt(g*cos(f*x + e))*sqrt(d*sin(f*x + e))/(b*sin(f*x + e) + a), x)
Timed out. \[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}\,\sqrt {d\,\sin \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \] Input:
int(((g*cos(e + f*x))^(1/2)*(d*sin(e + f*x))^(1/2))/(a + b*sin(e + f*x)),x )
Output:
int(((g*cos(e + f*x))^(1/2)*(d*sin(e + f*x))^(1/2))/(a + b*sin(e + f*x)), x)
\[ \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}}{\sin \left (f x +e \right ) b +a}d x \right ) \] Input:
int((g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x)
Output:
sqrt(g)*sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x)))/(sin(e + f*x)* b + a),x)