Integrand size = 15, antiderivative size = 240 \[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {3}{2} \sqrt {1-x} \sin (x) \] Output:
3*(1-x)^(1/2)*cos(x)-(1-x)^(3/2)*cos(x)-3/2*2^(1/2)*Pi^(1/2)*cos(1)*Fresne lC(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))+5/4*2^(1/2)*Pi^(1/2)*cos(1)*FresnelS(2^(1 /2)/Pi^(1/2)*(1-x)^(1/2))-5/4*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*( 1-x)^(1/2))*sin(1)-3/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*(1-x)^(1 /2))*sin(1)-3/2*(1-x)^(1/2)*sin(x)
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.77 \[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \sqrt {1+x} \left ((-6+5 i) \sqrt {2 \pi } \sqrt {-1+x} \text {erfi}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (\cos (1)+i \sin (1))+(2+2 i) \left ((-4-3 i)+(2+3 i) x+2 x^2\right ) (\cos (x)+i \sin (x))+\left ((2+2 i) \left ((-4+3 i)+(2-3 i) x+2 x^2\right ) (\cos (1)+i \sin (1))-(6+5 i) \sqrt {2 \pi } \sqrt {-1+x} \text {erf}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (\cos (x)+i \sin (x))\right ) (\cos (1+x)-i \sin (1+x))\right )}{\sqrt {1-x^2}} \] Input:
Integrate[E^ArcTanh[x]*x*Sqrt[1 + x]*Sin[x],x]
Output:
((-1/16 + I/16)*Sqrt[1 + x]*((-6 + 5*I)*Sqrt[2*Pi]*Sqrt[-1 + x]*Erfi[((1 + I)*Sqrt[-1 + x])/Sqrt[2]]*(Cos[1] + I*Sin[1]) + (2 + 2*I)*((-4 - 3*I) + ( 2 + 3*I)*x + 2*x^2)*(Cos[x] + I*Sin[x]) + ((2 + 2*I)*((-4 + 3*I) + (2 - 3* I)*x + 2*x^2)*(Cos[1] + I*Sin[1]) - (6 + 5*I)*Sqrt[2*Pi]*Sqrt[-1 + x]*Erf[ ((1 + I)*Sqrt[-1 + x])/Sqrt[2]]*(Cos[x] + I*Sin[x]))*(Cos[1 + x] - I*Sin[1 + x])))/Sqrt[1 - x^2]
Time = 0.76 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6679, 7267, 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.
\(\displaystyle \int x \sqrt {x+1} e^{\text {arctanh}(x)} \sin (x) \, dx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \int \frac {x (x+1) \sin (x)}{\sqrt {1-x}}dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle -2 \int x (x+1) \sin (x)d\sqrt {1-x}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\sin (x) (1-x)^2-3 \sin (x) (1-x)+2 \sin (x)\right )d\sqrt {1-x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\sqrt {2 \pi } \sin (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{4} \sqrt {\frac {\pi }{2}} \sin (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sin (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\sqrt {2 \pi } \cos (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{4} \sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{4} \sqrt {1-x} \sin (x)+\frac {1}{2} (1-x)^{3/2} \cos (x)-\frac {3}{2} \sqrt {1-x} \cos (x)\right )\) |
Input:
Int[E^ArcTanh[x]*x*Sqrt[1 + x]*Sin[x],x]
Output:
-2*((-3*Sqrt[1 - x]*Cos[x])/2 + ((1 - x)^(3/2)*Cos[x])/2 + (3*Sqrt[Pi/2]*C os[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]])/2 + (3*Sqrt[Pi/2]*Cos[1]*FresnelS[ Sqrt[2/Pi]*Sqrt[1 - x]])/4 - Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1])/4 + Sqrt[2* Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] + (3*Sqrt[Pi/2]*FresnelS[Sqrt[ 2/Pi]*Sqrt[1 - x]]*Sin[1])/2 + (3*Sqrt[1 - x]*Sin[x])/4)
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])
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]]
\[\int \frac {\left (1+x \right )^{\frac {3}{2}} x \sin \left (x \right )}{\sqrt {-x^{2}+1}}d x\]
Input:
int((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x)
Output:
int((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x)
