3.2.35 \(\int (a+b \text {ArcTan}(\frac {c}{x})) \, dx\) [135]

Optimal. Leaf size=27 \[ a x+b x \text {ArcTan}\left (\frac {c}{x}\right )+\frac {1}{2} b c \log \left (c^2+x^2\right ) \]

[Out]

a*x+b*x*arctan(c/x)+1/2*b*c*ln(c^2+x^2)

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4930, 269, 266} \begin {gather*} a x+b x \text {ArcTan}\left (\frac {c}{x}\right )+\frac {1}{2} b c \log \left (c^2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*ArcTan[c/x],x]

[Out]

a*x + b*x*ArcTan[c/x] + (b*c*Log[c^2 + x^2])/2

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rubi steps

\begin {align*} \int \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=a x+b \int \tan ^{-1}\left (\frac {c}{x}\right ) \, dx\\ &=a x+b x \tan ^{-1}\left (\frac {c}{x}\right )+(b c) \int \frac {1}{\left (1+\frac {c^2}{x^2}\right ) x} \, dx\\ &=a x+b x \tan ^{-1}\left (\frac {c}{x}\right )+(b c) \int \frac {x}{c^2+x^2} \, dx\\ &=a x+b x \tan ^{-1}\left (\frac {c}{x}\right )+\frac {1}{2} b c \log \left (c^2+x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} a x+b x \text {ArcTan}\left (\frac {c}{x}\right )+\frac {1}{2} b c \log \left (c^2+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcTan[c/x],x]

[Out]

a*x + b*x*ArcTan[c/x] + (b*c*Log[c^2 + x^2])/2

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Maple [A]
time = 0.07, size = 38, normalized size = 1.41

method	result	size
default	\(a x +b x \arctan \left (\frac {c}{x}\right )+\frac {b c \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}-b c \ln \left (\frac {c}{x}\right )\)	\(38\)
derivativedivides	\(-c \left (-\frac {a x}{c}-\frac {b x \arctan \left (\frac {c}{x}\right )}{c}-\frac {b \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+b \ln \left (\frac {c}{x}\right )\right )\)	\(46\)
risch	\(\text {Expression too large to display}\)	\(642\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arctan(c/x),x,method=_RETURNVERBOSE)

[Out]

a*x+b*x*arctan(c/x)+1/2*b*c*ln(1+c^2/x^2)-b*c*ln(c/x)

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Maxima [A]
time = 0.27, size = 27, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, {\left (2 \, x \arctan \left (\frac {c}{x}\right ) + c \log \left (c^{2} + x^{2}\right )\right )} b + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arctan(c/x),x, algorithm="maxima")

[Out]

1/2*(2*x*arctan(c/x) + c*log(c^2 + x^2))*b + a*x

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Fricas [A]
time = 0.76, size = 25, normalized size = 0.93 \begin {gather*} b x \arctan \left (\frac {c}{x}\right ) + \frac {1}{2} \, b c \log \left (c^{2} + x^{2}\right ) + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arctan(c/x),x, algorithm="fricas")

[Out]

b*x*arctan(c/x) + 1/2*b*c*log(c^2 + x^2) + a*x

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Sympy [A]
time = 0.08, size = 22, normalized size = 0.81 \begin {gather*} a x + b \left (\frac {c \log {\left (c^{2} + x^{2} \right )}}{2} + x \operatorname {atan}{\left (\frac {c}{x} \right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*atan(c/x),x)

[Out]

a*x + b*(c*log(c**2 + x**2)/2 + x*atan(c/x))

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Giac [A]
time = 0.45, size = 46, normalized size = 1.70 \begin {gather*} a x + \frac {{\left (c^{2} {\left (\log \left (\frac {c^{2}}{x^{2}} + 1\right ) - \log \left (\frac {c^{2}}{x^{2}}\right )\right )} + 2 \, c x \arctan \left (\frac {c}{x}\right )\right )} b}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*arctan(c/x),x, algorithm="giac")

[Out]

a*x + 1/2*(c^2*(log(c^2/x^2 + 1) - log(c^2/x^2)) + 2*c*x*arctan(c/x))*b/c

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Mupad [B]
time = 0.30, size = 25, normalized size = 0.93 \begin {gather*} a\,x+b\,x\,\mathrm {atan}\left (\frac {c}{x}\right )+\frac {b\,c\,\ln \left (c^2+x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*atan(c/x),x)

[Out]

a*x + b*x*atan(c/x) + (b*c*log(c^2 + x^2))/2

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