3.2.47 \(\int ((1+2 e^{x^2} x) \cos (x)-(e^{x^2}+x) \sin (x)) \, dx\) [147]

Optimal. Leaf size=10 \[ \left (e^{x^2}+x\right ) \cos (x) \]

[Out]

(x+exp(x^2))*cos(x)

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Rubi [F]
time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \left (\left (1+2 e^{x^2} x\right ) \cos (x)-\left (e^{x^2}+x\right ) \sin (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 2*E^x^2*x)*Cos[x] - (E^x^2 + x)*Sin[x],x]

[Out]

x*Cos[x] - (I/4)*E^(1/4)*Sqrt[Pi]*Erfi[(-I + 2*x)/2] + (I/4)*E^(1/4)*Sqrt[Pi]*Erfi[(I + 2*x)/2] + 2*Defer[Int]
[E^x^2*x*Cos[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (1+2 e^{x^2} x\right ) \cos (x) \, dx-\int \left (e^{x^2}+x\right ) \sin (x) \, dx\\ &=\int \left (\cos (x)+2 e^{x^2} x \cos (x)\right ) \, dx-\int \left (e^{x^2} \sin (x)+x \sin (x)\right ) \, dx\\ &=2 \int e^{x^2} x \cos (x) \, dx+\int \cos (x) \, dx-\int e^{x^2} \sin (x) \, dx-\int x \sin (x) \, dx\\ &=x \cos (x)+\sin (x)+2 \int e^{x^2} x \cos (x) \, dx-\int \left (\frac {1}{2} i e^{-i x+x^2}-\frac {1}{2} i e^{i x+x^2}\right ) \, dx-\int \cos (x) \, dx\\ &=x \cos (x)-\frac {1}{2} i \int e^{-i x+x^2} \, dx+\frac {1}{2} i \int e^{i x+x^2} \, dx+2 \int e^{x^2} x \cos (x) \, dx\\ &=x \cos (x)+2 \int e^{x^2} x \cos (x) \, dx-\frac {1}{2} \left (i \sqrt [4]{e}\right ) \int e^{\frac {1}{4} (-i+2 x)^2} \, dx+\frac {1}{2} \left (i \sqrt [4]{e}\right ) \int e^{\frac {1}{4} (i+2 x)^2} \, dx\\ &=x \cos (x)-\frac {1}{4} i \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i+2 x)\right )+\frac {1}{4} i \sqrt [4]{e} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i+2 x)\right )+2 \int e^{x^2} x \cos (x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.33, size = 27, normalized size = 2.70 \begin {gather*} \frac {1}{2} e^{-i x} \left (1+e^{2 i x}\right ) \left (e^{x^2}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*E^x^2*x)*Cos[x] - (E^x^2 + x)*Sin[x],x]

[Out]

((1 + E^((2*I)*x))*(E^x^2 + x))/(2*E^(I*x))

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Maple [A]
time = 0.20, size = 10, normalized size = 1.00

method	result	size
parallelrisch	\(\left (x +{\mathrm e}^{x^{2}}\right ) \cos \left (x \right )\)	\(10\)
risch	\(\frac {{\mathrm e}^{x \left (x -i\right )}}{2}+\frac {{\mathrm e}^{x \left (i+x \right )}}{2}+x \cos \left (x \right )\)	\(24\)
norman	\(\frac {x -x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-{\mathrm e}^{x^{2}} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+{\mathrm e}^{x^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x*exp(x^2))*cos(x)-(x+exp(x^2))*sin(x),x,method=_RETURNVERBOSE)

[Out]

(x+exp(x^2))*cos(x)

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.47, size = 473, normalized size = 47.30 \begin {gather*} -\frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erf}\left (i \, x + \frac {1}{2}\right ) - \operatorname {erf}\left (i \, x - \frac {1}{2}\right )\right )} e^{\frac {1}{4}} + x \cos \left (x\right ) + \frac {2 \, {\left (16 \, x^{4} + 8 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (e^{\left (x^{2} + i \, x - \frac {1}{4}\right )} + e^{\left (x^{2} - i \, x - \frac {1}{4}\right )} + e^{\left (\overline {x}^{2} + i \, \overline {x} - \frac {1}{4}\right )} + e^{\left (\overline {x}^{2} - i \, \overline {x} - \frac {1}{4}\right )}\right )} e^{\frac {1}{4}} - {\left ({\left (2 \, {\left (i \, \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right )} - 1\right )} - i \, \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right )} - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right ) - 1\right )}\right )} x - \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right )} - 1\right )} - \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right )} - 1\right )} - \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right ) - 1\right )} - \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right ) - 1\right )}\right )} \cos \left (\frac {1}{2} \, \arctan \left (4 \, x, -4 \, x^{2} + 1\right )\right ) - {\left (2 \, {\left (\sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right )} - 1\right )} + \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right )} - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right ) - 1\right )}\right )} x + i \, \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right )} - 1\right )} - i \, \sqrt {\pi } {\left (\overline {\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right )} - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} + i \, x + \frac {1}{4}}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-x^{2} - i \, x + \frac {1}{4}}\right ) - 1\right )}\right )} \sin \left (\frac {1}{2} \, \arctan \left (4 \, x, -4 \, x^{2} + 1\right )\right )\right )} e^{\frac {1}{4}}}{8 \, {\left (16 \, x^{4} + 8 \, x^{2} + 1\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(pi)*(erf(I*x + 1/2) - erf(I*x - 1/2))*e^(1/4) + x*cos(x) + 1/8*(2*(16*x^4 + 8*x^2 + 1)^(1/4)*(e^(x^2
+ I*x - 1/4) + e^(x^2 - I*x - 1/4) + e^(conjugate(x)^2 + I*conjugate(x) - 1/4) + e^(conjugate(x)^2 - I*conjug
ate(x) - 1/4))*e^(1/4) - ((2*(I*sqrt(pi)*(conjugate(erf(sqrt(-x^2 + I*x + 1/4))) - 1) - I*sqrt(pi)*(conjugate(
erf(sqrt(-x^2 - I*x + 1/4))) - 1) - I*sqrt(pi)*(erf(sqrt(-x^2 + I*x + 1/4)) - 1) + I*sqrt(pi)*(erf(sqrt(-x^2 -
I*x + 1/4)) - 1))*x - sqrt(pi)*(conjugate(erf(sqrt(-x^2 + I*x + 1/4))) - 1) - sqrt(pi)*(conjugate(erf(sqrt(-x
^2 - I*x + 1/4))) - 1) - sqrt(pi)*(erf(sqrt(-x^2 + I*x + 1/4)) - 1) - sqrt(pi)*(erf(sqrt(-x^2 - I*x + 1/4)) -
1))*cos(1/2*arctan2(4*x, -4*x^2 + 1)) - (2*(sqrt(pi)*(conjugate(erf(sqrt(-x^2 + I*x + 1/4))) - 1) + sqrt(pi)*(
conjugate(erf(sqrt(-x^2 - I*x + 1/4))) - 1) + sqrt(pi)*(erf(sqrt(-x^2 + I*x + 1/4)) - 1) + sqrt(pi)*(erf(sqrt(
-x^2 - I*x + 1/4)) - 1))*x + I*sqrt(pi)*(conjugate(erf(sqrt(-x^2 + I*x + 1/4))) - 1) - I*sqrt(pi)*(conjugate(e
rf(sqrt(-x^2 - I*x + 1/4))) - 1) - I*sqrt(pi)*(erf(sqrt(-x^2 + I*x + 1/4)) - 1) + I*sqrt(pi)*(erf(sqrt(-x^2 -
I*x + 1/4)) - 1))*sin(1/2*arctan2(4*x, -4*x^2 + 1)))*e^(1/4))/(16*x^4 + 8*x^2 + 1)^(1/4)

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Fricas [A]
time = 0.61, size = 12, normalized size = 1.20 \begin {gather*} x \cos \left (x\right ) + \cos \left (x\right ) e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*cos(x) + cos(x)*e^(x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (x + e^{x^{2}}\right ) \sin {\left (x \right )} + \left (2 x e^{x^{2}} + 1\right ) \cos {\left (x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x*exp(x**2))*cos(x)-(x+exp(x**2))*sin(x),x)

[Out]

Integral(-(x + exp(x**2))*sin(x) + (2*x*exp(x**2) + 1)*cos(x), x)

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Giac [A]
time = 0.42, size = 12, normalized size = 1.20 \begin {gather*} x \cos \left (x\right ) + \cos \left (x\right ) e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*cos(x) + cos(x)*e^(x^2)

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Mupad [B]
time = 0.07, size = 9, normalized size = 0.90 \begin {gather*} \cos \left (x\right )\,\left (x+{\mathrm {e}}^{x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x)*(x + exp(x^2))

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Chatgpt [F] Failed to verify
time = 1.00, size = 22, normalized size = 2.20 \begin {gather*} \frac {{\mathrm e}^{x^{2}} \sin \left (x \right )}{2}+\frac {\cos \left (x \right ) \left (2 x \,{\mathrm e}^{x^{2}}-1\right )}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

1/2*exp(x^2)*sin(x)+1/2*cos(x)*(2*x*exp(x^2)-1)

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