3.9.16 \(\int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx\) [816]

3.9.16.1 Optimal result

Integrand size = 15, antiderivative size = 140 \[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=\sqrt {1-x} \cos (x)-\sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\sqrt {2 \pi } \cos (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1) \]

output

-1/2*cos(1)*FresnelC(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))*2^(1/2)*Pi^(1/2)-1/2*Fr 
esnelS(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))*sin(1)*2^(1/2)*Pi^(1/2)+cos(1)*Fresne 
lS(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))*2^(1/2)*Pi^(1/2)-FresnelC(2^(1/2)/Pi^(1/2 
)*(1-x)^(1/2))*sin(1)*2^(1/2)*Pi^(1/2)+cos(x)*(1-x)^(1/2)
 

3.9.16.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt {1+x} \left ((-2-i) \sqrt {2 \pi } \sqrt {-1+x} \text {erfi}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (\cos (1)+i \sin (1))-(2-2 i) (-1+x) (\cos (x)+i \sin (x))+\left ((-2+2 i) (-1+x) (\cos (1)+i \sin (1))+(1+2 i) \sqrt {2 \pi } \sqrt {-1+x} \text {erf}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (-i \cos (x)+\sin (x))\right ) (\cos (1+x)-i \sin (1+x))\right )}{\sqrt {1-x^2}} \]

input

Integrate[(E^ArcTanh[x]*x*Sin[x])/Sqrt[1 + x],x]
 

output

((1/8 + I/8)*Sqrt[1 + x]*((-2 - I)*Sqrt[2*Pi]*Sqrt[-1 + x]*Erfi[((1 + I)*S 
qrt[-1 + x])/Sqrt[2]]*(Cos[1] + I*Sin[1]) - (2 - 2*I)*(-1 + x)*(Cos[x] + I 
*Sin[x]) + ((-2 + 2*I)*(-1 + x)*(Cos[1] + I*Sin[1]) + (1 + 2*I)*Sqrt[2*Pi] 
*Sqrt[-1 + x]*Erf[((1 + I)*Sqrt[-1 + x])/Sqrt[2]]*((-I)*Cos[x] + Sin[x]))* 
(Cos[1 + x] - I*Sin[1 + x])))/Sqrt[1 - x^2]
 

3.9.16.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6679, 7267, 25, 7293, 2009}

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.

\(\displaystyle \int \frac {x e^{\text {arctanh}(x)} \sin (x)}{\sqrt {x+1}} \, dx\)

\(\Big \downarrow \) 6679

\(\displaystyle \int \frac {x \sin (x)}{\sqrt {1-x}}dx\)

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int -x \sin (x)d\sqrt {1-x}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int x \sin (x)d\sqrt {1-x}\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int (\sin (x)-(1-x) \sin (x))d\sqrt {1-x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\sqrt {\frac {\pi }{2}} \sin (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sin (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {1}{2} \sqrt {1-x} \cos (x)\right )\)

input

Int[(E^ArcTanh[x]*x*Sin[x])/Sqrt[1 + x],x]
 

output

2*((Sqrt[1 - x]*Cos[x])/2 - (Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 
- x]])/2 + Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] - Sqrt[Pi/2] 
*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi] 
*Sqrt[1 - x]]*Sin[1])/2)
 

3.9.16.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6679

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol 
] :> Simp[c^p   Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] 
, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] 
|| GtQ[c, 0])
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.9.16.4 Maple [F]

\[\int \frac {\sqrt {1+x}\, x \sin \left (x \right )}{\sqrt {-x^{2}+1}}d x\]

input

int((1+x)^(1/2)/(-x^2+1)^(1/2)*x*sin(x),x)
 

output

int((1+x)^(1/2)/(-x^2+1)^(1/2)*x*sin(x),x)
 

3.9.16.5 Fricas [F]

\[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {x + 1} x \sin \left (x\right )}{\sqrt {-x^{2} + 1}} \,d x } \]

input

integrate((1+x)^(1/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="fricas")
 

output

integral(-sqrt(-x^2 + 1)*sqrt(x + 1)*x*sin(x)/(x^2 - 1), x)
 

