3.234 \(\int \frac{\text{csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

Optimal. Leaf size=184 \[ \frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{1}{4 a d (1-\cosh (c+d x))}-\frac{1}{4 a d (\cosh (c+d x)+1)}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d} \]

[Out]

(b^(3/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcT
anh[Cosh[c + d*x]]/(2*a*d) + (b^(3/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqr
t[Sqrt[a] + Sqrt[b]]*d) + 1/(4*a*d*(1 - Cosh[c + d*x])) - 1/(4*a*d*(1 + Cosh[c + d*x]))

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Rubi [A]  time = 0.200735, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3215, 1170, 207, 1093, 205, 208} \[ \frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/2} d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/2} d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{1}{4 a d (1-\cosh (c+d x))}-\frac{1}{4 a d (\cosh (c+d x)+1)}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]

[Out]

(b^(3/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcT
anh[Cosh[c + d*x]]/(2*a*d) + (b^(3/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqr
t[Sqrt[a] + Sqrt[b]]*d) + 1/(4*a*d*(1 - Cosh[c + d*x])) - 1/(4*a*d*(1 + Cosh[c + d*x]))

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

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])

Rule 1093

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

Rule 205

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

Rule 208

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

Rubi steps

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

Mathematica [C]  time = 0.378545, size = 265, normalized size = 1.44 \[ -\frac{4 b \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{2 \text{$\#$1}^3 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1}^3 c+\text{$\#$1}^3 d x-2 \text{$\#$1} \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\text{$\#$1} c-\text{$\#$1} d x}{-8 \text{$\#$1}^2 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b-b}\& \right ]+\text{csch}^2\left (\frac{1}{2} (c+d x)\right )+\text{sech}^2\left (\frac{1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{8 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3/(a - b*Sinh[c + d*x]^4),x]

[Out]

-(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + 4*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6
+ b*#1^8 & , (-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c
+ d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 -
Sinh[(c + d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ] + Sech[(c + d*x)/2]^2)/(8*a*d)

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Maple [A]  time = 0.066, size = 190, normalized size = 1. \begin{align*}{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{b}{2\,da}\arctan \left ({\frac{1}{4} \left ( -2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\,\sqrt{ab}+2\,a \right ){\frac{1}{\sqrt{-ab-\sqrt{ab}a}}}} \right ){\frac{1}{\sqrt{-ab-\sqrt{ab}a}}}}+{\frac{b}{2\,da}\arctan \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\,\sqrt{ab}-2\,a \right ){\frac{1}{\sqrt{-ab+\sqrt{ab}a}}}} \right ){\frac{1}{\sqrt{-ab+\sqrt{ab}a}}}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x)

[Out]

1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/2/d*b/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a
*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))+1/2/d*b/a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2
*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))-1/8/d/a/tanh(1/2*d*x+1/2*c)^2-1/2/d/a*ln(tanh(1/2*d*x+1
/2*c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{a d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + a d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, a d} - \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, a d} - 8 \, \int \frac{b e^{\left (5 \, d x + 5 \, c\right )} - b e^{\left (3 \, d x + 3 \, c\right )}}{a b e^{\left (8 \, d x + 8 \, c\right )} - 4 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a b - 2 \,{\left (8 \, a^{2} e^{\left (4 \, c\right )} - 3 \, a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-(e^(3*d*x + 3*c) + e^(d*x + c))/(a*d*e^(4*d*x + 4*c) - 2*a*d*e^(2*d*x + 2*c) + a*d) + 1/2*log((e^(d*x + c) +
1)*e^(-c))/(a*d) - 1/2*log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 8*integrate((b*e^(5*d*x + 5*c) - b*e^(3*d*x + 3*c
))/(a*b*e^(8*d*x + 8*c) - 4*a*b*e^(6*d*x + 6*c) - 4*a*b*e^(2*d*x + 2*c) + a*b - 2*(8*a^2*e^(4*c) - 3*a*b*e^(4*
c))*e^(4*d*x)), x)

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Fricas [B]  time = 2.58645, size = 4716, normalized size = 25.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/4*(4*cosh(d*x + c)^3 + 12*cosh(d*x + c)*sinh(d*x + c)^2 + 4*sinh(d*x + c)^3 - (a*d*cosh(d*x + c)^4 + 4*a*d*
cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*
sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sq
rt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b^2*cosh(d*x +
c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 + 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) - ((a^5 - a^4
*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)))*sqrt(-((a^
4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cosh(d*x + c)^4 +
4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2
- a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^4 - a^3*b)
*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b^2*cos
h(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 - 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) - ((a^
5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)))*sqr
t(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cosh(d*x +
c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x
+ c)^2 - a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^4 -
a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b
^2*cosh(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 + 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c)
+ ((a^5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)
))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cosh(
d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh
(d*x + c)^2 - a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a
^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2
+ 2*b^2*cosh(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 - 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x
+ c) + ((a^5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)
*d^4)))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) - 2*(co
sh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2
- 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*log(cosh(d*x + c) + sinh(d*x + c)
+ 1) + 2*(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sin
h(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*log(cosh(d*x + c) +
sinh(d*x + c) - 1) + 4*(3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + 4*cosh(d*x + c))/(a*d*cosh(d*x + c)^4 + 4*a*d*c
osh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*s
inh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*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(csch(d*x+c)**3/(a-b*sinh(d*x+c)**4),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3/(a-b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError