3.179 \(\int \frac{\text{csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\)

Optimal. Leaf size=304 \[ -\frac{2 b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac{\coth (c+d x)}{a d} \]

[Out]

(-2*b^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - (
-1)^(2/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*(-1)^(1/3)*b^(2/3)*ArcTa
n[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(4
/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) - (2*b^(2/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2
/3) + b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) + b^(2/3)]*d) - Coth[c + d*x]/(a*d)

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Rubi [A]  time = 0.478469, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3220, 3767, 8, 2660, 618, 204} \[ -\frac{2 b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} d \sqrt{a^{2/3}+b^{2/3}}}-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac{\coth (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - (
-1)^(2/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*(-1)^(1/3)*b^(2/3)*ArcTa
n[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(4
/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) - (2*b^(2/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sqrt[a^(2
/3) + b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) + b^(2/3)]*d) - Coth[c + d*x]/(a*d)

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\text{csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\int \left (-\frac{\text{csch}^2(c+d x)}{a}+\frac{b \sinh (c+d x)}{a \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=\frac{\int \text{csch}^2(c+d x) \, dx}{a}-\frac{b \int \frac{\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx}{a}\\ &=\frac{(i b) \int \left (\frac{\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}-\frac{(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)\right )}-\frac{1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a}-\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{a d}\\ &=-\frac{\coth (c+d x)}{a d}-\frac{\left (i b^{2/3}\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{4/3}}+\frac{\left (\sqrt [6]{-1} b^{2/3}\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{4/3}}+\frac{\left ((-1)^{5/6} b^{2/3}\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{4/3}}\\ &=-\frac{\coth (c+d x)}{a d}-\frac{\left (2 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}+\frac{\left (2 \sqrt [3]{-1} b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-2 \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}-\frac{\left (2 (-1)^{2/3} b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}\\ &=-\frac{\coth (c+d x)}{a d}+\frac{\left (4 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}-\frac{\left (4 \sqrt [3]{-1} b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}+\frac{\left (4 (-1)^{2/3} b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \sqrt [3]{-1} \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}\\ &=-\frac{2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{-1} \sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} \sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac{2 b^{2/3} \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} \sqrt{a^{2/3}+b^{2/3}} d}-\frac{\coth (c+d x)}{a d}\\ \end{align*}

Mathematica [C]  time = 0.390087, size = 230, normalized size = 0.76 \[ -\frac{2 b \text{RootSum}\left [8 \text{$\#$1}^3 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b-b\& ,\frac{2 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1}^2 c+\text{$\#$1}^2 d x-2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-c-d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\& \right ]+3 \tanh \left (\frac{1}{2} (c+d x)\right )+3 \coth \left (\frac{1}{2} (c+d x)\right )}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*Coth[(c + d*x)/2] + 2*b*RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-c - d*x - 2*Log[-Cosh[(
c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + c*#1^2 + d*x*#1^2 + 2*Log[-Co
sh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2*b*#1^
2 + b*#1^4) & ] + 3*Tanh[(c + d*x)/2])/(6*a*d)

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Maple [C]  time = 0.075, size = 123, normalized size = 0.4 \begin{align*} -{\frac{1}{2\,da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{2\,b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}-3\,a{{\it \_Z}}^{4}-8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}-a \right ) }{\frac{{{\it \_R}}^{3}-{\it \_R}}{{{\it \_R}}^{5}a-2\,{{\it \_R}}^{3}a-4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2/(a+b*sinh(d*x+c)^3),x)

[Out]

-1/2/d/a*tanh(1/2*d*x+1/2*c)-1/2/d/a/tanh(1/2*d*x+1/2*c)-2/3/d/a*b*sum((_R^3-_R)/(_R^5*a-2*_R^3*a-4*_R^2*b+_R*
a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a-3*_Z^4*a-8*_Z^3*b+3*_Z^2*a-a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} - 4 \, \int \frac{b e^{\left (4 \, d x + 4 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )}}{a b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a b}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2/(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

-2/(a*d*e^(2*d*x + 2*c) - a*d) - 4*integrate((b*e^(4*d*x + 4*c) - b*e^(2*d*x + 2*c))/(a*b*e^(6*d*x + 6*c) - 3*
a*b*e^(4*d*x + 4*c) + 8*a^2*e^(3*d*x + 3*c) + 3*a*b*e^(2*d*x + 2*c) - a*b), x)

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Fricas [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(csch(d*x+c)^2/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

Timed out

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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(csch(d*x+c)**2/(a+b*sinh(d*x+c)**3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2/(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(csch(d*x + c)^2/(b*sinh(d*x + c)^3 + a), x)