3.2.79 \(\int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [179]

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

[Out]

-coth(d*x+c)/a/d+2/3*(-1)^(1/3)*b^(2/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/(
(-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2))/a^(4/3)/d/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2)-2/3*b^(2/3)*arctanh((b^(1/3)-a
^(1/3)*tanh(1/2*d*x+1/2*c))/(a^(2/3)+b^(2/3))^(1/2))/a^(4/3)/d/(a^(2/3)+b^(2/3))^(1/2)-2/3*b^(2/3)*arctan((-1)
^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2))/a^(4/
3)/d/((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2)

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Rubi [A]
time = 0.44, antiderivative size = 304, normalized size of antiderivative = 1.00, 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 = {3299, 3852, 8, 2739, 632, 210} \begin {gather*} -\frac {2 b^{2/3} \text {ArcTan}\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} \text {ArcTan}\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 {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 {\coth (c+d x)}{a d} \end {gather*}

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 8

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

Rule 210

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

Rule 632

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 2739

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 3299

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 3852

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]

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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.30, size = 230, normalized size = 0.76 \begin {gather*} -\frac {3 \coth \left (\frac {1}{2} (c+d x)\right )+2 b \text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+c \text {$\#$1}^2+d x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \tanh \left (\frac {1}{2} (c+d x)\right )}{6 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(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[-Co
sh[(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
[-Cosh[(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])/(a*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.66, size = 118, normalized size = 0.39

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\)	\(118\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\)	\(118\)
risch	\(-\frac {2}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}+4 \left (\munderset {\textit {\_R} =\RootOf \left (\left (2985984 a^{10} d^{6}+2985984 a^{8} b^{2} d^{6}\right ) \textit {\_Z}^{6}+62208 a^{6} b^{2} d^{4} \textit {\_Z}^{4}-b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (-\frac {248832 d^{5} a^{10}}{a^{2} b^{3}-b^{5}}-\frac {248832 d^{5} a^{8} b^{2}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{5}+\left (\frac {20736 d^{4} a^{9}}{a^{2} b^{3}-b^{5}}+\frac {20736 d^{4} a^{7} b^{2}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{4}+\left (-\frac {3456 d^{3} a^{6} b^{2}}{a^{2} b^{3}-b^{5}}+\frac {1728 d^{3} b^{4} a^{4}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{3}+\left (\frac {288 d^{2} a^{5} b^{2}}{a^{2} b^{3}-b^{5}}-\frac {144 d^{2} b^{4} a^{3}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R}^{2}+\left (-\frac {12 d \,a^{4} b^{2}}{a^{2} b^{3}-b^{5}}+\frac {24 d \,b^{4} a^{2}}{a^{2} b^{3}-b^{5}}\right ) \textit {\_R} -\frac {a \,b^{4}}{a^{2} b^{3}-b^{5}}\right )\right )\)	\(377\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a*tanh(1/2*d*x+1/2*c)-1/2/a/tanh(1/2*d*x+1/2*c)-2/3/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 = 0.00, 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)^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 [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 21133 vs. \(2 (211) = 422\).
time = 1.24, size = 21133, normalized size = 69.52 \begin {gather*} \text {too large to display} \end {gather*}

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]

