3.3.33 \(\int \frac {\text {csch}(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [233]

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

[Out]

-arctanh(cosh(d*x+c))/a/d-1/2*b^(1/4)*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a/d/(a^(1/2)-b^(1/2)
)^(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a/d/(a^(1/2)+b^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 136, normalized size of antiderivative = 1.00, 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 = {3294, 1184, 213, 1180, 211, 214} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 211

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

Rule 213

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

Rule 214

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

Rule 1180

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 1184

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 3294

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]

Rubi steps

\begin {align*} \int \frac {\text {csch}(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=-\frac {\text {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 {\text {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 {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a d}-\frac {\text {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 \text {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 \text {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] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.18, size = 385, normalized size = 2.83 \begin {gather*} \frac {8 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 a d} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 1.98, size = 164, normalized size = 1.21

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\left (4096 a^{5} d^{4}-4096 a^{4} b \,d^{4}\right ) \textit {\_Z}^{4}+128 a^{2} b \,d^{2} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {1024 d^{3} a^{4}}{b}-1024 a^{3} d^{3}\right ) \textit {\_R}^{3}+32 a d \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(130\)
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+8 b \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{d}\)	\(164\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+8 b \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{d}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/a*ln(tanh(1/2*d*x+1/2*c))+8*b*(-1/16*(a*b)^(1/2)/a/b/(-(a*b)^(1/2)*a-a*b)^(1/2)*arctan(1/4*(-2*a*tanh(1
/2*d*x+1/2*c)^2+4*(a*b)^(1/2)+2*a)/(-(a*b)^(1/2)*a-a*b)^(1/2))-1/16*(a*b)^(1/2)/a/b/((a*b)^(1/2)*a-a*b)^(1/2)*
arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2+4*(a*b)^(1/2)-2*a)/((a*b)^(1/2)*a-a*b)^(1/2))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (100) = 200\).
time = 0.43, size = 1067, normalized size = 7.85 \begin {gather*} \frac {a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) - a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) + a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) + {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) - a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) + {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) - 4 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 4 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{4 \, a d} \end {gather*}

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)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (100) = 200\).
time = 0.47, size = 419, normalized size = 3.08 \begin {gather*} \frac {\frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b\right )} {\left | a \right |} {\left | b \right |} - {\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b + \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{4 \, a^{4} b^{2} + a^{3} b^{3} - 5 \, a^{2} b^{4}} - \frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b\right )} {\left | a \right |} {\left | b \right |} + {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b - \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{4 \, a^{4} b^{2} + a^{3} b^{3} - 5 \, a^{2} b^{4}} - \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a} + \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a}}{2 \, d} \end {gather*}

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]

1/2*(((4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a + 5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b)*abs(a)*abs(b) - (4*sqr
t(-b^2 - sqrt(a*b)*b)*a^2*b + 5*sqrt(-b^2 - sqrt(a*b)*b)*a*b^2)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c)
)/sqrt(-(a*b + sqrt(a^2*b^2 + (a^2 - a*b)*a*b))/(a*b)))/(4*a^4*b^2 + a^3*b^3 - 5*a^2*b^4) - ((4*sqrt(a*b)*sqrt
(-b^2 + sqrt(a*b)*b)*a + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b)*abs(a)*abs(b) + (4*sqrt(-b^2 + sqrt(a*b)*b)*a
^2*b + 5*sqrt(-b^2 + sqrt(a*b)*b)*a*b^2)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a*b - sqrt(a^2
*b^2 + (a^2 - a*b)*a*b))/(a*b)))/(4*a^4*b^2 + a^3*b^3 - 5*a^2*b^4) - log(e^(d*x + c) + e^(-d*x - c) + 2)/a + l
og(e^(d*x + c) + e^(-d*x - c) - 2)/a)/d

________________________________________________________________________________________

Mupad [B]
time = 9.19, size = 1243, normalized size = 9.14 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}+\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}-\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}-\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (65536\,a^2\,\sqrt {-a^2\,d^2}+50625\,b^2\,\sqrt {-a^2\,d^2}-115200\,a\,b\,\sqrt {-a^2\,d^2}\right )}{65536\,d\,a^3-115200\,d\,a^2\,b+50625\,d\,a\,b^2}\right )}{\sqrt {-a^2\,d^2}}-\ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}-\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}+\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}-\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*(a - b*sinh(c + d*x)^4)),x)

