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

Optimal. Leaf size=84 \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac{b^2 (3 a+b) \text{sech}^2(c+d x)}{2 d}+\frac{(a+b)^3 \log (\sinh (c+d x))}{d}+\frac{b^3 \text{sech}^4(c+d x)}{4 d} \]

[Out]

-((b*(3*a^2 + 3*a*b + b^2)*Log[Cosh[c + d*x]])/d) + ((a + b)^3*Log[Sinh[c + d*x]])/d + (b^2*(3*a + b)*Sech[c +
d*x]^2)/(2*d) + (b^3*Sech[c + d*x]^4)/(4*d)

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Rubi [A]  time = 0.106783, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4138, 446, 88} \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac{b^2 (3 a+b) \text{sech}^2(c+d x)}{2 d}+\frac{(a+b)^3 \log (\sinh (c+d x))}{d}+\frac{b^3 \text{sech}^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(3*a^2 + 3*a*b + b^2)*Log[Cosh[c + d*x]])/d) + ((a + b)^3*Log[Sinh[c + d*x]])/d + (b^2*(3*a + b)*Sech[c +
d*x]^2)/(2*d) + (b^3*Sech[c + d*x]^4)/(4*d)

Rule 4138

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \coth (c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (b+a x^2\right )^3}{x^5 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^3}{(1-x) x^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^3}{-1+x}+\frac{b^3}{x^3}+\frac{b^2 (3 a+b)}{x^2}+\frac{b \left (3 a^2+3 a b+b^2\right )}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))}{d}+\frac{(a+b)^3 \log (\sinh (c+d x))}{d}+\frac{b^2 (3 a+b) \text{sech}^2(c+d x)}{2 d}+\frac{b^3 \text{sech}^4(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.619703, size = 114, normalized size = 1.36 \[ -\frac{2 \cosh ^6(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \left (4 b \left (3 a^2+3 a b+b^2\right ) \log (\cosh (c+d x))-2 b^2 (3 a+b) \text{sech}^2(c+d x)-4 (a+b)^3 \log (\sinh (c+d x))-b^3 \text{sech}^4(c+d x)\right )}{d (a \cosh (2 c+2 d x)+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(4*b*(3*a^2 + 3*a*b + b^2)*Log[Cosh[c + d*x]] - 4*(a + b)^3*Log[
Sinh[c + d*x]] - 2*b^2*(3*a + b)*Sech[c + d*x]^2 - b^3*Sech[c + d*x]^4))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)

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Maple [A]  time = 0.048, size = 111, normalized size = 1.3 \begin{align*}{\frac{{a}^{3}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{2}b\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,a{b}^{2}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{a{b}^{2}\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{3}}{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{3}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.62742, size = 405, normalized size = 4.82 \begin{align*} b^{3}{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 4 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + 3 \, a b^{2}{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + 3 \, a^{2} b{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \frac{a^{3} \log \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.72209, size = 5819, normalized size = 69.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*d*x*cosh(d*x + c)^8 + 8*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*d*x*sinh(d*x + c)^8 + 2*(2*a^3*d*x -
3*a*b^2 - b^3)*cosh(d*x + c)^6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 + 2*a^3*d*x - 3*a*b^2 - b^3)*sinh(d*x + c)^6 +
4*(14*a^3*d*x*cosh(d*x + c)^3 + 3*(2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + a^3*d*x + 2*(3
*a^3*d*x - 6*a*b^2 - 4*b^3)*cosh(d*x + c)^4 + 2*(35*a^3*d*x*cosh(d*x + c)^4 + 3*a^3*d*x - 6*a*b^2 - 4*b^3 + 15
*(2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^3*d*x*cosh(d*x + c)^5 + 5*(2*a^3*d*x -
3*a*b^2 - b^3)*cosh(d*x + c)^3 + (3*a^3*d*x - 6*a*b^2 - 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(2*a^3*d*x -
3*a*b^2 - b^3)*cosh(d*x + c)^2 + 2*(14*a^3*d*x*cosh(d*x + c)^6 + 2*a^3*d*x + 15*(2*a^3*d*x - 3*a*b^2 - b^3)*c
osh(d*x + c)^4 - 3*a*b^2 - b^3 + 6*(3*a^3*d*x - 6*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^2*b
+ 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 8*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b + 3*a*
b^2 + b^3)*sinh(d*x + c)^8 + 4*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 4*(3*a^2*b + 3*a*b^2 + b^3 + 7*(3*a
^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(3
*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(
3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 9*a^2*b + 9*a*b^2 + 3*b^3 + 30*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 10*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*
x + c)^3 + 3*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a^2*b + 3*a*b^2 + b^3 + 4*(3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 15*(3*a^2*b + 3*a*b^2 + b^3
)*cosh(d*x + c)^4 + 3*a^2*b + 3*a*b^2 + b^3 + 9*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8
*((3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 3*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 3*(3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^3 + (3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(
d*x + c) - sinh(d*x + c))) - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 8*(a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^8 + 4*(a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^6 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^6 + 8*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*(35*(a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*cosh(d*x + c)^4 + 3*a^3 + 9*a^2*b + 9*a*b^2 + 3*b^3 + 30*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*
x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 10*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 3*a^2*b + 3
*a*b^2 + b^3 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2 + 4*(7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d
*x + c)^6 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 9*(a^3 + 3*a^
2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 3
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 +
3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) +
4*(2*a^3*d*x*cosh(d*x + c)^7 + 3*(2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(3*a^3*d*x - 6*a*b^2 - 4*b^3)
*cosh(d*x + c)^3 + (2*a^3*d*x - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x
+ c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6
+ 8*(7*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4
+ 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 3*d*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 + 15*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)
^2 + d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 + 3*d*cosh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*
sinh(d*x + c) + d)

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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)*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.20856, size = 439, normalized size = 5.23 \begin{align*} -\frac{12 \, a^{3} d x + 12 \,{\left (3 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 12 \,{\left (a^{3} e^{\left (2 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) - \frac{75 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 75 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 25 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 300 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 372 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 124 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 450 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 594 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 246 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 300 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 372 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 124 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 75 \, a^{2} b + 75 \, a b^{2} + 25 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*a^3*d*x + 12*(3*a^2*b*e^(2*c) + 3*a*b^2*e^(2*c) + b^3*e^(2*c))*e^(-2*c)*log(e^(2*d*x + 2*c) + 1) - 1
2*(a^3*e^(2*c) + 3*a^2*b*e^(2*c) + 3*a*b^2*e^(2*c) + b^3*e^(2*c))*e^(-2*c)*log(abs(e^(2*d*x + 2*c) - 1)) - (75
*a^2*b*e^(8*d*x + 8*c) + 75*a*b^2*e^(8*d*x + 8*c) + 25*b^3*e^(8*d*x + 8*c) + 300*a^2*b*e^(6*d*x + 6*c) + 372*a
*b^2*e^(6*d*x + 6*c) + 124*b^3*e^(6*d*x + 6*c) + 450*a^2*b*e^(4*d*x + 4*c) + 594*a*b^2*e^(4*d*x + 4*c) + 246*b
^3*e^(4*d*x + 4*c) + 300*a^2*b*e^(2*d*x + 2*c) + 372*a*b^2*e^(2*d*x + 2*c) + 124*b^3*e^(2*d*x + 2*c) + 75*a^2*
b + 75*a*b^2 + 25*b^3)/(e^(2*d*x + 2*c) + 1)^4)/d