3.91 \(\int \frac{1}{(1+(c+d x)^2)^3} \, dx\)

Optimal. Leaf size=60 \[ \frac{3 (c+d x)}{8 d \left ((c+d x)^2+1\right )}+\frac{c+d x}{4 d \left ((c+d x)^2+1\right )^2}+\frac{3 \tan ^{-1}(c+d x)}{8 d} \]

[Out]

(c + d*x)/(4*d*(1 + (c + d*x)^2)^2) + (3*(c + d*x))/(8*d*(1 + (c + d*x)^2)) + (3*ArcTan[c + d*x])/(8*d)

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Rubi [A]  time = 0.0163135, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 199, 203} \[ \frac{3 (c+d x)}{8 d \left ((c+d x)^2+1\right )}+\frac{c+d x}{4 d \left ((c+d x)^2+1\right )^2}+\frac{3 \tan ^{-1}(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[(1 + (c + d*x)^2)^(-3),x]

[Out]

(c + d*x)/(4*d*(1 + (c + d*x)^2)^2) + (3*(c + d*x))/(8*d*(1 + (c + d*x)^2)) + (3*ArcTan[c + d*x])/(8*d)

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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])

Rubi steps

\begin{align*} \int \frac{1}{\left (1+(c+d x)^2\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{c+d x}{4 d \left (1+(c+d x)^2\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{c+d x}{4 d \left (1+(c+d x)^2\right )^2}+\frac{3 (c+d x)}{8 d \left (1+(c+d x)^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{8 d}\\ &=\frac{c+d x}{4 d \left (1+(c+d x)^2\right )^2}+\frac{3 (c+d x)}{8 d \left (1+(c+d x)^2\right )}+\frac{3 \tan ^{-1}(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.0152632, size = 52, normalized size = 0.87 \[ \frac{\frac{3 (c+d x)}{(c+d x)^2+1}+\frac{2 (c+d x)}{\left ((c+d x)^2+1\right )^2}+3 \tan ^{-1}(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (c + d*x)^2)^(-3),x]

[Out]

((2*(c + d*x))/(1 + (c + d*x)^2)^2 + (3*(c + d*x))/(1 + (c + d*x)^2) + 3*ArcTan[c + d*x])/(8*d)

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Maple [A]  time = 0.003, size = 94, normalized size = 1.6 \begin{align*}{\frac{2\,{d}^{2}x+2\,cd}{8\,{d}^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}+1 \right ) ^{2}}}+{\frac{6\,{d}^{2}x+6\,cd}{16\,{d}^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}+1 \right ) }}+{\frac{3}{8\,d}\arctan \left ({\frac{2\,{d}^{2}x+2\,cd}{2\,d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+(d*x+c)^2)^3,x)

[Out]

1/8*(2*d^2*x+2*c*d)/d^2/(d^2*x^2+2*c*d*x+c^2+1)^2+3/16*(2*d^2*x+2*c*d)/d^2/(d^2*x^2+2*c*d*x+c^2+1)+3/8/d*arcta
n(1/2*(2*d^2*x+2*c*d)/d)

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Maxima [B]  time = 1.48005, size = 155, normalized size = 2.58 \begin{align*} \frac{3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} +{\left (9 \, c^{2} + 5\right )} d x + 5 \, c}{8 \,{\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \,{\left (3 \, c^{2} + 1\right )} d^{3} x^{2} + 4 \,{\left (c^{3} + c\right )} d^{2} x +{\left (c^{4} + 2 \, c^{2} + 1\right )} d\right )}} + \frac{3 \, \arctan \left (\frac{d^{2} x + c d}{d}\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*d^3*x^3 + 9*c*d^2*x^2 + 3*c^3 + (9*c^2 + 5)*d*x + 5*c)/(d^5*x^4 + 4*c*d^4*x^3 + 2*(3*c^2 + 1)*d^3*x^2 +
4*(c^3 + c)*d^2*x + (c^4 + 2*c^2 + 1)*d) + 3/8*arctan((d^2*x + c*d)/d)/d

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Fricas [B]  time = 1.74752, size = 347, normalized size = 5.78 \begin{align*} \frac{3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} +{\left (9 \, c^{2} + 5\right )} d x + 3 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \,{\left (3 \, c^{2} + 1\right )} d^{2} x^{2} + c^{4} + 4 \,{\left (c^{3} + c\right )} d x + 2 \, c^{2} + 1\right )} \arctan \left (d x + c\right ) + 5 \, c}{8 \,{\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \,{\left (3 \, c^{2} + 1\right )} d^{3} x^{2} + 4 \,{\left (c^{3} + c\right )} d^{2} x +{\left (c^{4} + 2 \, c^{2} + 1\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*d^3*x^3 + 9*c*d^2*x^2 + 3*c^3 + (9*c^2 + 5)*d*x + 3*(d^4*x^4 + 4*c*d^3*x^3 + 2*(3*c^2 + 1)*d^2*x^2 + c^
4 + 4*(c^3 + c)*d*x + 2*c^2 + 1)*arctan(d*x + c) + 5*c)/(d^5*x^4 + 4*c*d^4*x^3 + 2*(3*c^2 + 1)*d^3*x^2 + 4*(c^
3 + c)*d^2*x + (c^4 + 2*c^2 + 1)*d)

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Sympy [C]  time = 1.2406, size = 146, normalized size = 2.43 \begin{align*} \frac{3 c^{3} + 9 c d^{2} x^{2} + 5 c + 3 d^{3} x^{3} + x \left (9 c^{2} d + 5 d\right )}{8 c^{4} d + 16 c^{2} d + 32 c d^{4} x^{3} + 8 d^{5} x^{4} + 8 d + x^{2} \left (48 c^{2} d^{3} + 16 d^{3}\right ) + x \left (32 c^{3} d^{2} + 32 c d^{2}\right )} + \frac{- \frac{3 i \log{\left (x + \frac{3 c - 3 i}{3 d} \right )}}{16} + \frac{3 i \log{\left (x + \frac{3 c + 3 i}{3 d} \right )}}{16}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(d*x+c)**2)**3,x)

[Out]

(3*c**3 + 9*c*d**2*x**2 + 5*c + 3*d**3*x**3 + x*(9*c**2*d + 5*d))/(8*c**4*d + 16*c**2*d + 32*c*d**4*x**3 + 8*d
**5*x**4 + 8*d + x**2*(48*c**2*d**3 + 16*d**3) + x*(32*c**3*d**2 + 32*c*d**2)) + (-3*I*log(x + (3*c - 3*I)/(3*
d))/16 + 3*I*log(x + (3*c + 3*I)/(3*d))/16)/d

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Giac [A]  time = 1.11363, size = 99, normalized size = 1.65 \begin{align*} \frac{3 \, \arctan \left (d x + c\right )}{8 \, d} + \frac{3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 9 \, c^{2} d x + 3 \, c^{3} + 5 \, d x + 5 \, c}{8 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*arctan(d*x + c)/d + 1/8*(3*d^3*x^3 + 9*c*d^2*x^2 + 9*c^2*d*x + 3*c^3 + 5*d*x + 5*c)/((d^2*x^2 + 2*c*d*x +
c^2 + 1)^2*d)