3.9.16.6 Sympy [F]

\[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=\int \frac {x \sqrt {x + 1} \sin {\left (x \right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

input

integrate((1+x)**(1/2)/(-x**2+1)**(1/2)*x*sin(x),x)
 

output

Integral(x*sqrt(x + 1)*sin(x)/sqrt(-(x - 1)*(x + 1)), x)
 

3.9.16.7 Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.48 \[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=-\frac {{\left ({\left ({\left ({\left (-i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right ) - {\left ({\left (\cos \left (1\right ) - i \, \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (\cos \left (1\right ) + i \, \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} x + {\left ({\left ({\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \cos \left (1\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \sin \left (1\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (x - 1, 0\right )\right ) + {\left ({\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \cos \left (1\right ) + {\left (-i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x - i}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x + i}\right ) - 1\right )}\right )} \sin \left (1\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} {\left | x - 1 \right |} + {\left ({\left (i \, \cos \left (1\right ) + \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (-i \, \cos \left (1\right ) + \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right ) + {\left ({\left (\cos \left (1\right ) - i \, \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, i \, x - i\right ) + {\left (\cos \left (1\right ) + i \, \sin \left (1\right )\right )} \Gamma \left (\frac {3}{2}, -i \, x + i\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} \sqrt {-x + 1} \sqrt {{\left | x - 1 \right |}}}{2 \, {\left (x - 1\right )}^{2}} \]

input

integrate((1+x)^(1/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="maxima")
 

output

-1/2*((((-I*cos(1) - sin(1))*gamma(3/2, I*x - I) + (I*cos(1) - sin(1))*gam 
ma(3/2, -I*x + I))*cos(3/2*arctan2(x - 1, 0)) - ((cos(1) - I*sin(1))*gamma 
(3/2, I*x - I) + (cos(1) + I*sin(1))*gamma(3/2, -I*x + I))*sin(3/2*arctan2 
(x - 1, 0)))*x + (((I*sqrt(pi)*(erf(sqrt(I*x - I)) - 1) - I*sqrt(pi)*(erf( 
sqrt(-I*x + I)) - 1))*cos(1) + (sqrt(pi)*(erf(sqrt(I*x - I)) - 1) + sqrt(p 
i)*(erf(sqrt(-I*x + I)) - 1))*sin(1))*cos(1/2*arctan2(x - 1, 0)) + ((sqrt( 
pi)*(erf(sqrt(I*x - I)) - 1) + sqrt(pi)*(erf(sqrt(-I*x + I)) - 1))*cos(1) 
+ (-I*sqrt(pi)*(erf(sqrt(I*x - I)) - 1) + I*sqrt(pi)*(erf(sqrt(-I*x + I)) 
- 1))*sin(1))*sin(1/2*arctan2(x - 1, 0)))*abs(x - 1) + ((I*cos(1) + sin(1) 
)*gamma(3/2, I*x - I) + (-I*cos(1) + sin(1))*gamma(3/2, -I*x + I))*cos(3/2 
*arctan2(x - 1, 0)) + ((cos(1) - I*sin(1))*gamma(3/2, I*x - I) + (cos(1) + 
 I*sin(1))*gamma(3/2, -I*x + I))*sin(3/2*arctan2(x - 1, 0)))*sqrt(-x + 1)* 
sqrt(abs(x - 1))/(x - 1)^2
 

3.9.16.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=-\left (\frac {3}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{i} + \left (\frac {3}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{\left (-i\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (-i \, x\right )} + 1.16622538328000 \]

input

integrate((1+x)^(1/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="giac")
 

output

-(3/8*I + 1/8)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(-x + 1))*e 
^I + (3/8*I - 1/8)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(-x + 1) 
)*e^(-I) + 1/2*sqrt(-x + 1)*e^(I*x) + 1/2*sqrt(-x + 1)*e^(-I*x) + 1.166225 
38328000
 

3.9.16.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\text {arctanh}(x)} x \sin (x)}{\sqrt {1+x}} \, dx=\int \frac {x\,\sin \left (x\right )\,\sqrt {x+1}}{\sqrt {1-x^2}} \,d x \]

input

int((x*sin(x)*(x + 1)^(1/2))/(1 - x^2)^(1/2),x)
 

output

int((x*sin(x)*(x + 1)^(1/2))/(1 - x^2)^(1/2), x)