Integrand size = 25, antiderivative size = 356 \[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{3/2} \sqrt [4]{a^2-b^2} d}-\frac {b \sqrt {e} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{3/2} \sqrt [4]{a^2-b^2} d}-\frac {b^2 e \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^2 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {b^2 e \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^2 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}} \]
b*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))*e^(1/2)/a^( 3/2)/(a^2-b^2)^(1/4)/d-b*arctanh(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1 /4)/e^(1/2))*e^(1/2)/a^(3/2)/(a^2-b^2)^(1/4)/d+b^2*e*(sin(1/2*c+1/4*Pi+1/2 *d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d *x),2*a/(a-(a^2-b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/a^2/d/(a-(a^2-b^2)^( 1/2))/(e*sin(d*x+c))^(1/2)+b^2*e*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1 /2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a+(a^2-b^2) ^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/a^2/d/(a+(a^2-b^2)^(1/2))/(e*sin(d*x+c)) ^(1/2)-2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*Ell ipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/a/d/sin(d*x +c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\frac {\left (b+a \sqrt {\cos ^2(c+d x)}\right ) \sqrt {e \sin (c+d x)} \left (3 \sqrt {2} b \left (-a^2+b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )\right )+8 a^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{12 a^{3/2} \left (a^2-b^2\right ) d (b+a \cos (c+d x)) \sqrt {\sin (c+d x)}} \]
((b + a*Sqrt[Cos[c + d*x]^2])*Sqrt[e*Sin[c + d*x]]*(3*Sqrt[2]*b*(-a^2 + b^ 2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^( 1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/ 4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/ 4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2)))/(12*a^(3/2)*(a^2 - b^2)*d*(b + a*Cos[c + d*x])*Sqrt[Sin[c + d*x]])
Time = 1.62 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.98, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4360, 25, 25, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 25, 266, 827, 218, 221, 3042, 3286, 3042, 3284}
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.
| \(\displaystyle \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cos (c+d x) \sqrt {e \sin (c+d x)}}{-a \cos (c+d x)-b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cos (c+d x) \sqrt {e \sin (c+d x)}}{a \cos (c+d x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )}}{a \sin \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
| \(\Big \downarrow \) 3346 |
| \(\displaystyle \frac {\int \sqrt {e \sin (c+d x)}dx}{a}-\frac {b \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {e \sin (c+d x)}dx}{a}-\frac {b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{a}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{a \sqrt {\sin (c+d x)}}-\frac {b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{a \sqrt {\sin (c+d x)}}-\frac {b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{a}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{b-a \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{a}\) |
| \(\Big \downarrow \) 3180 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (-\frac {a e \int -\frac {\sqrt {e \sin (c+d x)}}{\left (a^2-b^2\right ) e^2-a^2 e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (\frac {a e \int \frac {\sqrt {e \sin (c+d x)}}{\left (a^2-b^2\right ) e^2-a^2 e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (\frac {2 a e \int \frac {e^2 \sin ^2(c+d x)}{\left (a^2-b^2\right ) e^2-a^2 e^4 \sin ^4(c+d x)}d\sqrt {e \sin (c+d x)}}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (\frac {2 a e \left (\frac {\int \frac {1}{\sqrt {a^2-b^2} e-a e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 a}-\frac {\int \frac {1}{a e^2 \sin ^2(c+d x)+\sqrt {a^2-b^2} e}d\sqrt {e \sin (c+d x)}}{2 a}\right )}{d}-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}\right )}{a}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}+\frac {2 a e \left (\frac {\int \frac {1}{\sqrt {a^2-b^2} e-a e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 a}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (-\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a}+\frac {b e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (-\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (-\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {b e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (a \sin (c+d x)+\sqrt {a^2-b^2}\right )}dx}{2 a \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )}{a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a d \sqrt {\sin (c+d x)}}-\frac {b \left (\frac {b e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \left (a-\sqrt {a^2-b^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {b e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a d \left (\sqrt {a^2-b^2}+a\right ) \sqrt {e \sin (c+d x)}}+\frac {2 a e \left (\frac {\text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}-\frac {\arctan \left (\frac {\sqrt {a} \sqrt {e} \sin (c+d x)}{\sqrt [4]{a^2-b^2}}\right )}{2 a^{3/2} \sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{d}\right )}{a}\) |
(2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a*d*Sqrt[Sin[c + d*x]]) - (b*((2*a*e*(-1/2*ArcTan[(Sqrt[a]*Sqrt[e]*Sin[c + d*x])/(a^2 - b ^2)^(1/4)]/(a^(3/2)*(a^2 - b^2)^(1/4)*Sqrt[e]) + ArcTanh[(Sqrt[a]*Sqrt[e]* Sin[c + d*x])/(a^2 - b^2)^(1/4)]/(2*a^(3/2)*(a^2 - b^2)^(1/4)*Sqrt[e])))/d + (b*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqr t[Sin[c + d*x]])/(a*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b*e*E llipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]])))/a
3.3.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(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), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ )]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b)) Int[1/(S qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b)) In t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f) Su bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / ; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* (x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b Int [(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b Int[(g*Cos[e + f*x])^p/( a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 6.26 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.23
| method | result | size |
| default | \(\frac {\frac {e b \left (2 \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {e \sin \left (d x +c \right )}+\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}{\sqrt {e \sin \left (d x +c \right )}-\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}\right )\right )}{2 a^{2} \left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}-\frac {e \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, b^{2} \left (\operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, -\frac {a}{\sqrt {a^{2}-b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) \sqrt {a^{2}-b^{2}}-\operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {a}{a +\sqrt {a^{2}-b^{2}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {a^{2}-b^{2}}-4 \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a +2 \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a +\operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, -\frac {a}{\sqrt {a^{2}-b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) a +\operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {a}{a +\sqrt {a^{2}-b^{2}}}, \frac {\sqrt {2}}{2}\right ) a \right )}{2 a^{2} \left (\sqrt {a^{2}-b^{2}}-a \right ) \left (a +\sqrt {a^{2}-b^{2}}\right ) \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(439\) |
(1/2*e*b/a^2/(e^2*(a^2-b^2)/a^2)^(1/4)*(2*arctan((e*sin(d*x+c))^(1/2)/(e^2 *(a^2-b^2)/a^2)^(1/4))-ln(((e*sin(d*x+c))^(1/2)+(e^2*(a^2-b^2)/a^2)^(1/4)) /((e*sin(d*x+c))^(1/2)-(e^2*(a^2-b^2)/a^2)^(1/4))))-1/2*e*(-sin(d*x+c)+1)^ (1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*b^2*(EllipticPi((-sin(d*x+c) +1)^(1/2),-a/((a^2-b^2)^(1/2)-a),1/2*2^(1/2))*(a^2-b^2)^(1/2)-EllipticPi(( -sin(d*x+c)+1)^(1/2),a/(a+(a^2-b^2)^(1/2)),1/2*2^(1/2))*(a^2-b^2)^(1/2)-4* EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a+2*EllipticF((-sin(d*x+c)+1) ^(1/2),1/2*2^(1/2))*a+EllipticPi((-sin(d*x+c)+1)^(1/2),-a/((a^2-b^2)^(1/2) -a),1/2*2^(1/2))*a+EllipticPi((-sin(d*x+c)+1)^(1/2),a/(a+(a^2-b^2)^(1/2)), 1/2*2^(1/2))*a)/a^2/((a^2-b^2)^(1/2)-a)/(a+(a^2-b^2)^(1/2))/cos(d*x+c)/(e* sin(d*x+c))^(1/2))/d
Timed out. \[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\int \frac {\sqrt {e \sin {\left (c + d x \right )}}}{a + b \sec {\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \sin \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \sin \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\sqrt {e\,\sin \left (c+d\,x\right )}}{b+a\,\cos \left (c+d\,x\right )} \,d x \]