Integrand size = 16, antiderivative size = 72 \[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=-\sqrt {1+x} \cos (x)+\sqrt {\frac {\pi }{2}} \cos (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right )+\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {1+x}\right ) \sin (1) \]
1/2*cos(1)*FresnelC(2^(1/2)/Pi^(1/2)*(1+x)^(1/2))*2^(1/2)*Pi^(1/2)+1/2*Fre snelS(2^(1/2)/Pi^(1/2)*(1+x)^(1/2))*sin(1)*2^(1/2)*Pi^(1/2)-cos(x)*(1+x)^( 1/2)
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=\frac {i e^{-i} \left (-\sqrt {-i (1+x)} \Gamma \left (\frac {3}{2},-i (1+x)\right )+e^{2 i} \sqrt {i (1+x)} \Gamma \left (\frac {3}{2},i (1+x)\right )\right )}{2 \sqrt {1+x}} \]
((I/2)*(-(Sqrt[(-I)*(1 + x)]*Gamma[3/2, (-I)*(1 + x)]) + E^(2*I)*Sqrt[I*(1 + x)]*Gamma[3/2, I*(1 + x)]))/(E^I*Sqrt[1 + x])
Time = 0.54 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6679, 3042, 3777, 3042, 3787, 3042, 3785, 3786, 3832, 3833}
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.
\(\displaystyle \int \sqrt {1-x} e^{\text {arctanh}(x)} \sin (x) \, dx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \int \sqrt {x+1} \sin (x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {x+1} \sin (x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{2} \int \frac {\cos (x)}{\sqrt {x+1}}dx-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {\sin \left (x+\frac {\pi }{2}\right )}{\sqrt {x+1}}dx-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle \frac {1}{2} \left (\sin (1) \int \frac {\sin (x+1)}{\sqrt {x+1}}dx+\cos (1) \int \frac {\cos (x+1)}{\sqrt {x+1}}dx\right )-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\sin (1) \int \frac {\sin (x+1)}{\sqrt {x+1}}dx+\cos (1) \int \frac {\sin \left (x+\frac {\pi }{2}+1\right )}{\sqrt {x+1}}dx\right )-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {1}{2} \left (\sin (1) \int \frac {\sin (x+1)}{\sqrt {x+1}}dx+2 \cos (1) \int \cos (x+1)d\sqrt {x+1}\right )-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {1}{2} \left (2 \sin (1) \int \sin (x+1)d\sqrt {x+1}+2 \cos (1) \int \cos (x+1)d\sqrt {x+1}\right )-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {1}{2} \left (2 \cos (1) \int \cos (x+1)d\sqrt {x+1}+\sqrt {2 \pi } \sin (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )\right )-\sqrt {x+1} \cos (x)\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {1}{2} \left (\sqrt {2 \pi } \cos (1) \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )+\sqrt {2 \pi } \sin (1) \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {x+1}\right )\right )-\sqrt {x+1} \cos (x)\) |
-(Sqrt[1 + x]*Cos[x]) + (Sqrt[2*Pi]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x] ] + Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1])/2
3.9.11.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
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])
\[\int \frac {\left (1+x \right ) \sqrt {1-x}\, \sin \left (x \right )}{\sqrt {-x^{2}+1}}d x\]
\[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=\int { \frac {{\left (x + 1\right )} \sqrt {-x + 1} \sin \left (x\right )}{\sqrt {-x^{2} + 1}} \,d x } \]
\[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=\int \frac {\sqrt {1 - x} \left (x + 1\right ) \sin {\left (x \right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 498, normalized size of antiderivative = 6.92 \[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=\text {Too large to display} \]
-1/2*(((-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(sqr t(-I*x - I)) - 1))*sin(1))*cos(1/2*arctan2(x + 1, 0)) - ((sqrt(pi)*(erf(sq rt(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)))*sqrt(x + 1)/sqrt(abs(x + 1)) - 1/2*(((I*sqr t(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(sqrt(-I*x - I)) - 1))*sin(1))*abs(x + 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))*a bs(x + 1)*sin(1/2*arctan2(x + 1, 0)) + (((I*cos(1) - sin(1))*gamma(3/2, I* x + I) + (-I*cos(1) - sin(1))*gamma(3/2, -I*x - I))*x + (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))*x + (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(abs(x + 1))/(x + 1)^(3/2)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{i} - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x + 1}\right ) e^{\left (-i\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (i \, x\right )} - \frac {1}{2} \, \sqrt {x + 1} e^{\left (-i \, x\right )} - 0.339605729125000 \]
(1/8*I - 1/8)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(x + 1))*e^I - (1/8*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) - 0.33960572912 5000
Timed out. \[ \int e^{\text {arctanh}(x)} \sqrt {1-x} \sin (x) \, dx=\int \frac {\sin \left (x\right )\,\sqrt {1-x}\,\left (x+1\right )}{\sqrt {1-x^2}} \,d x \]