3.133 \(\int \coth ^5(c+d x) (a+b \text{sech}^2(c+d x))^3 \, dx\)

Optimal. Leaf size=81 \[ \frac{\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^4(c+d x)}{4 d}-\frac{(2 a-b) (a+b)^2 \text{csch}^2(c+d x)}{2 d}-\frac{b^3 \log (\cosh (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.120995, antiderivative size = 81, 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 = {4138, 446, 88} \[ \frac{\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}-\frac{(a+b)^3 \text{csch}^4(c+d x)}{4 d}-\frac{(2 a-b) (a+b)^2 \text{csch}^2(c+d x)}{2 d}-\frac{b^3 \log (\cosh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

-((2*a - b)*(a + b)^2*Csch[c + d*x]^2)/(2*d) - ((a + b)^3*Csch[c + d*x]^4)/(4*d) - (b^3*Log[Cosh[c + d*x]])/d
+ ((a^3 + b^3)*Log[Sinh[c + d*x]])/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 ^5(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 \left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^3}{(1-x)^3 x} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^3}{(-1+x)^3}-\frac{(2 a-b) (a+b)^2}{(-1+x)^2}+\frac{-a^3-b^3}{-1+x}+\frac{b^3}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{(2 a-b) (a+b)^2 \text{csch}^2(c+d x)}{2 d}-\frac{(a+b)^3 \text{csch}^4(c+d x)}{4 d}-\frac{b^3 \log (\cosh (c+d x))}{d}+\frac{\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.858402, size = 101, normalized size = 1.25 \[ -\frac{2 \left (a \cosh ^2(c+d x)+b\right )^3 \left (-4 \left (a^3+b^3\right ) \log (\sinh (c+d x))+(a+b)^3 \text{csch}^4(c+d x)+2 (2 a-b) (a+b)^2 \text{csch}^2(c+d x)+4 b^3 \log (\cosh (c+d x))\right )}{d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3,x]

[Out]

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

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Maple [B]  time = 0.047, size = 194, normalized size = 2.4 \begin{align*}{\frac{{a}^{3}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}}{4\,d}}-{\frac{3\,{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{2}b \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,a{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a{b}^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3}}{4\,d \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{3}}{2\,d \left ( \sinh \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)^5*(a+b*sech(d*x+c)^2)^3,x)

[Out]

