3.1.4 \(\int (a+b \coth ^2(c+d x))^2 \, dx\) [4]

Optimal. Leaf size=43 \[ (a+b)^2 x-\frac {b (2 a+b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d} \]

[Out]

(a+b)^2*x-b*(2*a+b)*coth(d*x+c)/d-1/3*b^2*coth(d*x+c)^3/d

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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 398, 212} \begin {gather*} -\frac {b (2 a+b) \coth (c+d x)}{d}+x (a+b)^2-\frac {b^2 \coth ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[c + d*x]^2)^2,x]

[Out]

(a + b)^2*x - (b*(2*a + b)*Coth[c + d*x])/d - (b^2*Coth[c + d*x]^3)/(3*d)

Rule 212

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

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[c*(ff/f), Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \left (a+b \coth ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-b (2 a+b)-b^2 x^2+\frac {(a+b)^2}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac {b (2 a+b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 65, normalized size = 1.51 \begin {gather*} -\frac {\coth (c+d x) \left (b \left (6 a+3 b+b \coth ^2(c+d x)\right )-3 (a+b)^2 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \sqrt {\tanh ^2(c+d x)}\right )}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[c + d*x]^2)^2,x]

[Out]

-1/3*(Coth[c + d*x]*(b*(6*a + 3*b + b*Coth[c + d*x]^2) - 3*(a + b)^2*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Sqrt[Tanh[
c + d*x]^2]))/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(41)=82\).
time = 0.34, size = 84, normalized size = 1.95

method	result	size
derivativedivides	\(\frac {-\frac {b^{2} \left (\coth ^{3}\left (d x +c \right )\right )}{3}-2 a b \coth \left (d x +c \right )-b^{2} \coth \left (d x +c \right )-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\coth \left (d x +c \right )-1\right )}{2}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\coth \left (d x +c \right )+1\right )}{2}}{d}\)	\(84\)
default	\(\frac {-\frac {b^{2} \left (\coth ^{3}\left (d x +c \right )\right )}{3}-2 a b \coth \left (d x +c \right )-b^{2} \coth \left (d x +c \right )-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\coth \left (d x +c \right )-1\right )}{2}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\coth \left (d x +c \right )+1\right )}{2}}{d}\)	\(84\)
risch	\(a^{2} x +2 a b x +b^{2} x -\frac {4 b \left (3 a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}-6 a \,{\mathrm e}^{2 d x +2 c}-3 b \,{\mathrm e}^{2 d x +2 c}+3 a +2 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*coth(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3*b^2*coth(d*x+c)^3-2*a*b*coth(d*x+c)-b^2*coth(d*x+c)-1/2*(a^2+2*a*b+b^2)*ln(coth(d*x+c)-1)+1/2*(a^2+2
*a*b+b^2)*ln(coth(d*x+c)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (41) = 82\).
time = 0.27, size = 114, normalized size = 2.65 \begin {gather*} \frac {1}{3} \, b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 2 \, a b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/3*b^2*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) - 2)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x -
4*c) + e^(-6*d*x - 6*c) - 1))) + 2*a*b*(x + c/d + 2/(d*(e^(-2*d*x - 2*c) - 1))) + a^2*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (41) = 82\).
time = 0.39, size = 197, normalized size = 4.58 \begin {gather*} -\frac {2 \, {\left (3 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (3 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 6 \, a b + 4 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} - 6 \, a b \cosh \left (d x + c\right ) + 3 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x - {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 6 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 6 \, a b + 4 \, b^{2}\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/3*(2*(3*a*b + 2*b^2)*cosh(d*x + c)^3 + 6*(3*a*b + 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^2 - (3*(a^2 + 2*a*b +
b^2)*d*x + 6*a*b + 4*b^2)*sinh(d*x + c)^3 - 6*a*b*cosh(d*x + c) + 3*(3*(a^2 + 2*a*b + b^2)*d*x - (3*(a^2 + 2*a
*b + b^2)*d*x + 6*a*b + 4*b^2)*cosh(d*x + c)^2 + 6*a*b + 4*b^2)*sinh(d*x + c))/(d*sinh(d*x + c)^3 + 3*(d*cosh(
d*x + c)^2 - d)*sinh(d*x + c))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (36) = 72\).
time = 1.34, size = 238, normalized size = 5.53 \begin {gather*} \begin {cases} x \left (a + b \coth ^{2}{\left (c \right )}\right )^{2} & \text {for}\: d = 0 \\- \frac {a^{2} \log {\left (- e^{- d x} \right )}}{d} - \frac {2 a b \log {\left (- e^{- d x} \right )} \coth ^{2}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} - \frac {b^{2} \log {\left (- e^{- d x} \right )} \coth ^{4}{\left (d x + \log {\left (- e^{- d x} \right )} \right )}}{d} & \text {for}\: c = \log {\left (- e^{- d x} \right )} \\- \frac {a^{2} \log {\left (e^{- d x} \right )}}{d} - \frac {2 a b \log {\left (e^{- d x} \right )} \coth ^{2}{\left (d x + \log {\left (e^{- d x} \right )} \right )}}{d} - \frac {b^{2} \log {\left (e^{- d x} \right )} \coth ^{4}{\left (d x + \log {\left (e^{- d x} \right )} \right )}}{d} & \text {for}\: c = \log {\left (e^{- d x} \right )} \\a^{2} x + 2 a b x - \frac {2 a b}{d \tanh {\left (c + d x \right )}} + b^{2} x - \frac {b^{2}}{d \tanh {\left (c + d x \right )}} - \frac {b^{2}}{3 d \tanh ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)**2)**2,x)

[Out]

Piecewise((x*(a + b*coth(c)**2)**2, Eq(d, 0)), (-a**2*log(-exp(-d*x))/d - 2*a*b*log(-exp(-d*x))*coth(d*x + log
(-exp(-d*x)))**2/d - b**2*log(-exp(-d*x))*coth(d*x + log(-exp(-d*x)))**4/d, Eq(c, log(-exp(-d*x)))), (-a**2*lo
g(exp(-d*x))/d - 2*a*b*log(exp(-d*x))*coth(d*x + log(exp(-d*x)))**2/d - b**2*log(exp(-d*x))*coth(d*x + log(exp
(-d*x)))**4/d, Eq(c, log(exp(-d*x)))), (a**2*x + 2*a*b*x - 2*a*b/(d*tanh(c + d*x)) + b**2*x - b**2/(d*tanh(c +
d*x)) - b**2/(3*d*tanh(c + d*x)**3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (41) = 82\).
time = 0.41, size = 103, normalized size = 2.40 \begin {gather*} \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b + 2 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/3*(3*(a^2 + 2*a*b + b^2)*(d*x + c) - 4*(3*a*b*e^(4*d*x + 4*c) + 3*b^2*e^(4*d*x + 4*c) - 6*a*b*e^(2*d*x + 2*c
) - 3*b^2*e^(2*d*x + 2*c) + 3*a*b + 2*b^2)/(e^(2*d*x + 2*c) - 1)^3)/d

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Mupad [B]
time = 0.11, size = 41, normalized size = 0.95 \begin {gather*} x\,{\left (a+b\right )}^2-\frac {b^2\,{\mathrm {coth}\left (c+d\,x\right )}^3}{3\,d}-\frac {b\,\mathrm {coth}\left (c+d\,x\right )\,\left (2\,a+b\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(c + d*x)^2)^2,x)

[Out]

x*(a + b)^2 - (b^2*coth(c + d*x)^3)/(3*d) - (b*coth(c + d*x)*(2*a + b))/d

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