3.816 \(\int \frac{e^{\tanh ^{-1}(x)} x \sin (x)}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=140 \[ -\sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{1-x} \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.187591, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {6129, 6742, 3353, 3352, 3351, 3385, 3354} \[ -\sqrt{2 \pi } \sin (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \cos (1) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{\frac{\pi }{2}} \sin (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{1-x} \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[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 6742

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

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3385

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3354

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(x)} x \sin (x)}{\sqrt{1+x}} \, dx &=\int \frac{x \sin (x)}{\sqrt{1-x}} \, dx\\ &=2 \operatorname{Subst}\left (\int \left (-1+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\sin \left (1-x^2\right )+x^2 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt{1-x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\right )+2 \operatorname{Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=\sqrt{1-x} \cos (x)+(2 \cos (1)) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-(2 \sin (1)) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-\operatorname{Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=\sqrt{1-x} \cos (x)+\sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-\cos (1) \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )-\sin (1) \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{1-x}\right )\\ &=\sqrt{1-x} \cos (x)-\sqrt{\frac{\pi }{2}} \cos (1) C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )+\sqrt{2 \pi } \cos (1) S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right )-\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)-\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{1-x}\right ) \sin (1)\\ \end{align*}

Mathematica [C]  time = 7.88062, size = 162, normalized size = 1.16 \[ \frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \sqrt{x+1} \left ((\cos (x+1)-i \sin (x+1)) \left ((1+2 i) \sqrt{2 \pi } \sqrt{x-1} \text{Erf}\left (\frac{(1+i) \sqrt{x-1}}{\sqrt{2}}\right ) (\sin (x)-i \cos (x))-(2-2 i) (x-1) (\cos (1)+i \sin (1))\right )+(-2-i) \sqrt{2 \pi } \sqrt{x-1} \text{Erfi}\left (\frac{(1+i) \sqrt{x-1}}{\sqrt{2}}\right ) (\cos (1)+i \sin (1))-(2-2 i) (x-1) (\cos (x)+i \sin (x))\right )}{\sqrt{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1/8 + I/8)*Sqrt[1 + x]*((-2 - I)*Sqrt[2*Pi]*Sqrt[-1 + x]*Erfi[((1 + I)*Sqrt[-1 + x])/Sqrt[2]]*(Cos[1] + I*Si
n[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*P
i]*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]

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Maple [F]  time = 0.27, size = 0, normalized size = 0. \begin{align*} \int{x\sin \left ( x \right ) \sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.21693, size = 468, normalized size = 3.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((((-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*arcta
n2(x - 1, 0)))*x + (((I*sqrt(pi)*(erf(sqrt(I*x - I)) - 1) - I*sqrt(pi)*(erf(sqrt(-I*x + I)) - 1))*cos(1) + (sq
rt(pi)*(erf(sqrt(I*x - I)) - 1) + sqrt(pi)*(erf(sqrt(-I*x + I)) - 1))*sin(1))*cos(1/2*arctan2(x - 1, 0)) + ((s
qrt(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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} + 1} \sqrt{x + 1} x \sin \left (x\right )}{x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sqrt{x + 1} \sin{\left (x \right )}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.21159, size = 100, normalized size = 0.71 \begin{align*} -\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 )} + 1.166225383276 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(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) + 1.1662
25383276