3.9.8 \(\int e^{\tanh ^{-1}(x)} x \sqrt {1+x} \sin (x) \, dx\) [808]

Optimal. Leaf size=240 \[ 3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {3}{2} \sqrt {1-x} \sin (x) \]

[Out]

-(1-x)^(3/2)*cos(x)-3/2*cos(1)*FresnelC(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))*2^(1/2)*Pi^(1/2)+5/4*cos(1)*FresnelS(2^(
1/2)/Pi^(1/2)*(1-x)^(1/2))*2^(1/2)*Pi^(1/2)-5/4*FresnelC(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))*sin(1)*2^(1/2)*Pi^(1/2)
-3/2*FresnelS(2^(1/2)/Pi^(1/2)*(1-x)^(1/2))*sin(1)*2^(1/2)*Pi^(1/2)+3*cos(x)*(1-x)^(1/2)-3/2*sin(x)*(1-x)^(1/2
)

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Rubi [A]
time = 0.27, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6264, 6874, 3434, 3433, 3432, 3466, 3435, 3467} \begin {gather*} -2 \sqrt {2 \pi } \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sin (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-3 \sqrt {\frac {\pi }{2}} \cos (1) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-3 \sqrt {\frac {\pi }{2}} \sin (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {1-x} \sin (x)+(1-x)^{3/2} (-\cos (x))+3 \sqrt {1-x} \cos (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*Sqrt[1 - x]*Cos[x] - (1 - x)^(3/2)*Cos[x] - 3*Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] - (3*Sqrt[P
i/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]])/2 + 2*Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + (3*Sq
rt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1])/2 - 2*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] - (3*Sqrt[1 - x]*Sin[x])/2

Rule 3432

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]

Rule 3433

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]

Rule 3434

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 3435

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]

Rule 3466

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 3467

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

Rule 6264

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 6874

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

Rubi steps

\begin {align*} \int e^{\tanh ^{-1}(x)} x \sqrt {1+x} \sin (x) \, dx &=\int \frac {x (1+x) \sin (x)}{\sqrt {1-x}} \, dx\\ &=-\left (2 \text {Subst}\left (\int \left (-2+x^2\right ) \left (-1+x^2\right ) \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\right )\\ &=-\left (2 \text {Subst}\left (\int \left (2 \sin \left (1-x^2\right )-3 x^2 \sin \left (1-x^2\right )+x^4 \sin \left (1-x^2\right )\right ) \, dx,x,\sqrt {1-x}\right )\right )\\ &=-\left (2 \text {Subst}\left (\int x^4 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\right )-4 \text {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+6 \text {Subst}\left (\int x^2 \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \text {Subst}\left (\int \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+3 \text {Subst}\left (\int x^2 \cos \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )+(4 \cos (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )-(4 \sin (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {3}{2} \sqrt {1-x} \sin (x)+\frac {3}{2} \text {Subst}\left (\int \sin \left (1-x^2\right ) \, dx,x,\sqrt {1-x}\right )-(3 \cos (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )-(3 \sin (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {3}{2} \sqrt {1-x} \sin (x)-\frac {1}{2} (3 \cos (1)) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )+\frac {1}{2} (3 \sin (1)) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {1-x}\right )\\ &=3 \sqrt {1-x} \cos (x)-(1-x)^{3/2} \cos (x)-3 \sqrt {\frac {\pi }{2}} \cos (1) C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+2 \sqrt {2 \pi } \cos (1) S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {1-x}\right ) \sin (1)-\frac {3}{2} \sqrt {1-x} \sin (x)\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 3.42, size = 185, normalized size = 0.77 \begin {gather*} \frac {i \sqrt {1+x} \left ((-11-i) \sqrt {\frac {\pi }{2}} \sqrt {-1+x} \text {Erfi}\left (\frac {(1+i) \sqrt {-1+x}}{\sqrt {2}}\right ) (\cos (1)+i \sin (1))+\left ((-4-3 i)+(2+3 i) x+2 x^2\right ) (2 i \cos (x)-2 \sin (x))+\left (2 \left ((-3-4 i)+(3+2 i) x+2 i x^2\right ) (\cos (1)+i \sin (1))-(1+11 i) \sqrt {\frac {\pi }{2}} \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 )}{8 \sqrt {1-x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/8)*Sqrt[1 + x]*((-11 - I)*Sqrt[Pi/2]*Sqrt[-1 + x]*Erfi[((1 + I)*Sqrt[-1 + x])/Sqrt[2]]*(Cos[1] + I*Sin[1])
+ ((-4 - 3*I) + (2 + 3*I)*x + 2*x^2)*((2*I)*Cos[x] - 2*Sin[x]) + (2*((-3 - 4*I) + (3 + 2*I)*x + (2*I)*x^2)*(C
os[1] + I*Sin[1]) - (1 + 11*I)*Sqrt[Pi/2]*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]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (1+x \right )^{\frac {3}{2}} x \sin \left (x \right )}{\sqrt {-x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x)

