3.2.30 \(\int \coth ^2(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [130]

Optimal. Leaf size=61 \[ a^3 x-\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 61, 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*} a^3 x-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

a^3*x - ((a + b)^3*Coth[c + d*x])/d - (b^2*(3*a + 2*b)*Tanh[c + d*x])/d + (b^3*Tanh[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 ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \left (1-x^2\right )\right )^3}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-b^2 (3 a+2 b)+\frac {(a+b)^3}{x^2}+b^3 x^2-\frac {a^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d}-\frac {a^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^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. \(126\) vs. \(2(61)=122\).
time = 1.21, size = 126, normalized size = 2.07 \begin {gather*} \frac {8 (a \cosh (c+d x)+b \text {sech}(c+d x))^3 \left (3 a^3 d x \cosh ^3(c+d x)-b^3 \text {sech}(c) \sinh (d x)+\cosh ^2(c+d x) \left (3 (a+b)^3 \coth (c+d x) \text {csch}(c)-b^2 (9 a+5 b) \text {sech}(c)\right ) \sinh (d x)-b^3 \cosh (c+d x) \tanh (c)\right )}{3 d (a+2 b+a \cosh (2 (c+d x)))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(8*(a*Cosh[c + d*x] + b*Sech[c + d*x])^3*(3*a^3*d*x*Cosh[c + d*x]^3 - b^3*Sech[c]*Sinh[d*x] + Cosh[c + d*x]^2*
(3*(a + b)^3*Coth[c + d*x]*Csch[c] - b^2*(9*a + 5*b)*Sech[c])*Sinh[d*x] - b^3*Cosh[c + d*x]*Tanh[c]))/(3*d*(a
+ 2*b + a*Cosh[2*(c + d*x)])^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(191\) vs. \(2(59)=118\).
time = 2.67, size = 192, normalized size = 3.15

method	result	size
risch	\(a^{3} x -\frac {2 \left (3 a^{3} {\mathrm e}^{6 d x +6 c}+9 a^{2} b \,{\mathrm e}^{6 d x +6 c}+9 a^{3} {\mathrm e}^{4 d x +4 c}+27 a^{2} b \,{\mathrm e}^{4 d x +4 c}+18 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 a^{3} {\mathrm e}^{2 d x +2 c}+27 a^{2} b \,{\mathrm e}^{2 d x +2 c}+36 a \,b^{2} {\mathrm e}^{2 d x +2 c}+16 b^{3} {\mathrm e}^{2 d x +2 c}+3 a^{3}+9 a^{2} b +18 a \,b^{2}+8 b^{3}\right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^2*(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

a^3*x-2/3*(3*a^3*exp(6*d*x+6*c)+9*a^2*b*exp(6*d*x+6*c)+9*a^3*exp(4*d*x+4*c)+27*a^2*b*exp(4*d*x+4*c)+18*a*b^2*e
xp(4*d*x+4*c)+9*a^3*exp(2*d*x+2*c)+27*a^2*b*exp(2*d*x+2*c)+36*a*b^2*exp(2*d*x+2*c)+16*b^3*exp(2*d*x+2*c)+3*a^3
+9*a^2*b+18*a*b^2+8*b^3)/d/(1+exp(2*d*x+2*c))^3/(exp(2*d*x+2*c)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (59) = 118\).
time = 0.27, size = 172, normalized size = 2.82 \begin {gather*} a^{3} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} - \frac {16}{3} \, b^{3} {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}} + \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac {12 \, a b^{2}}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

a^3*(x + c/d + 2/(d*(e^(-2*d*x - 2*c) - 1))) - 16/3*b^3*(2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d
*x - 6*c) - e^(-8*d*x - 8*c) + 1)) + 1/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) + 1))) +
6*a^2*b/(d*(e^(-2*d*x - 2*c) - 1)) + 12*a*b^2/(d*(e^(-4*d*x - 4*c) - 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (59) = 118\).
time = 0.49, size = 359, normalized size = 5.89 \begin {gather*} -\frac {{\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 4 \, {\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 9 \, a^{3} + 27 \, a^{2} b + 18 \, a b^{2} + 4 \, {\left (3 \, a^{3} + 9 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{3} + 18 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3} + 3 \, {\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 4 \, {\left ({\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{12 \, {\left (d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\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)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/12*((3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)^4 - 4*(3*a^3*d*x + 3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^
3)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*sinh(d*x + c)^4 + 9*a^3 + 27*a^2*b + 1
8*a*b^2 + 4*(3*a^3 + 9*a^2*b + 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 2*(6*a^3 + 18*a^2*b + 18*a*b^2 + 8*b^3 + 3*(
3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 4*((3*a^3*d*x + 3*a^3 + 9*a^2*b + 18*a*
b^2 + 8*b^3)*cosh(d*x + c)^3 + (3*a^3*d*x + 3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c))/
(d*cosh(d*x + c)*sinh(d*x + c)^3 + (d*cosh(d*x + c)^3 + d*cosh(d*x + c))*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)**2*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (59) = 118\).
time = 0.43, size = 135, normalized size = 2.21 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a^{3} - \frac {6 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {2 \, {\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 5 \, b^{3}\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)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/3*(3*(d*x + c)*a^3 - 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(e^(2*d*x + 2*c) - 1) + 2*(9*a*b^2*e^(4*d*x + 4*c) +
3*b^3*e^(4*d*x + 4*c) + 18*a*b^2*e^(2*d*x + 2*c) + 12*b^3*e^(2*d*x + 2*c) + 9*a*b^2 + 5*b^3)/(e^(2*d*x + 2*c)
+ 1)^3)/d

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Mupad [B]
time = 0.14, size = 234, normalized size = 3.84 \begin {gather*} \frac {\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (b^3+3\,a\,b^2\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}+a^3\,x+\frac {\frac {2\,\left (b^3+a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{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)^2*(a + b/cosh(c + d*x)^2)^3,x)

[Out]

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

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