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

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Maxima [B]  time = 1.70119, size = 570, normalized size = 7.04 \begin{align*} a^{3}{\left (x + \frac{c}{d} + \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{4 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 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 )} + 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 )} + 6 \, a^{2} b{\left (\frac{e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac{e^{\left (-6 \, d x - 6 \, c\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 )} - \frac{12 \, a b^{2}}{d{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.74878, size = 4574, normalized size = 56.47 \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)^5*(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 -
2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 - 2*a^3*d*x + 2*a^3 + 3*a^2*b - b^3)*s
inh(d*x + c)^6 + 4*(14*a^3*d*x*cosh(d*x + c)^3 - 3*(2*a^3*d*x - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c))*sinh(d*x
+ c)^5 + a^3*d*x + 2*(3*a^3*d*x - 2*a^3 + 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 - 2*a^3 + 6*a*b^2 + 4*b^3 - 15*(2*a^3*d*x - 2*a^3 - 3*a^2*b + 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 - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^3 + (3*a^3*d*x - 2*a^3 +
6*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a^3*d*x - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^2 + 2*(1
4*a^3*d*x*cosh(d*x + c)^6 - 2*a^3*d*x - 15*(2*a^3*d*x - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^4 + 2*a^3 + 3*a^2
*b - b^3 + 6*(3*a^3*d*x - 2*a^3 + 6*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + (b^3*cosh(d*x + c)^8 + 8
*b^3*cosh(d*x + c)*sinh(d*x + c)^7 + b^3*sinh(d*x + c)^8 - 4*b^3*cosh(d*x + c)^6 + 6*b^3*cosh(d*x + c)^4 + 4*(
7*b^3*cosh(d*x + c)^2 - b^3)*sinh(d*x + c)^6 + 8*(7*b^3*cosh(d*x + c)^3 - 3*b^3*cosh(d*x + c))*sinh(d*x + c)^5
- 4*b^3*cosh(d*x + c)^2 + 2*(35*b^3*cosh(d*x + c)^4 - 30*b^3*cosh(d*x + c)^2 + 3*b^3)*sinh(d*x + c)^4 + 8*(7*
b^3*cosh(d*x + c)^5 - 10*b^3*cosh(d*x + c)^3 + 3*b^3*cosh(d*x + c))*sinh(d*x + c)^3 + b^3 + 4*(7*b^3*cosh(d*x
+ c)^6 - 15*b^3*cosh(d*x + c)^4 + 9*b^3*cosh(d*x + c)^2 - b^3)*sinh(d*x + c)^2 + 8*(b^3*cosh(d*x + c)^7 - 3*b^
3*cosh(d*x + c)^5 + 3*b^3*cosh(d*x + c)^3 - 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 + b^3)*cosh(d*x + c)^8 + 8*(a^3 + b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 + b^3
)*sinh(d*x + c)^8 - 4*(a^3 + b^3)*cosh(d*x + c)^6 - 4*(a^3 + b^3 - 7*(a^3 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^6 + 8*(7*(a^3 + b^3)*cosh(d*x + c)^3 - 3*(a^3 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(a^3 + b^3)*cosh(d*x
+ c)^4 + 2*(35*(a^3 + b^3)*cosh(d*x + c)^4 + 3*a^3 + 3*b^3 - 30*(a^3 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4
+ 8*(7*(a^3 + b^3)*cosh(d*x + c)^5 - 10*(a^3 + b^3)*cosh(d*x + c)^3 + 3*(a^3 + b^3)*cosh(d*x + c))*sinh(d*x +
c)^3 + a^3 + b^3 - 4*(a^3 + b^3)*cosh(d*x + c)^2 + 4*(7*(a^3 + b^3)*cosh(d*x + c)^6 - 15*(a^3 + b^3)*cosh(d*x
+ c)^4 - a^3 - b^3 + 9*(a^3 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^3 + b^3)*cosh(d*x + c)^7 - 3*(a^3
+ b^3)*cosh(d*x + c)^5 + 3*(a^3 + b^3)*cosh(d*x + c)^3 - (a^3 + 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 - 2*a^3 - 3*a^2*b + b^
3)*cosh(d*x + c)^5 + 2*(3*a^3*d*x - 2*a^3 + 6*a*b^2 + 4*b^3)*cosh(d*x + c)^3 - (2*a^3*d*x - 2*a^3 - 3*a^2*b +
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)**5*(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.58104, size = 335, normalized size = 4.14 \begin{align*} -\frac{12 \, a^{3} d x + 12 \, b^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 12 \,{\left (a^{3} 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{25 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 25 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 52 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 72 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 124 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 102 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 246 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 52 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 72 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 124 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{3} + 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)^5*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/12*(12*a^3*d*x + 12*b^3*log(e^(2*d*x + 2*c) + 1) - 12*(a^3*e^(2*c) + b^3*e^(2*c))*e^(-2*c)*log(abs(e^(2*d*x
+ 2*c) - 1)) + (25*a^3*e^(8*d*x + 8*c) + 25*b^3*e^(8*d*x + 8*c) - 52*a^3*e^(6*d*x + 6*c) + 72*a^2*b*e^(6*d*x
+ 6*c) - 124*b^3*e^(6*d*x + 6*c) + 102*a^3*e^(4*d*x + 4*c) + 144*a*b^2*e^(4*d*x + 4*c) + 246*b^3*e^(4*d*x + 4*
c) - 52*a^3*e^(2*d*x + 2*c) + 72*a^2*b*e^(2*d*x + 2*c) - 124*b^3*e^(2*d*x + 2*c) + 25*a^3 + 25*b^3)/(e^(2*d*x
+ 2*c) - 1)^4)/d