3.123 \(\int \frac{c+d x}{1-x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right ) \]

[Out]

(c*ArcTan[x])/2 + (c*ArcTanh[x])/2 + (d*ArcTanh[x^2])/2

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Rubi [A]  time = 0.0180047, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1876, 212, 206, 203, 275} \[ \frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(1 - x^4),x]

[Out]

(c*ArcTan[x])/2 + (c*ArcTanh[x])/2 + (d*ArcTanh[x^2])/2

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{c+d x}{1-x^4} \, dx &=\int \left (\frac{c}{1-x^4}+\frac{d x}{1-x^4}\right ) \, dx\\ &=c \int \frac{1}{1-x^4} \, dx+d \int \frac{x}{1-x^4} \, dx\\ &=\frac{1}{2} c \int \frac{1}{1-x^2} \, dx+\frac{1}{2} c \int \frac{1}{1+x^2} \, dx+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0147418, size = 42, normalized size = 1.75 \[ \frac{1}{4} \left (-(c+d) \log (1-x)+c \log (x+1)+2 c \tan ^{-1}(x)+d \log \left (x^2+1\right )-d \log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(1 - x^4),x]

[Out]

(2*c*ArcTan[x] - (c + d)*Log[1 - x] + c*Log[1 + x] - d*Log[1 + x] + d*Log[1 + x^2])/4

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Maple [B]  time = 0.004, size = 44, normalized size = 1.8 \begin{align*} -{\frac{\ln \left ( -1+x \right ) c}{4}}-{\frac{\ln \left ( -1+x \right ) d}{4}}+{\frac{\ln \left ( 1+x \right ) c}{4}}-{\frac{\ln \left ( 1+x \right ) d}{4}}+{\frac{d\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{c\arctan \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(-x^4+1),x)

[Out]

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

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Maxima [A]  time = 1.4422, size = 47, normalized size = 1.96 \begin{align*} \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left (x + 1\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-x^4+1),x, algorithm="maxima")

[Out]

1/2*c*arctan(x) + 1/4*d*log(x^2 + 1) + 1/4*(c - d)*log(x + 1) - 1/4*(c + d)*log(x - 1)

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Fricas [A]  time = 1.29288, size = 119, normalized size = 4.96 \begin{align*} \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left (x + 1\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-x^4+1),x, algorithm="fricas")

[Out]

1/2*c*arctan(x) + 1/4*d*log(x^2 + 1) + 1/4*(c - d)*log(x + 1) - 1/4*(c + d)*log(x - 1)

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Sympy [C]  time = 0.459214, size = 313, normalized size = 13.04 \begin{align*} \frac{\left (c - d\right ) \log{\left (x + \frac{c^{4} \left (c - d\right ) + 5 c^{2} d^{3} + c^{2} d \left (c - d\right )^{2} - 2 d^{4} \left (c - d\right ) + 2 d^{2} \left (c - d\right )^{3}}{c^{5} + 4 c d^{4}} \right )}}{4} - \frac{\left (c + d\right ) \log{\left (x + \frac{- c^{4} \left (c + d\right ) + 5 c^{2} d^{3} + c^{2} d \left (c + d\right )^{2} + 2 d^{4} \left (c + d\right ) - 2 d^{2} \left (c + d\right )^{3}}{c^{5} + 4 c d^{4}} \right )}}{4} - \left (- \frac{i c}{4} - \frac{d}{4}\right ) \log{\left (x + \frac{- 4 c^{4} \left (- \frac{i c}{4} - \frac{d}{4}\right ) + 5 c^{2} d^{3} + 16 c^{2} d \left (- \frac{i c}{4} - \frac{d}{4}\right )^{2} + 8 d^{4} \left (- \frac{i c}{4} - \frac{d}{4}\right ) - 128 d^{2} \left (- \frac{i c}{4} - \frac{d}{4}\right )^{3}}{c^{5} + 4 c d^{4}} \right )} - \left (\frac{i c}{4} - \frac{d}{4}\right ) \log{\left (x + \frac{- 4 c^{4} \left (\frac{i c}{4} - \frac{d}{4}\right ) + 5 c^{2} d^{3} + 16 c^{2} d \left (\frac{i c}{4} - \frac{d}{4}\right )^{2} + 8 d^{4} \left (\frac{i c}{4} - \frac{d}{4}\right ) - 128 d^{2} \left (\frac{i c}{4} - \frac{d}{4}\right )^{3}}{c^{5} + 4 c d^{4}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-x**4+1),x)

[Out]

(c - d)*log(x + (c**4*(c - d) + 5*c**2*d**3 + c**2*d*(c - d)**2 - 2*d**4*(c - d) + 2*d**2*(c - d)**3)/(c**5 +
4*c*d**4))/4 - (c + d)*log(x + (-c**4*(c + d) + 5*c**2*d**3 + c**2*d*(c + d)**2 + 2*d**4*(c + d) - 2*d**2*(c +
d)**3)/(c**5 + 4*c*d**4))/4 - (-I*c/4 - d/4)*log(x + (-4*c**4*(-I*c/4 - d/4) + 5*c**2*d**3 + 16*c**2*d*(-I*c/
4 - d/4)**2 + 8*d**4*(-I*c/4 - d/4) - 128*d**2*(-I*c/4 - d/4)**3)/(c**5 + 4*c*d**4)) - (I*c/4 - d/4)*log(x + (
-4*c**4*(I*c/4 - d/4) + 5*c**2*d**3 + 16*c**2*d*(I*c/4 - d/4)**2 + 8*d**4*(I*c/4 - d/4) - 128*d**2*(I*c/4 - d/
4)**3)/(c**5 + 4*c*d**4))

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Giac [B]  time = 1.05346, size = 50, normalized size = 2.08 \begin{align*} \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-x^4+1),x, algorithm="giac")

[Out]

1/2*c*arctan(x) + 1/4*d*log(x^2 + 1) + 1/4*(c - d)*log(abs(x + 1)) - 1/4*(c + d)*log(abs(x - 1))