-1/2*(sqrt(2/3)*sqrt(1/6)*(a*d*cosh(d*x + c)^2 + 2*a*d*cosh(d*x + c)*sinh(d*x + c) + a*d*sinh(d*x + c)^2 - a*d
)*sqrt(((a^4 + a^2*b^2)*(2*(1/2)^(2/3)*b^4*(-I*sqrt(3) + 1)/((a^4*d^2 + a^2*b^2*d^2)^2*(b^4/(a^10*d^6 + a^8*b^
2*d^6) - 2*b^6/(a^4*d^2 + a^2*b^2*d^2)^3 - (a^2 - b^2)*b^4/((a^2 + b^2)^2*a^8*d^6))^(1/3)) + (1/2)^(1/3)*(b^4/
(a^10*d^6 + a^8*b^2*d^6) - 2*b^6/(a^4*d^2 + a^2*b^2*d^2)^3 - (a^2 - b^2)*b^4/((a^2 + b^2)^2*a^8*d^6))^(1/3)*(I
*sqrt(3) + 1) + 2*b^2/(a^4*d^2 + a^2*b^2*d^2))*d^2 + 3*sqrt(1/3)*(a^4 + a^2*b^2)*d^2*sqrt(-((a^8 + 2*a^6*b^2 +
a^4*b^4)*(2*(1/2)^(2/3)*b^4*(-I*sqrt(3) + 1)/((a^4*d^2 + a^2*b^2*d^2)^2*(b^4/(a^10*d^6 + a^8*b^2*d^6) - 2*b^6
/(a^4*d^2 + a^2*b^2*d^2)^3 - (a^2 - b^2)*b^4/((a^2 + b^2)^2*a^8*d^6))^(1/3)) + (1/2)^(1/3)*(b^4/(a^10*d^6 + a^
8*b^2*d^6) - 2*b^6/(a^4*d^2 + a^2*b^2*d^2)^3 - (a^2 - b^2)*b^4/((a^2 + b^2)^2*a^8*d^6))^(1/3)*(I*sqrt(3) + 1)
+ 2*b^2/(a^4*d^2 + a^2*b^2*d^2))^2*d^4 - 12*b^4 - 4*(a^4*b^2 + a^2*b^4)*(2*(1/2)^(2/3)*b^4*(-I*sqrt(3) + 1)/((
a^4*d^2 +  ...

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 23.06, size = 1293, normalized size = 4.25 \begin {gather*} \left (\sum _{k=1}^6\ln \left (-\frac {8192\,b^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}-65536\,a\,b^3-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^2\,a^3\,b^3\,d^2\,294912-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^4\,b^3\,d^3\,221184-\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\,a^2\,b^3\,d\,196608+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^8\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,10616832+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^9\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,7962624-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^6\,b\,d^3\,1769472+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^7\,b\,d^4\,2654208-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^8\,b\,d^5\,1990656+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^2\,a^4\,b^2\,d^2\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,2064384+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^5\,b^2\,d^3\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,5529600+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^6\,b^2\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,7299072+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^7\,b^2\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,9953280+\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\,a^3\,b^2\,d\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,393216}{a^4\,b^5}\right )\,\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\right )+\frac {2}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^2*(a + b*sinh(c + d*x)^3)),x)

[Out]

symsum(log(-(8192*b^4*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x
) - 65536*a*b^3 - 294912*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^2*a^3*
b^3*d^2 - 221184*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^3*a^4*b^3*d^3
- 196608*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)*a^2*b^3*d + 10616832*r
oot(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^4*a^8*d^4*exp(root(729*a^8*b^2*d
^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) + 7962624*root(729*a^8*b^2*d^6*z^6 + 729*a
^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^5*a^9*d^5*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243
*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) - 1769472*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z
^4 - b^4, z, k)^3*a^6*b*d^3 + 2654208*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4,
z, k)^4*a^7*b*d^4 - 1990656*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^5*
a^8*b*d^5 + 2064384*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^2*a^4*b^2*d
^2*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) + 5529600*root(72
9*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^3*a^5*b^2*d^3*exp(root(729*a^8*b^2*d^6
*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) + 7299072*root(729*a^8*b^2*d^6*z^6 + 729*a^1
0*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)^4*a^6*b^2*d^4*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 2
43*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x) + 9953280*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4
*z^4 - b^4, z, k)^5*a^7*b^2*d^5*exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z
, k) + d*x) + 393216*root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k)*a^3*b^2*d*
exp(root(729*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k) + d*x))/(a^4*b^5))*root(729
*a^8*b^2*d^6*z^6 + 729*a^10*d^6*z^6 + 243*a^6*b^2*d^4*z^4 - b^4, z, k), k, 1, 6) + 2/(a*d - a*d*exp(2*c + 2*d*
x))

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