\[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {3}{2}} x \sin \left (x\right )}{\sqrt {-x^{2} + 1}} \,d x } \] Input:
integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="fricas")
Output:
integral(-sqrt(-x^2 + 1)*sqrt(x + 1)*x*sin(x)/(x - 1), x)
Timed out. \[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=\text {Timed out} \] Input:
integrate((1+x)**(3/2)/(-x**2+1)**(1/2)*x*sin(x),x)
Output:
Timed out
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.62 \[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=\text {Too large to display} \] Input:
integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="maxima")
Output:
1/2*(2*(((-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(pi)*(erf(s qrt(-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)))*(x - 1)^2 - (((-I*cos(1) - sin(1))*gamma(5 /2, I*x - I) + (I*cos(1) - sin(1))*gamma(5/2, -I*x + I))*cos(5/2*arctan2(x - 1, 0)) - ((cos(1) - I*sin(1))*gamma(5/2, I*x - I) + (cos(1) + I*sin(1)) *gamma(5/2, -I*x + I))*sin(5/2*arctan2(x - 1, 0)))*x^2 + (3*(((I*cos(1) + sin(1))*gamma(3/2, I*x - I) + (-I*cos(1) + sin(1))*gamma(3/2, -I*x + I))*c os(3/2*arctan2(x - 1, 0)) + ((cos(1) - I*sin(1))*gamma(3/2, I*x - I) + (co s(1) + I*sin(1))*gamma(3/2, -I*x + I))*sin(3/2*arctan2(x - 1, 0)))*abs(x - 1) + 2*((-I*cos(1) - sin(1))*gamma(5/2, I*x - I) + (I*cos(1) - sin(1))*ga mma(5/2, -I*x + I))*cos(5/2*arctan2(x - 1, 0)) - 2*((cos(1) - I*sin(1))*ga mma(5/2, I*x - I) + (cos(1) + I*sin(1))*gamma(5/2, -I*x + I))*sin(5/2*arct an2(x - 1, 0)))*x + 3*(((-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)))*abs(x - 1) - ((-I*cos(1) - sin(1))*gamma(5/2, I*x - I) + (I*cos(1) - sin(1))*gamma(5/2, -I*x + I))*cos(5/2*arctan2(x...
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.51 \[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=-\left (\frac {11}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{i} + \left (\frac {11}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {-x + 1}\right ) e^{\left (-i\right )} - \frac {1}{4} i \, {\left (-2 i \, {\left (-x + 1\right )}^{\frac {3}{2}} + \left (4 i - 3\right ) \, \sqrt {-x + 1}\right )} e^{\left (i \, x\right )} - \frac {1}{4} i \, {\left (-2 i \, {\left (-x + 1\right )}^{\frac {3}{2}} + \left (4 i + 3\right ) \, \sqrt {-x + 1}\right )} e^{\left (-i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (i \, x\right )} + \frac {1}{2} \, \sqrt {-x + 1} e^{\left (-i \, x\right )} \] Input:
integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="giac")
Output:
-(11/16*I - 1/16)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(-x + 1) )*e^I + (11/16*I + 1/16)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(- x + 1))*e^(-I) - 1/4*I*(-2*I*(-x + 1)^(3/2) + (4*I - 3)*sqrt(-x + 1))*e^(I *x) - 1/4*I*(-2*I*(-x + 1)^(3/2) + (4*I + 3)*sqrt(-x + 1))*e^(-I*x) + 1/2* sqrt(-x + 1)*e^(I*x) + 1/2*sqrt(-x + 1)*e^(-I*x)
Timed out. \[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=\int \frac {x\,\sin \left (x\right )\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x^2}} \,d x \] Input:
int((x*sin(x)*(x + 1)^(3/2))/(1 - x^2)^(1/2),x)
Output:
int((x*sin(x)*(x + 1)^(3/2))/(1 - x^2)^(1/2), x)
\[ \int e^{\text {arctanh}(x)} x \sqrt {1+x} \sin (x) \, dx=\int \frac {\sin \left (x \right ) x^{2}}{\sqrt {1-x}}d x +\int \frac {\sin \left (x \right ) x}{\sqrt {1-x}}d x \] Input:
int((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x)
Output:
int((sin(x)*x**2)/sqrt( - x + 1),x) + int((sin(x)*x)/sqrt( - x + 1),x)