\(\int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\) [1421]

Optimal result

Integrand size = 37, antiderivative size = 536 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {g^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b d^{3/2} f}+\frac {g^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b d^{3/2} f}-\frac {g^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\left (\sqrt {g}+\sqrt {g} \cot (e+f x)\right ) \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b d^{3/2} f}-\frac {2 g (g \cos (e+f x))^{3/2}}{a d f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a b d f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a b d f \sqrt {d \sin (e+f x)}}-\frac {2 g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a d^2 f \sqrt {\sin (2 e+2 f x)}} \] Output:

1/2*g^(5/2)*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/d^(3/2)/f+1/2*g^(5/2)*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/d^(3/2)/f-1/2*g 
^(5/2)*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/d^(3/2)/f-2*g*(g*cos(f*x+e))^(3/2)/ 
a/d/f/(d*sin(f*x+e))^(1/2)-2*2^(1/2)*(-a+b)^(1/2)*(a+b)^(1/2)*g^(5/2)*Elli 
pticPi((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)/a/b/d/f/(d*sin(f*x+e))^(1/2)+2*2^(1/2)*(-a+b) 
^(1/2)*(a+b)^(1/2)*g^(5/2)*EllipticPi((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)/a/b/d/f/(d*sin( 
f*x+e))^(1/2)+2*g^2*(g*cos(f*x+e))^(1/2)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/ 
2))*(d*sin(f*x+e))^(1/2)/a/d^2/f/sin(2*f*x+2*e)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 29.57 (sec) , antiderivative size = 1611, normalized size of antiderivative = 3.01 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \] Input:

Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f* 
x])),x]
 

Output:

(-2*(g*Cos[e + f*x])^(5/2)*Tan[e + f*x])/(a*f*(d*Sin[e + f*x])^(3/2)) + (( 
g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^(3/2)*((2*a*(-(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)])*Cos[e + 
 f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/((a^2 - b 
^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (b*Sqrt[Tan[e + f*x 
]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[ 
e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + 
 f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + 
 f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^ 
2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^ 
(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[ 
e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + 
f*x]^2]))/(6*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x] 
)*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Ta 
n[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4 
, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x 
]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, ( 
-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a 
^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTa...
 

Rubi [A] (warning: unable to verify)

Time = 2.81 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.11, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.649, Rules used = {3042, 3379, 3042, 3317, 3042, 3050, 3042, 3052, 3042, 3055, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 3119, 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.

\(\displaystyle \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))}dx\)

\(\Big \downarrow \) 3379

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \int \frac {\sqrt {g \cos (e+f x)} (b-a \sin (e+f x))}{(d \sin (e+f x))^{3/2}}dx}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \int \frac {\sqrt {g \cos (e+f x)} (b-a \sin (e+f x))}{(d \sin (e+f x))^{3/2}}dx}{a b}\)

\(\Big \downarrow \) 3317

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3050

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3052

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )-\frac {a \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3055

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \int \frac {g \cot (e+f x)}{d \left (\cot ^2(e+f x) g^2+g^2\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\int \frac {\cot (e+f x) g+g}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}+\frac {\int \frac {1}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {g \cot (e+f x)}{d}-1}d\left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\int \frac {g-g \cot (e+f x)}{\cot ^2(e+f x) g^2+g^2}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\sqrt {d} \left (\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {g}-\frac {2 \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}-\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 \sqrt {2} d \sqrt {g}}+\frac {\int \frac {\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{\frac {\cot (e+f x) g}{d}+\frac {g}{d}+\frac {\sqrt {2} \sqrt {g \cos (e+f x)} \sqrt {g}}{\sqrt {d} \sqrt {d \sin (e+f x)}}}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}}{2 d \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )+\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 3385

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d \sqrt {d \sin (e+f x)}}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a b d \sqrt {d \sin (e+f x)}}+\frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}\)

\(\Big \downarrow \) 3384

\(\displaystyle \frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}-\frac {4 \sqrt {2} g^3 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{a b d f \sqrt {d \sin (e+f x)}}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}-\frac {4 \sqrt {2} g^3 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{a b d f \sqrt {d \sin (e+f x)}}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {g^2 \left (\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}+g \cot (e+f x)+g\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 d}\right )}{f}+b \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )\right )}{a b}-\frac {4 \sqrt {2} g^3 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{a b d f \sqrt {d \sin (e+f x)}}\)

