∫ csch(c+dx)__ a−b sinh4(c+dx)dx

3.233 $\int \frac{\text{csch}(c+d x)}{a-b \sinh ^4(c+d x)} \, dx$

Optimal. Leaf size=136 \[ -\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d} \]

[Out]

-(b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[C

osh[c + d*x]]/(a*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + S

qrt[b]]*d)

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Rubi [A] time = 0.163084, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.273, Rules used = {3215, 1170, 207, 1166, 205, 208} \[ -\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[C

osh[c + d*x]]/(a*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + S

qrt[b]]*d)

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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di

st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +

q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 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}(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \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}{a \left (-1+x^2\right )}+\frac{b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{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))}{a d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a d}\\ &=-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a \sqrt{\sqrt{a}-\sqrt{b}} d}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a \sqrt{\sqrt{a}+\sqrt{b}} d}\\ \end{align*}

Mathematica [C] time = 0.248951, size = 385, normalized size = 2.83 \[ \frac{8 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-b \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{2 \text{$\#$1}^6 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-6 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+6 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1}^6 c-3 \text{$\#$1}^4 c+3 \text{$\#$1}^2 c+\text{$\#$1}^6 d x-3 \text{$\#$1}^4 d x+3 \text{$\#$1}^2 d x-2 \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-c-d x}{-8 \text{$\#$1}^3 a+\text{$\#$1}^7 b-3 \text{$\#$1}^5 b+3 \text{$\#$1}^3 b-\text{$\#$1} b}\& \right ]}{8 a d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

*#1^2 + 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 3*c

*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#

1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c +

d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(8*a*d)

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Maple [A] time = 0.053, size = 159, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,da}\sqrt{ab}\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}{2\,da}\sqrt{ab}\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}{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)/(a-b*sinh(d*x+c)^4),x)

[Out]

-1/2/d*(a*b)^(1/2)/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*(a*b)^(1/2)/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/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{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a d} - 2 \, \int \frac{b e^{\left (7 \, d x + 7 \, c\right )} - 3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 3 \, b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + 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)/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-log((e^(d*x + c) + 1)*e^(-c))/(a*d) + log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 2*integrate((b*e^(7*d*x + 7*c) -

3*b*e^(5*d*x + 5*c) + 3*b*e^(3*d*x + 3*c) - b*e^(d*x + 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 = 2.26058, size = 2431, normalized size = 17.88 \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)/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

1/4*(a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(b*cosh

(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*(a*b*d*cosh(d*x + c) + a*b*d*sinh(d*x +

c) - ((a^4 - a^3*b)*d^3*cosh(d*x + c) + (a^4 - a^3*b)*d^3*sinh(d*x + c))*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4

)))*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2)) + b) - a*d*sqrt

(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(b*cosh(d*x + c)^2 +

2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - 2*(a*b*d*cosh(d*x + c) + a*b*d*sinh(d*x + c) - ((a^4 -

a^3*b)*d^3*cosh(d*x + c) + (a^4 - a^3*b)*d^3*sinh(d*x + c))*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)))*sqrt(-((a

^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2)) + b) + a*d*sqrt(((a^3 - a^2*

b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(b*cosh(d*x + c)^2 + 2*b*cosh(d*x

+ c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*(a*b*d*cosh(d*x + c) + a*b*d*sinh(d*x + c) + ((a^4 - a^3*b)*d^3*cos

h(d*x + c) + (a^4 - a^3*b)*d^3*sinh(d*x + c))*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)))*sqrt(((a^3 - a^2*b)*d^2

*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2)) + b) - a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/(

(a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x +

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

^4 - a^3*b)*d^3*sinh(d*x + c))*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)))*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 -

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

sh(d*x + c) + sinh(d*x + c) - 1))/(a*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(csch(d*x+c)/(a-b*sinh(d*x+c)**4),x)

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError

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