3.4.17 \(\int \sin (4 \arctan (x)) \, dx\) [317]

Optimal. Leaf size=18 \[ -\frac {4}{1+x^2}-2 \log \left (1+x^2\right ) \]

[Out]

-4/(x^2+1)-2*ln(x^2+1)

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Rubi [F]
time = 0.00, 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 \sin (4 \arctan (x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sin[4*ArcTan[x]],x]

[Out]

Defer[Int][Sin[4*ArcTan[x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \sin (4 \arctan (x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4}{1+x^2}-2 \log \left (1+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[4*ArcTan[x]],x]

[Out]

-4/(1 + x^2) - 2*Log[1 + x^2]

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Maple [C] Result contains complex when optimal does not.
time = 0.25, size = 34, normalized size = 1.89

method	result	size
default	\(-\frac {2 i}{i+x}-2 \ln \left (i+x \right )+\frac {2 i}{x -i}-2 \ln \left (x -i\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(4*arctan(x)),x,method=_RETURNVERBOSE)

[Out]

-2*I/(I+x)-2*ln(I+x)+2*I/(x-I)-2*ln(x-I)

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Maxima [A]
time = 0.42, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left ({\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2\right )}}{x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*arctan(x)),x, algorithm="maxima")

[Out]

-2*((x^2 + 1)*log(x^2 + 1) + 2)/(x^2 + 1)

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Fricas [A]
time = 0.58, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left ({\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2\right )}}{x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*arctan(x)),x, algorithm="fricas")

[Out]

-2*((x^2 + 1)*log(x^2 + 1) + 2)/(x^2 + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sin {\left (4 \operatorname {atan}{\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*atan(x)),x)

[Out]

Integral(sin(4*atan(x)), x)

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Giac [A]
time = 0.73, size = 23, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (x^{2} - 1\right )}}{x^{2} + 1} - 2 \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*arctan(x)),x, algorithm="giac")

[Out]

2*(x^2 - 1)/(x^2 + 1) - 2*log(x^2 + 1)

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Mupad [B]
time = 0.19, size = 18, normalized size = 1.00 \begin {gather*} -2\,\ln \left (x^2+1\right )-\frac {4}{x^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(4*atan(x)),x)

[Out]

- 2*log(x^2 + 1) - 4/(x^2 + 1)

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Chatgpt [F] Failed to verify
time = 1.00, size = 7, normalized size = 0.39 \begin {gather*} -\frac {\cos \left (4 \arctan \left (x \right )\right )}{4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(sin(4*arctan(x)),x)

[Out]

-1/4*cos(4*arctan(x))

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