[Out]

log((((((4294967296*a*d^2*(exp(2*c + 2*d*x) + 1)*(26*a*b - 49*a^2 + 15*b^2))/(b^6*(a - b)^3) + (8589934592*a^2
*d^3*exp(c + d*x)*(3*a*b + 16*a^2 - 15*b^2)*(-(a^2*b + (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/(b^7*(a - b)^2
))*(-(a^2*b + (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 - (2147483648*d*exp(c + d*x)*(17*a - 15*b))/(b^6*(a -
b)^2))*(-(a^2*b + (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 + (268435456*(exp(2*c + 2*d*x) + 1)*(3*a*b + 16*
a^2 - 15*b^2))/(a*b^6*(a - b)^3))*(-(a^2*b + (a^5*b)^(1/2))/(16*(a^5*d^2 - a^4*b*d^2)))^(1/2) - log((((((42949
67296*a*d^2*(exp(2*c + 2*d*x) + 1)*(26*a*b - 49*a^2 + 15*b^2))/(b^6*(a - b)^3) - (8589934592*a^2*d^3*exp(c + d
*x)*(3*a*b + 16*a^2 - 15*b^2)*(-(a^2*b + (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/(b^7*(a - b)^2))*(-(a^2*b +
(a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 + (2147483648*d*exp(c + d*x)*(17*a - 15*b))/(b^6*(a - b)^2))*(-(a^2
*b + (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 + (268435456*(exp(2*c + 2*d*x) + 1)*(3*a*b + 16*a^2 - 15*b^2))
/(a*b^6*(a - b)^3))*(-(a^2*b + (a^5*b)^(1/2))/(16*(a^5*d^2 - a^4*b*d^2)))^(1/2) - (2*atan((exp(d*x)*exp(c)*(65
536*a^2*(-a^2*d^2)^(1/2) + 50625*b^2*(-a^2*d^2)^(1/2) - 115200*a*b*(-a^2*d^2)^(1/2)))/(65536*a^3*d + 50625*a*b
^2*d - 115200*a^2*b*d)))/(-a^2*d^2)^(1/2) - log((((((4294967296*a*d^2*(exp(2*c + 2*d*x) + 1)*(26*a*b - 49*a^2
+ 15*b^2))/(b^6*(a - b)^3) - (8589934592*a^2*d^3*exp(c + d*x)*(3*a*b + 16*a^2 - 15*b^2)*(-(a^2*b - (a^5*b)^(1/
2))/(a^4*d^2*(a - b)))^(1/2))/(b^7*(a - b)^2))*(-(a^2*b - (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 + (214748
3648*d*exp(c + d*x)*(17*a - 15*b))/(b^6*(a - b)^2))*(-(a^2*b - (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 + (2
68435456*(exp(2*c + 2*d*x) + 1)*(3*a*b + 16*a^2 - 15*b^2))/(a*b^6*(a - b)^3))*(-(a^2*b - (a^5*b)^(1/2))/(16*(a
^5*d^2 - a^4*b*d^2)))^(1/2) + log((((((4294967296*a*d^2*(exp(2*c + 2*d*x) + 1)*(26*a*b - 49*a^2 + 15*b^2))/(b^
6*(a - b)^3) + (8589934592*a^2*d^3*exp(c + d*x)*(3*a*b + 16*a^2 - 15*b^2)*(-(a^2*b - (a^5*b)^(1/2))/(a^4*d^2*(
a - b)))^(1/2))/(b^7*(a - b)^2))*(-(a^2*b - (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 - (2147483648*d*exp(c +
d*x)*(17*a - 15*b))/(b^6*(a - b)^2))*(-(a^2*b - (a^5*b)^(1/2))/(a^4*d^2*(a - b)))^(1/2))/4 + (268435456*(exp(
2*c + 2*d*x) + 1)*(3*a*b + 16*a^2 - 15*b^2))/(a*b^6*(a - b)^3))*(-(a^2*b - (a^5*b)^(1/2))/(16*(a^5*d^2 - a^4*b
*d^2)))^(1/2)

________________________________________________________________________________________