Integrand size = 37, antiderivative size = 208 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=-\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}}+\frac {2 \sqrt {2} \sqrt {g} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{\sqrt {-a+b} \sqrt {a+b} f \sqrt {d \cos (e+f x)}} \]
-2*EllipticPi((g*sin(f*x+e))^(1/2)/g^(1/2)/(1+cos(f*x+e))^(1/2),-(-a+b)^(1 /2)/(a+b)^(1/2),I)*2^(1/2)*g^(1/2)*cos(f*x+e)^(1/2)/f/(-a+b)^(1/2)/(a+b)^( 1/2)/(d*cos(f*x+e))^(1/2)+2*EllipticPi((g*sin(f*x+e))^(1/2)/g^(1/2)/(1+cos (f*x+e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*g^(1/2)*cos(f*x+e)^(1/2 )/f/(-a+b)^(1/2)/(a+b)^(1/2)/(d*cos(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 2.89 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.80 \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\frac {2 \left (b+a \sqrt {\sec ^2(e+f x)}\right ) \sqrt {g \sin (e+f x)} \left (\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )-\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+a \tan (e+f x)\right )}{4 \sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2}}+\frac {b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),-\frac {a^2 \tan ^2(e+f x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(e+f x)}{3 \left (-a^2+b^2\right )}\right )}{f \sqrt {d \cos (e+f x)} (a+b \cos (e+f x)) \sqrt {\sec ^2(e+f x)} \sqrt {\tan (e+f x)}} \]
(2*(b + a*Sqrt[Sec[e + f*x]^2])*Sqrt[g*Sin[e + f*x]]*((-2*ArcTan[1 - (Sqrt [2]*Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2] *Sqrt[a]*Sqrt[Tan[e + f*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - Sq rt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]] - Log [Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + a*Tan[e + f*x]])/(4*Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, -((a^2*Tan[e + f*x]^2)/(a^2 - b^2))]*Tan[e + f*x]^(3/2))/(3*(-a^2 + b^2))))/(f*Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x ])*Sqrt[Sec[e + f*x]^2]*Sqrt[Tan[e + f*x]])
Time = 0.70 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {3042, 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.
| \(\displaystyle \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {d \sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3385 |
| \(\displaystyle \frac {\sqrt {\cos (e+f x)} \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \cos (e+f x))}dx}{\sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (e+f x)} \int \frac {\sqrt {-g \cos \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )} \left (a+b \sin \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{\sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 3384 |
| \(\displaystyle \frac {4 \sqrt {2} g \sqrt {\cos (e+f x)} \int \frac {g \sin (e+f x)}{(\cos (e+f x)+1) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}} \left (\frac {(a-b) \sin ^2(e+f x) g^2}{(\cos (e+f x)+1)^2}+(a+b) g^2\right )}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f \sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {4 \sqrt {2} g \sqrt {\cos (e+f x)} \left (\frac {\int \frac {1}{\left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \sin (e+f x)}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\left (\frac {\sqrt {b-a} \sin (e+f x) g}{\cos (e+f x)+1}+\sqrt {a+b} g\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {g \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{f \sqrt {d \cos (e+f x)}}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {4 \sqrt {2} g \sqrt {\cos (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{f \sqrt {d \cos (e+f x)}}\) |
(4*Sqrt[2]*g*Sqrt[Cos[e + f*x]]*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b ]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1]/(Sq rt[-a + b]*Sqrt[a + b]*Sqrt[g]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b], Arc Sin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1]/(2*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[g])))/(f*Sqrt[d*Cos[e + f*x]])
3.1.3.3.1 Defintions of rubi rules used
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[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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(164)=328\).
Time = 4.15 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.23
| method | result | size |
| default | \(-\frac {\sqrt {g \sin \left (f x +e \right )}\, \left (\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -a \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b \right ) \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \sqrt {2}}{f \sqrt {\cos \left (f x +e \right ) d}\, \left (b +\sqrt {-a^{2}+b^{2}}-a \right ) \left (-b +\sqrt {-a^{2}+b^{2}}+a \right )}\) | \(464\) |
-1/f*(g*sin(f*x+e))^(1/2)*((-a^2+b^2)^(1/2)*EllipticPi((-cot(f*x+e)+csc(f* x+e)+1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))-(-a^2+b^2)^(1/ 2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b)) ^(1/2)),1/2*2^(1/2))-a*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),(a-b)/( a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))+EllipticPi((-cot(f*x+e)+csc(f*x+e)+ 1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b-a*EllipticPi((-co t(f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2 ))+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b)) ^(1/2)),1/2*2^(1/2))*b)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f* x+e)+1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)/(cos(f*x+e)*d)^(1/2)*(cot(f *x+e)+csc(f*x+e))*2^(1/2)/(b+(-a^2+b^2)^(1/2)-a)/(-b+(-a^2+b^2)^(1/2)+a)
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {d \cos {\left (e + f x \right )}} \left (a + b \cos {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \cos \left (f x + e\right )}} \,d x } \]
\[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int { \frac {\sqrt {g \sin \left (f x + e\right )}}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \cos \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {d \cos (e+f x)} (a+b \cos (e+f x))} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\sqrt {d\,\cos \left (e+f\,x\right )}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \]