Integrand size = 19, antiderivative size = 47 \[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \] Output:
-EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))*sin(b*x+a)/b/sin(2*b*x+2*a)^(1/2)/(d *tan(b*x+a))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.28 \[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)} \sin (a+b x) \sqrt {d \tan (a+b x)}}{3 b d} \] Input:
Integrate[Sin[a + b*x]/Sqrt[d*Tan[a + b*x]],x]
Output:
(2*Hypergeometric2F1[3/4, 3/2, 7/4, -Tan[a + b*x]^2]*Sqrt[Sec[a + b*x]^2]* Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/(3*b*d)
Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 3081, 3042, 3052, 3042, 3119}
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.
\(\displaystyle \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}}dx\) |
\(\Big \downarrow \) 3081 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle \frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (a+b x) \int \sqrt {\sin (2 a+2 b x)}dx}{\sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\) |
Input:
Int[Sin[a + b*x]/Sqrt[d*Tan[a + b*x]],x]
Output:
(EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt [d*Tan[a + b*x]])
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]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ n) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 1)]) || IntegersQ[m - 1/2, n - 1/2])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(206\) vs. \(2(45)=90\).
Time = 0.75 (sec) , antiderivative size = 207, normalized size of antiderivative = 4.40
method | result | size |
default | \(\frac {-\frac {\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (2+2 \sec \left (b x +a \right )\right )}{2}-\frac {\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-1-\sec \left (b x +a \right )\right )}{2}+1-\cos \left (b x +a \right )}{b \sqrt {d \tan \left (b x +a \right )}}\) | \(207\) |
Input:
int(sin(b*x+a)/(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/b*(-1/2*(csc(b*x+a)-cot(b*x+a)+1)^(1/2)*(-2*csc(b*x+a)+2*cot(b*x+a)+2)^( 1/2)*(-csc(b*x+a)+cot(b*x+a))^(1/2)*EllipticE((csc(b*x+a)-cot(b*x+a)+1)^(1 /2),1/2*2^(1/2))*(2+2*sec(b*x+a))-1/2*(csc(b*x+a)-cot(b*x+a)+1)^(1/2)*(-2* csc(b*x+a)+2*cot(b*x+a)+2)^(1/2)*(-csc(b*x+a)+cot(b*x+a))^(1/2)*EllipticF( (csc(b*x+a)-cot(b*x+a)+1)^(1/2),1/2*2^(1/2))*(-1-sec(b*x+a))+1-cos(b*x+a)) /(d*tan(b*x+a))^(1/2)
\[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\sqrt {d \tan \left (b x + a\right )}} \,d x } \] Input:
integrate(sin(b*x+a)/(d*tan(b*x+a))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(d*tan(b*x + a))*sin(b*x + a)/(d*tan(b*x + a)), x)
\[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {\sin {\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \] Input:
integrate(sin(b*x+a)/(d*tan(b*x+a))**(1/2),x)
Output:
Integral(sin(a + b*x)/sqrt(d*tan(a + b*x)), x)
\[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\sqrt {d \tan \left (b x + a\right )}} \,d x } \] Input:
integrate(sin(b*x+a)/(d*tan(b*x+a))^(1/2),x, algorithm="maxima")
Output:
integrate(sin(b*x + a)/sqrt(d*tan(b*x + a)), x)
\[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\sqrt {d \tan \left (b x + a\right )}} \,d x } \] Input:
integrate(sin(b*x+a)/(d*tan(b*x+a))^(1/2),x, algorithm="giac")
Output:
integrate(sin(b*x + a)/sqrt(d*tan(b*x + a)), x)
Timed out. \[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {\sin \left (a+b\,x\right )}{\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \,d x \] Input:
int(sin(a + b*x)/(d*tan(a + b*x))^(1/2),x)
Output:
int(sin(a + b*x)/(d*tan(a + b*x))^(1/2), x)
\[ \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\tan \left (b x +a \right )}\, \sin \left (b x +a \right )}{\tan \left (b x +a \right )}d x \right )}{d} \] Input:
int(sin(b*x+a)/(d*tan(b*x+a))^(1/2),x)
Output:
(sqrt(d)*int((sqrt(tan(a + b*x))*sin(a + b*x))/tan(a + b*x),x))/d