3.2.20 \(\int \coth ^4(c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [120]

Optimal. Leaf size=46 \[ a^2 x-\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4226, 1816, 213} \begin {gather*} -\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}+a^2 x-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2,x]

[Out]

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

Rule 213

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

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 4226

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2
*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rubi steps

\begin {align*} \int \coth ^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \left (1-x^2\right )\right )^2}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {(a+b)^2}{x^4}+\frac {a^2-b^2}{x^2}-\frac {a^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}-\frac {a^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x-\frac {\left (a^2-b^2\right ) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(46)=92\).
time = 0.56, size = 160, normalized size = 3.48 \begin {gather*} \frac {\text {csch}(c) \text {csch}^3(c+d x) \left (9 a^2 d x \cosh (d x)-9 a^2 d x \cosh (2 c+d x)-3 a^2 d x \cosh (2 c+3 d x)+3 a^2 d x \cosh (4 c+3 d x)-12 a^2 \sinh (d x)+12 b^2 \sinh (d x)-12 a^2 \sinh (2 c+d x)-12 a b \sinh (2 c+d x)+8 a^2 \sinh (2 c+3 d x)+4 a b \sinh (2 c+3 d x)-4 b^2 \sinh (2 c+3 d x)\right )}{24 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2,x]

[Out]

(Csch[c]*Csch[c + d*x]^3*(9*a^2*d*x*Cosh[d*x] - 9*a^2*d*x*Cosh[2*c + d*x] - 3*a^2*d*x*Cosh[2*c + 3*d*x] + 3*a^
2*d*x*Cosh[4*c + 3*d*x] - 12*a^2*Sinh[d*x] + 12*b^2*Sinh[d*x] - 12*a^2*Sinh[2*c + d*x] - 12*a*b*Sinh[2*c + d*x
] + 8*a^2*Sinh[2*c + 3*d*x] + 4*a*b*Sinh[2*c + 3*d*x] - 4*b^2*Sinh[2*c + 3*d*x]))/(24*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(44)=88\).
time = 2.16, size = 94, normalized size = 2.04

method	result	size
risch	\(a^{2} x -\frac {4 \left (3 a^{2} {\mathrm e}^{4 d x +4 c}+3 a b \,{\mathrm e}^{4 d x +4 c}-3 a^{2} {\mathrm e}^{2 d x +2 c}+3 b^{2} {\mathrm e}^{2 d x +2 c}+2 a^{2}+a b -b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^4*(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (44) = 88\).
time = 0.27, size = 268, normalized size = 5.83 \begin {gather*} \frac {1}{3} \, a^{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 )} + \frac {4}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\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 )}} - \frac {1}{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 )} + \frac {4}{3} \, a b {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\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 )}} + \frac {1}{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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^4*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (44) = 88\).
time = 0.39, size = 201, normalized size = 4.37 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} + a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (2 \, a^{2} + a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a^{2} d x + 4 \, a^{2} + 2 \, a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (a b + b^{2}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (3 \, a^{2} d x - {\left (3 \, a^{2} d x + 4 \, a^{2} + 2 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, a^{2} + 2 \, a b - 2 \, 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(coth(d*x+c)^4*(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/3*(2*(2*a^2 + a*b - b^2)*cosh(d*x + c)^3 + 6*(2*a^2 + a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^2 - (3*a^2*d*x
+ 4*a^2 + 2*a*b - 2*b^2)*sinh(d*x + c)^3 + 6*(a*b + b^2)*cosh(d*x + c) + 3*(3*a^2*d*x - (3*a^2*d*x + 4*a^2 +
2*a*b - 2*b^2)*cosh(d*x + c)^2 + 4*a^2 + 2*a*b - 2*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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**4*(a+b*sech(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (44) = 88\).
time = 0.45, size = 100, normalized size = 2.17 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a^{2} - \frac {4 \, {\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} + a b - 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(coth(d*x+c)^4*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.42, size = 183, normalized size = 3.98 \begin {gather*} a^2\,x-\frac {\frac {4\,\left (a^2+b\,a\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2+b\,a\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+a\,b\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {4\,\left (b^2+a\,b\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2+b\,a\right )}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {4\,\left (a^2+b\,a\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^4*(a + b/cosh(c + d*x)^2)^2,x)

[Out]

a^2*x - ((4*(a*b + a^2))/(3*d) + (4*exp(4*c + 4*d*x)*(a*b + a^2))/(3*d) + (8*exp(2*c + 2*d*x)*(a*b + b^2))/(3*
d))/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((4*(a*b + b^2))/(3*d) + (4*exp(2*c + 2
*d*x)*(a*b + a^2))/(3*d))/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - (4*(a*b + a^2))/(3*d*(exp(2*c + 2*d*x)
- 1))

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