\(\int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\) [100]

Optimal result

Integrand size = 19, antiderivative size = 79 \[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sin (a+b x)}{b d \sqrt {d \tan (a+b x)}}+\frac {\operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b d \sqrt {d \tan (a+b x)}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.59 \[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\cos (2 (a+b x)) \sec (a+b x) \left (\sqrt [4]{-1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sec ^2(a+b x)-\sqrt {\sec ^2(a+b x)} \sqrt {\tan (a+b x)}\right ) \tan ^{\frac {3}{2}}(a+b x)}{b \sqrt {\sec ^2(a+b x)} (d \tan (a+b x))^{3/2} \left (-1+\tan ^2(a+b x)\right )} \] Input:

Integrate[Sin[a + b*x]/(d*Tan[a + b*x])^(3/2),x]
 

Output:

(Cos[2*(a + b*x)]*Sec[a + b*x]*((-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)* 
Sqrt[Tan[a + b*x]]], -1]*Sec[a + b*x]^2 - Sqrt[Sec[a + b*x]^2]*Sqrt[Tan[a 
+ b*x]])*Tan[a + b*x]^(3/2))/(b*Sqrt[Sec[a + b*x]^2]*(d*Tan[a + b*x])^(3/2 
)*(-1 + Tan[a + b*x]^2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.43, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3042, 3082, 3042, 3049, 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.

\(\displaystyle \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}}dx\)

\(\Big \downarrow \) 3082

\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sqrt {\sin (a+b x)}}dx}{d \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\sin (a+b x)} \int \frac {\cos (a+b x)^{3/2}}{\sqrt {\sin (a+b x)}}dx}{d \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\)

\(\Big \downarrow \) 3049

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3053

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

Input:

Int[Sin[a + b*x]/(d*Tan[a + b*x])^(3/2),x]
 

Output:

(Sqrt[Sin[a + b*x]]*((Sqrt[Cos[a + b*x]]*Sqrt[Sin[a + b*x]])/b + (Elliptic 
F[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*Sqrt[Cos[a + b*x]]*Sqrt[ 
Sin[a + b*x]])))/(d*Sqrt[Cos[a + b*x]]*Sqrt[d*Tan[a + b*x]])
 

Defintions of rubi rules used

rule 3042

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

rule 3049

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[a*(b*Sin[e + f*x])^(n + 1)*((a*Cos[e + f*x])^(m - 1)/ 
(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n))   Int[(b*Sin[e + f*x])^n*(a 
*Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && 
 NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
 

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 3082

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[a*Cos[e + f*x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b* 
(a*Sin[e + f*x])^(n + 1)))   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]
 

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 = 2.74 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {\frac {\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}\, \sqrt {-2 \csc \left (b x +a \right )+2 \cot \left (b x +a \right )+2}\, \sqrt {-\csc \left (b x +a \right )+\cot \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (b x +a \right )-\cot \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (1+\sec \left (b x +a \right )\right )}{2}+\sin \left (b x +a \right )}{b \sqrt {d \tan \left (b x +a \right )}\, d}\)	\(114\)

Input:

int(sin(b*x+a)/(d*tan(b*x+a))^(3/2),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))*(1+sec(b*x+a))+sin(b*x+a))/(d*tan(b*x+a))^(1/2)/d
 

Fricas [F]

\[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(sin(b*x+a)/(d*tan(b*x+a))^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(d*tan(b*x + a))*sin(b*x + a)/(d^2*tan(b*x + a)^2), x)
 

Sympy [F]

\[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {\sin {\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(sin(b*x+a)/(d*tan(b*x+a))**(3/2),x)
 

Output:

Integral(sin(a + b*x)/(d*tan(a + b*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(sin(b*x+a)/(d*tan(b*x+a))^(3/2),x, algorithm="maxima")
 

Output:

integrate(sin(b*x + a)/(d*tan(b*x + a))^(3/2), x)
 

Giac [F]

\[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(sin(b*x+a)/(d*tan(b*x+a))^(3/2),x, algorithm="giac")
 

Output:

integrate(sin(b*x + a)/(d*tan(b*x + a))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {\sin \left (a+b\,x\right )}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \] Input:

int(sin(a + b*x)/(d*tan(a + b*x))^(3/2),x)
 

Output:

int(sin(a + b*x)/(d*tan(a + b*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\sin (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\tan \left (b x +a \right )}\, \sin \left (b x +a \right )}{\tan \left (b x +a \right )^{2}}d x \right )}{d^{2}} \] Input:

int(sin(b*x+a)/(d*tan(b*x+a))^(3/2),x)
 

Output:

(sqrt(d)*int((sqrt(tan(a + b*x))*sin(a + b*x))/tan(a + b*x)**2,x))/d**2