Integrand size = 25, antiderivative size = 204 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=-\frac {2 \sqrt {2} \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{\sqrt {-a+b} \sqrt {a+b} d \sqrt {\sin (c+d x)}}+\frac {2 \sqrt {2} \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{\sqrt {-a+b} \sqrt {a+b} d \sqrt {\sin (c+d x)}} \]
-2*EllipticPi(sin(d*x+c)^(1/2)/(1+cos(d*x+c))^(1/2),-(-a+b)^(1/2)/(a+b)^(1 /2),I)*2^(1/2)*cos(d*x+c)^(1/2)*(e*tan(d*x+c))^(1/2)/d/(-a+b)^(1/2)/(a+b)^ (1/2)/sin(d*x+c)^(1/2)+2*EllipticPi(sin(d*x+c)^(1/2)/(1+cos(d*x+c))^(1/2), (-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*cos(d*x+c)^(1/2)*(e*tan(d*x+c))^(1/2)/ d/(-a+b)^(1/2)/(a+b)^(1/2)/sin(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 14.92 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=\frac {2 \left (b+a \sqrt {\sec ^2(c+d x)}\right ) \sqrt {e \tan (c+d x)} \left (\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+a \tan (c+d x)\right )-\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+a \tan (c+d x)\right )}{4 \sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2}}+\frac {b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),-\frac {a^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 \left (-a^2+b^2\right )}\right )}{d (a+b \cos (c+d x)) \sqrt {\sec ^2(c+d x)} \sqrt {\tan (c+d x)}} \]
(2*(b + a*Sqrt[Sec[c + d*x]^2])*Sqrt[e*Tan[c + d*x]]*((-2*ArcTan[1 - (Sqrt [2]*Sqrt[a]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2] *Sqrt[a]*Sqrt[Tan[c + d*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[c + d*x]] + a*Tan[c + d*x]] - Log [Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + a*Tan[c + d*x]])/(4*Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Tan[c + d*x]^(3/2))/(3*(-a^2 + b^2))))/(d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]^ 2]*Sqrt[Tan[c + d*x]])
Time = 0.93 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3212, 3042, 3209, 3042, 3385, 3042, 3384, 993, 1537, 412}
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.
\(\displaystyle \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3212 |
\(\displaystyle \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \cot (c+d x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {e \tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {1}{\left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right ) \sqrt {-e \tan \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3209 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {-\cos (c+d x)} (a+b \cos (c+d x))}dx}{\sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{\sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3385 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{\sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{\sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3384 |
\(\displaystyle \frac {4 \sqrt {2} \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \frac {\sin (c+d x)}{(\cos (c+d x)+1) \sqrt {1-\frac {\sin ^2(c+d x)}{(\cos (c+d x)+1)^2}} \left (\frac {(a-b) \sin ^2(c+d x)}{(\cos (c+d x)+1)^2}+a+b\right )}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 993 |
\(\displaystyle \frac {4 \sqrt {2} \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\int \frac {1}{\left (\sqrt {a+b}-\frac {\sqrt {b-a} \sin (c+d x)}{\cos (c+d x)+1}\right ) \sqrt {1-\frac {\sin ^2(c+d x)}{(\cos (c+d x)+1)^2}}}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\left (\frac {\sqrt {b-a} \sin (c+d x)}{\cos (c+d x)+1}+\sqrt {a+b}\right ) \sqrt {1-\frac {\sin ^2(c+d x)}{(\cos (c+d x)+1)^2}}}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {b-a}}\right )}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 1537 |
\(\displaystyle \frac {4 \sqrt {2} \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\sin (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {\sin (c+d x)}{\cos (c+d x)+1}+1} \left (\sqrt {a+b}-\frac {\sqrt {b-a} \sin (c+d x)}{\cos (c+d x)+1}\right )}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\sin (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {\sin (c+d x)}{\cos (c+d x)+1}+1} \left (\frac {\sqrt {b-a} \sin (c+d x)}{\cos (c+d x)+1}+\sqrt {a+b}\right )}d\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}}{2 \sqrt {b-a}}\right )}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {4 \sqrt {2} \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{2 \sqrt {b-a} \sqrt {a+b}}\right )}{d \sqrt {\sin (c+d x)}}\) |
(4*Sqrt[2]*Sqrt[Cos[c + d*x]]*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]) , ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]/(Sqrt[-a + b]*Sqr t[a + b]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]] /Sqrt[1 + Cos[c + d*x]]], -1]/(2*Sqrt[-a + b]*Sqrt[a + b]))*Sqrt[e*Tan[c + d*x]])/(d*Sqrt[Sin[c + d*x]])
3.1.21.3.1 Defintions of rubi rules used
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
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[(-a)*c, 2]}, Simp[Sqrt[-c] Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqr t[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] & & GtQ[a, 0] && LtQ[c, 0]
Int[1/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(g_)*tan[(e_.) + (f_.)*( x_)]]), x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[g*Tan [e + f*x]]) Int[Sqrt[Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x ])), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.), x_Symbol] :> Simp[g^(2*IntPart[p])*(g*Cot[e + f*x])^FracPart[p] *(g*Tan[e + f*x])^FracPart[p] Int[(a + b*Sin[e + f*x])^m/(g*Tan[e + f*x]) ^p, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
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. \(453\) vs. \(2(164)=328\).
Time = 3.76 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.23
method | result | size |
default | \(-\frac {\left (\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {-a^{2}+b^{2}}\, \Pi \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )-a \Pi \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {a -b}{a -b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b -a \Pi \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right )+\Pi \left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, -\frac {a -b}{-a +b +\sqrt {-\left (a -b \right ) \left (a +b \right )}}, \frac {\sqrt {2}}{2}\right ) b \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {e \tan \left (d x +c \right )}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {2}}{d \left (b +\sqrt {-a^{2}+b^{2}}-a \right ) \left (-b +\sqrt {-a^{2}+b^{2}}+a \right )}\) | \(454\) |
-1/d*((-a^2+b^2)^(1/2)*EllipticPi((-cot(d*x+c)+csc(d*x+c)+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(d *x+c)+csc(d*x+c)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))- a*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1 /2)),1/2*2^(1/2))+EllipticPi((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),(a-b)/(a-b+( -(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b-a*EllipticPi((-cot(d*x+c)+csc(d*x+c)+1 )^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))+EllipticPi((-cot(d *x+c)+csc(d*x+c)+1)^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))* b)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(-cot(d*x +c)+csc(d*x+c)+1)^(1/2)*(e*tan(d*x+c))^(1/2)*(cot(d*x+c)+csc(d*x+c))*2^(1/ 2)/(b+(-a^2+b^2)^(1/2)-a)/(-b+(-a^2+b^2)^(1/2)+a)
Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=\int \frac {\sqrt {e \tan {\left (c + d x \right )}}}{a + b \cos {\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{b \cos \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{b \cos \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx=\int \frac {\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}}{a+b\,\cos \left (c+d\,x\right )} \,d x \]