3.489 \(\int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=80 \[ \frac {b \text {csch}(c+d x)}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d}-\frac {\text {csch}^2(c+d x)}{2 a d} \]

[Out]

b*csch(d*x+c)/a^2/d-1/2*csch(d*x+c)^2/a/d+(a^2+b^2)*ln(sinh(d*x+c))/a^3/d-(a^2+b^2)*ln(a+b*sinh(d*x+c))/a^3/d

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Rubi [A]  time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2721, 894} \[ \frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 64, normalized size = 0.80 \[ \frac {2 \left (a^2+b^2\right ) (\log (\sinh (c+d x))-\log (a+b \sinh (c+d x)))-a^2 \text {csch}^2(c+d x)+2 a b \text {csch}(c+d x)}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 617, normalized size = 7.71 \[ \frac {2 \, a b \cosh \left (d x + c\right )^{3} + 2 \, a b \sinh \left (d x + c\right )^{3} - 2 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )}{a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} - 2 \, a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d + 2 \, {\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} - a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - a^{3} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*b*cosh(d*x + c)^3 + 2*a*b*sinh(d*x + c)^3 - 2*a^2*cosh(d*x + c)^2 - 2*a*b*cosh(d*x + c) + 2*(3*a*b*cosh(d
*x + c) - a^2)*sinh(d*x + c)^2 - ((a^2 + b^2)*cosh(d*x + c)^4 + 4*(a^2 + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 +
(a^2 + b^2)*sinh(d*x + c)^4 - 2*(a^2 + b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + b^2)*cosh(d*x + c)^2 - a^2 - b^2)*si
nh(d*x + c)^2 + a^2 + b^2 + 4*((a^2 + b^2)*cosh(d*x + c)^3 - (a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*(
b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + ((a^2 + b^2)*cosh(d*x + c)^4 + 4*(a^2 + b^2)*cosh(d*x
+ c)*sinh(d*x + c)^3 + (a^2 + b^2)*sinh(d*x + c)^4 - 2*(a^2 + b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + b^2)*cosh(d*x
+ c)^2 - a^2 - b^2)*sinh(d*x + c)^2 + a^2 + b^2 + 4*((a^2 + b^2)*cosh(d*x + c)^3 - (a^2 + b^2)*cosh(d*x + c))
*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(3*a*b*cosh(d*x + c)^2 - 2*a^2*cosh(d
*x + c) - a*b)*sinh(d*x + c))/(a^3*d*cosh(d*x + c)^4 + 4*a^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d*sinh(d*x
+ c)^4 - 2*a^3*d*cosh(d*x + c)^2 + a^3*d + 2*(3*a^3*d*cosh(d*x + c)^2 - a^3*d)*sinh(d*x + c)^2 + 4*(a^3*d*cosh
(d*x + c)^3 - a^3*d*cosh(d*x + c))*sinh(d*x + c))

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giac [B]  time = 1.72, size = 166, normalized size = 2.08 \[ \frac {\frac {{\left (a^{2} e^{c} + b^{2} e^{c}\right )} e^{\left (-c\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} e^{c} + b^{2} e^{c}\right )} e^{\left (-c\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b \right |}\right )}{a^{3}} + \frac {2 \, {\left (a b e^{\left (3 \, d x + 3 \, c\right )} - a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b e^{\left (d x + c\right )}\right )}}{a^{3} {\left (e^{\left (d x + c\right )} + 1\right )}^{2} {\left (e^{\left (d x + c\right )} - 1\right )}^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 194, normalized size = 2.42 \[ -\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{2 d \,a^{2}}-\frac {\ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d a}-\frac {\ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right ) b^{2}}{d \,a^{3}}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{d \,a^{3}}+\frac {b}{2 d \,a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

-1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/2/d/a^2*tanh(1/2*d*x+1/2*c)*b-1/d/a*ln(tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x
+1/2*c)*b-a)-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)*b-a)*b^2-1/8/d/a/tanh(1/2*d*x+1/2*c)^2+1
/d/a*ln(tanh(1/2*d*x+1/2*c))+1/d/a^3*ln(tanh(1/2*d*x+1/2*c))*b^2+1/2/d*b/a^2/tanh(1/2*d*x+1/2*c)

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maxima [B]  time = 0.34, size = 173, normalized size = 2.16 \[ -\frac {2 \, {\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{3} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.01, size = 1329, normalized size = 16.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^3/(a + b*sinh(c + d*x)),x)

[Out]

((2*atan((a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^
(1/2))/(2*a^3*d*(a^2 + b^2)^2) + ((a^7*d + a^5*b^2*d)*(-a^6*d^2)^(1/2))/(2*a^6*d^2*((a^2 + b^2)^2)^(1/2)*(a^2
+ b^2)) - (a^6*b^2*exp(2*c)*exp(2*d*x)*(-a^6*d^2)^(1/2)*((4*(a^2 + 2*b^2)*(a^4 + b^4 + 2*a^2*b^2))/(a^9*b^2*d*
(a^2 + b^2)^2) + (2*(2*a^4*b^3*d + 2*a^6*b*d)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^11*b^3*d^2*((a^2 + b^2)^2)^(1/
2)*(a^2 + b^2)) + (4*(a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 +
2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^9*b^2*d*(a^2 + b^2)^2*(-a^6*d^2)^(1/2)) + (4*(a^7*d + a^5
*b^2*d)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^12*b^2*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2))))/(8*(a^4 + b^4 + 2*a^
2*b^2)^(1/2)) + (a^6*b^2*exp(3*c)*exp(3*d*x)*((2*(a^7*d + a^5*b^2*d)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^11*b^3*
d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2)
+ 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^10*b^3*d*(a^2 + b^2)
^2*(-a^6*d^2)^(1/2)))*(-a^6*d^2)^(1/2))/(8*(a^4 + b^4 + 2*a^2*b^2)^(1/2)) - (a^6*b^2*exp(d*x)*exp(c)*(-a^6*d^2
)^(1/2)*((8*(a^4 + b^4 + 2*a^2*b^2))/(a^8*b*d*(a^2 + b^2)^2) - (4*(2*a^4*b^3*d + 2*a^6*b*d)*(a^4 + b^4 + 2*a^2
*b^2)^(1/2))/(a^12*b^2*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) + (2*(a^7*d + a^5*b^2*d)*(a^4 + b^4 + 2*a^2*b^2)
^(1/2))/(a^11*b^3*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 +
2*a^2*b^2)^(1/2) + 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^10
*b^3*d*(a^2 + b^2)^2*(-a^6*d^2)^(1/2))))/(8*(a^4 + b^4 + 2*a^2*b^2)^(1/2))) - 2*atan((4*a^6*b*d*(a^2 + b^2)^2*
(-a^6*d^2)^(1/2) + 4*a^4*b^3*d*(a^2 + b^2)^2*(-a^6*d^2)^(1/2))*(1/(8*a^5*b*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^
2)^3) - exp(d*x)*exp(c)*(1/(16*a^4*b^2*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3) - (a^2 + 2*b^2)^2/(16*a^8*b^2*
d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3)) + (a^2 + 2*b^2)/(8*a^7*b*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3))))
*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(-a^6*d^2)^(1/2) - (2/(a*d) - (2*b*exp(c + d*x))/(a^2*d))/(exp(2*c + 2*d*x) -
1) - 2/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Integral(coth(c + d*x)**3/(a + b*sinh(c + d*x)), x)

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