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

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

[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)

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Rubi [A]  time = 0.102519, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4141, 1802, 207} \[ a^3 x-\frac{b^2 (3 a+2 b) \tanh (c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d}+\frac{b^3 \tanh ^3(c+d x)}{3 d} \]

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 4141

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])

Rule 1802

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 207

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

Rubi steps

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

Mathematica [B]  time = 1.76005, size = 126, normalized size = 2.07 \[ \frac{8 (a \cosh (c+d x)+b \text{sech}(c+d x))^3 \left (3 a^3 d x \cosh ^3(c+d x)+\sinh (d x) \cosh ^2(c+d x) \left (3 (a+b)^3 \text{csch}(c) \coth (c+d x)-b^2 (9 a+5 b) \text{sech}(c)\right )-b^3 \tanh (c) \cosh (c+d x)+b^3 (-\text{sech}(c)) \sinh (d x)\right )}{3 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

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 [A]  time = 0.043, size = 111, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c-{\rm coth} \left (dx+c\right ) \right ) -3\,{a}^{2}b{\rm coth} \left (dx+c\right )+3\,a{b}^{2} \left ( -{\frac{1}{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}-2\,\tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{1}{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-4\, \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) \right ) \right ) } \end{align*}

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)

[Out]

1/d*(a^3*(d*x+c-coth(d*x+c))-3*a^2*b*coth(d*x+c)+3*a*b^2*(-1/sinh(d*x+c)/cosh(d*x+c)-2*tanh(d*x+c))+b^3*(-1/si
nh(d*x+c)/cosh(d*x+c)^3-4*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)))

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Maxima [B]  time = 1.17231, size = 232, normalized size = 3.8 \begin{align*} 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{align*}

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]  time = 2.4562, size = 863, normalized size = 14.15 \begin{align*} -\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{align*}

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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]  time = 1.26706, size = 178, normalized size = 2.92 \begin{align*} \frac{3 \, a^{3} d x - \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{align*}

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*a^3*d*x - 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