Input:

Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x 
]
 

Output:

(-4*Sqrt[2]*(a^2 - b^2)*g^3*(-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]*Sqrt[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]])/(a*b*d*f*Sqrt[d*Sin[e + f*x]]) 
 + (g^2*((2*a*g*((-(ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqr 
t[g]*Sqrt[d*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g])) + ArcTan[1 + (Sqrt[ 
2]*Sqrt[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 
]])/Sqrt[d*Sin[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*Sqrt[g]))/(2*d)))/f + b*((-2* 
(g*Cos[e + f*x])^(3/2))/(d*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[g*Cos[e + f 
*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(d^2*f*Sqrt[Sin[2* 
e + 2*f*x]]))))/(a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 826

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]]))
 

rule 993

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

rule 1542

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3050

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a 
*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1))   Int[(b*Cos[e + f*x])^ 
n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 
1] && IntegersQ[2*m, 2*n]
 

rule 3052

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]
 

rule 3055

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]
 

rule 3119

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]
 

rule 3317

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]
 

rule 3379

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/(a*b) 
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(b - a*Sin[e + f*x]), x], x 
] + Simp[g^2*((a^2 - b^2)/(a*b*d))   Int[(g*Cos[e + f*x])^(p - 2)*((d*Sin[e 
 + f*x])^(n + 1)/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, 
 x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LtQ[n, -1] 
 || (EqQ[p, 3/2] && EqQ[n, -2^(-1)]))
 

rule 3384

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]
 

rule 3385

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]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3083 vs. \(2 (433 ) = 866\).

Time = 4.97 (sec) , antiderivative size = 3084, normalized size of antiderivative = 5.75

method	result	size
default	\(\text {Expression too large to display}\)	\(3084\)

Input:

int((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x,method=_R 
ETURNVERBOSE)
 

Output:

-4/f*cot(f*x+e)^2/d/(d*sin(f*x+e))^(1/2)*(1-cos(f*x+e))*(csc(f*x+e)^2*(1-c 
os(f*x+e))^2+1)*g^2*(g*cos(f*x+e))^(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^3*(-a^2+b^2)^(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)+EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a 
^3*(-a^2+b^2)^(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)-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^3*(-a^2+b^2)^(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)*(-cs 
c(f*x+e)+cot(f*x+e))^(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^3*(-a^2+b^2)^(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)+I*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*(-a^2+b^2)^(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)-I*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*(-a^2+b^2)^(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)+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))*(-2*csc(f*x+e)+2*cot(f*x+e)+2)^(1/2)*(-csc(f 
*x+e)+cot(f*x+e))^(1/2)*(csc(f*x+e)-cot(f*x+e)+1)^(1/2)*a^2*b^2-2*(-2*c...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al 
gorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))**(5/2)/(d*sin(f*x+e))**(3/2)/(a+b*sin(f*x+e)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al 
gorithm="maxima")
 

Output:

integrate((g*cos(f*x + e))^(5/2)/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(3 
/2)), x)
 

Giac [F]

\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, al 
gorithm="giac")
 

Output:

integrate((g*cos(f*x + e))^(5/2)/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(3 
/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \] Input:

int((g*cos(e + f*x))^(5/2)/((d*sin(e + f*x))^(3/2)*(a + b*sin(e + f*x))),x 
)
 

Output:

int((g*cos(e + f*x))^(5/2)/((d*sin(e + f*x))^(3/2)*(a + b*sin(e + f*x))), 
x)
 

Reduce [F]

\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\frac {\sqrt {g}\, \sqrt {d}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3} b +\sin \left (f x +e \right )^{2} a}d x \right ) g^{2}}{d^{2}} \] Input:

int((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x)
 

Output:

(sqrt(g)*sqrt(d)*int((sqrt(sin(e + f*x))*sqrt(cos(e + f*x))*cos(e + f*x)** 
2)/(sin(e + f*x)**3*b + sin(e + f*x)**2*a),x)*g**2)/d**2