\(\int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx\) [60]

Optimal result

Integrand size = 19, antiderivative size = 75 \[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}+\frac {\csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{2 b} \] Output:

-d*sin(b*x+a)/b/(d*tan(b*x+a))^(1/2)+1/2*csc(b*x+a)*InverseJacobiAM(a-1/4* 
Pi+b*x,2^(1/2))*sin(2*b*x+2*a)^(1/2)*(d*tan(b*x+a))^(1/2)/b
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 3.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {\cos (a+b x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sqrt {d \tan (a+b x)}}{b} \] Input:

Integrate[Sin[a + b*x]*Sqrt[d*Tan[a + b*x]],x]
 

Output:

(Cos[a + b*x]*(-1 + Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[a + b*x]^2]*Sqrt 
[Sec[a + b*x]^2])*Sqrt[d*Tan[a + b*x]])/b
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3078, 3042, 3081, 3042, 3053, 3042, 3120}

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.

\(\displaystyle \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (a+b x) \sqrt {d \tan (a+b x)}dx\)

\(\Big \downarrow \) 3078

\(\displaystyle \frac {1}{2} \int \csc (a+b x) \sqrt {d \tan (a+b x)}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {d \tan (a+b x)}}{\sin (a+b x)}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3081

\(\displaystyle \frac {\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{2 \sqrt {\sin (a+b x)}}-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}}dx}{2 \sqrt {\sin (a+b x)}}-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3053

\(\displaystyle \frac {1}{2} \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \sqrt {\sin (2 a+2 b x)} \csc (a+b x) \sqrt {d \tan (a+b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{2 b}-\frac {d \sin (a+b x)}{b \sqrt {d \tan (a+b x)}}\)

Input:

Int[Sin[a + b*x]*Sqrt[d*Tan[a + b*x]],x]
 

Output:

-((d*Sin[a + b*x])/(b*Sqrt[d*Tan[a + b*x]])) + (Csc[a + b*x]*EllipticF[a - 
 Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(2*b)
 

Defintions of rubi rules used

rule 3042

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

rule 3053

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ 
)]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b 
*Cos[e + f*x]])   Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f 
}, x]
 

rule 3078

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/( 
f*m)), x] + Simp[a^2*((m + n - 1)/m)   Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ 
e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1 
] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
 

rule 3081

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])
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 2.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63

method	result	size
default	\(\frac {\left (-\frac {\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (b x +a \right )-\csc \left (b x +a \right )\right )}{2}-\cos \left (b x +a \right )\right ) \sqrt {d \tan \left (b x +a \right )}}{b}\)	\(122\)

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)*EllipticF((csc(b*x+a)-cot(b*x+a)+1)^(1 
/2),1/2*2^(1/2))*(-cot(b*x+a)-csc(b*x+a))-cos(b*x+a))*(d*tan(b*x+a))^(1/2)
 

Fricas [F]

\[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right ) \,d x } \] 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), x)
 

Sympy [F]

\[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \sqrt {d \tan {\left (a + b x \right )}} \sin {\left (a + b x \right )}\, dx \] Input:

integrate(sin(b*x+a)*(d*tan(b*x+a))**(1/2),x)
 

Output:

Integral(sqrt(d*tan(a + b*x))*sin(a + b*x), x)
 

Maxima [F]

\[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(sin(b*x+a)*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*tan(b*x + a))*sin(b*x + a), x)
 

Giac [F]

\[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right ) \,d x } \] Input:

integrate(sin(b*x+a)*(d*tan(b*x+a))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*tan(b*x + a))*sin(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \sin \left (a+b\,x\right )\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )} \,d x \] 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)
 

Reduce [F]

\[ \int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (b x +a \right )}\, \sin \left (b x +a \right )d x \right ) \] 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),x)