[Out]

int((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x)

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Maxima [C] Result contains complex when optimal does not.
time = 0.34, size = 628, normalized size = 2.62 \begin {gather*} \frac {{\left (2 \, {\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 )}^{2} - {\left ({\left ({\left (-i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, i \, x - i\right ) + {\left (i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, -i \, x + i\right )\right )} \cos \left (\frac {5}{2} \, \arctan \left (x - 1, 0\right )\right ) - {\left ({\left (\cos \left (1\right ) - i \, \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, i \, x - i\right ) + {\left (\cos \left (1\right ) + i \, \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, -i \, x + i\right )\right )} \sin \left (\frac {5}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} x^{2} + {\left (3 \, {\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 )} {\left | x - 1 \right |} + 2 \, {\left ({\left (-i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, i \, x - i\right ) + {\left (i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, -i \, x + i\right )\right )} \cos \left (\frac {5}{2} \, \arctan \left (x - 1, 0\right )\right ) - 2 \, {\left ({\left (\cos \left (1\right ) - i \, \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, i \, x - i\right ) + {\left (\cos \left (1\right ) + i \, \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, -i \, x + i\right )\right )} \sin \left (\frac {5}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} x + 3 \, {\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 )} {\left | x - 1 \right |} - {\left ({\left (-i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, i \, x - i\right ) + {\left (i \, \cos \left (1\right ) - \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, -i \, x + i\right )\right )} \cos \left (\frac {5}{2} \, \arctan \left (x - 1, 0\right )\right ) + {\left ({\left (\cos \left (1\right ) - i \, \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, i \, x - i\right ) + {\left (\cos \left (1\right ) + i \, \sin \left (1\right )\right )} \Gamma \left (\frac {5}{2}, -i \, x + i\right )\right )} \sin \left (\frac {5}{2} \, \arctan \left (x - 1, 0\right )\right )\right )} \sqrt {-x + 1}}{2 \, {\left (x - 1\right )}^{2} \sqrt {{\left | x - 1 \right |}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="maxima")

[Out]

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(sqrt(-I*x + I)) - 1))*sin(1))*cos(1/2*arctan2(x - 1, 0)) - ((sqrt(pi)*(er
f(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))*gam
ma(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))*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) + 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)) - 2*((cos(1) - I*sin(1))*gamma(5/2, I*x - I) + (cos(1) + I*sin(1))*gamma(5/2, -I*x + I))*si
n(5/2*arctan2(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 - 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)))*sqrt(-x + 1)/((x - 1)^2*sqrt(abs(x - 1)))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/(-x**2+1)**(1/2)*x*sin(x),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 124, normalized size = 0.52 \begin {gather*} -\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 )} + 1.79526793396000 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(-x^2+1)^(1/2)*x*sin(x),x, algorithm="giac")

[Out]

-(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)*sqr
t(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) + 1.79526793396000

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\sin \left (x\right )\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*sin(x)*(x + 1)^(3/2))/(1 - x^2)^(1/2),x)

[Out]

int((x*sin(x)*(x + 1)^(3/2))/(1 - x^2)^(1/2), x)

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