3.2.65 \(\int \frac {\coth (c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [165]

Optimal. Leaf size=130 \[ -\frac {b^3}{4 a^3 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b^2 (3 a+2 b)}{2 a^3 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 (a+b)^3 d}+\frac {\log (\sinh (c+d x))}{(a+b)^3 d} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4223, 457, 90} \begin {gather*} -\frac {b^3}{4 a^3 d (a+b) \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d (a+b)^3}+\frac {\log (\sinh (c+d x))}{d (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4223

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, Dist[-(f*ff^(m + n*p - 1))^(-1), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*
(ff*x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\coth (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^7}{\left (1-x^2\right ) \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {x^3}{(1-x) (b+a x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^3 (-1+x)}-\frac {b^3}{a^2 (a+b) (b+a x)^3}+\frac {b^2 (3 a+2 b)}{a^2 (a+b)^2 (b+a x)^2}-\frac {b \left (3 a^2+3 a b+b^2\right )}{a^2 (a+b)^3 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b^3}{4 a^3 (a+b) d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b^2 (3 a+2 b)}{2 a^3 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 (a+b)^3 d}+\frac {\log (\sinh (c+d x))}{(a+b)^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.69, size = 155, normalized size = 1.19 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \left (4 \log (\sinh (c+d x))+\frac {2 b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b+a \sinh ^2(c+d x)\right )}{a^3}-\frac {b^3 (a+b)^2}{a^3 \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {2 b^2 (a+b) (3 a+2 b)}{a^3 \left (a+b+a \sinh ^2(c+d x)\right )}\right )}{32 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*(4*Log[Sinh[c + d*x]] + (2*b*(3*a^2 + 3*a*b + b^2)*Log[a +
b + a*Sinh[c + d*x]^2])/a^3 - (b^3*(a + b)^2)/(a^3*(a + b + a*Sinh[c + d*x]^2)^2) + (2*b^2*(a + b)*(3*a + 2*b)
)/(a^3*(a + b + a*Sinh[c + d*x]^2))))/(32*(a + b)^3*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(291\) vs. \(2(124)=248\).
time = 2.88, size = 292, normalized size = 2.25

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}+\frac {b \left (\frac {\left (-6 a^{3} b -8 a^{2} b^{2}-2 a \,b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (3 a^{2}-a b -b^{2}\right ) a b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (3 a^{2}+4 a b +b^{2}\right ) a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (3 a^{2}+3 a b +b^{2}\right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{3} \left (a +b \right )^{3}}}{d}\)	\(292\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}+\frac {b \left (\frac {\left (-6 a^{3} b -8 a^{2} b^{2}-2 a \,b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (3 a^{2}-a b -b^{2}\right ) a b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (3 a^{2}+4 a b +b^{2}\right ) a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (3 a^{2}+3 a b +b^{2}\right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{3} \left (a +b \right )^{3}}}{d}\)	\(292\)
risch	\(\frac {x}{a^{3}}-\frac {2 x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b x}{a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b c}{a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{2} x}{a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{2} c}{a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 b^{3} x}{a^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 b^{3} c}{a^{3} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 b^{2} {\mathrm e}^{2 d x +2 c} \left (3 a^{2} {\mathrm e}^{4 d x +4 c}+2 a b \,{\mathrm e}^{4 d x +4 c}+6 a^{2} {\mathrm e}^{2 d x +2 c}+14 a b \,{\mathrm e}^{2 d x +2 c}+6 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}+2 a b \right )}{a^{3} \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 b \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{3} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\)	\(609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/(a+b)^3*ln(tanh(1/2*d*x+1/2*c))-1/a^3*ln(tanh(1/2*d*x+1/2*c)+1)+b/a^3/
(a+b)^3*(((-6*a^3*b-8*a^2*b^2-2*a*b^3)*tanh(1/2*d*x+1/2*c)^6-4*(3*a^2-a*b-b^2)*a*b*tanh(1/2*d*x+1/2*c)^4-2*(3*
a^2+4*a*b+b^2)*a*b*tanh(1/2*d*x+1/2*c)^2)/(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/
2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)^2+1/2*(3*a^2+3*a*b+b^2)*ln(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)
^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (124) = 248\).
time = 0.30, size = 419, normalized size = 3.22 \begin {gather*} \frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} d} + \frac {2 \, {\left ({\left (3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{2} b^{2} + 7 \, a b^{3} + 3 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2} + 4 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{7} + 14 \, a^{6} b + 27 \, a^{5} b^{2} + 24 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {d x + c}{a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*a^2*b + 3*a*b^2 + b^3)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^6 + 3*a^5*b + 3*a
^4*b^2 + a^3*b^3)*d) + 2*((3*a^2*b^2 + 2*a*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^2*b^2 + 7*a*b^3 + 3*b^4)*e^(-4*d*x -
4*c) + (3*a^2*b^2 + 2*a*b^3)*e^(-6*d*x - 6*c))/((a^7 + 2*a^6*b + a^5*b^2 + 4*(a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a
^4*b^3)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 14*a^6*b + 27*a^5*b^2 + 24*a^4*b^3 + 8*a^3*b^4)*e^(-4*d*x - 4*c) + 4*(a^
7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*e^(-6*d*x - 6*c) + (a^7 + 2*a^6*b + a^5*b^2)*e^(-8*d*x - 8*c))*d) + log(e
^(-d*x - c) + 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + log(e^(-d*x - c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*
d) + (d*x + c)/(a^3*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4132 vs. \(2 (124) = 248\).
time = 0.63, size = 4132, normalized size = 31.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/2*(2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^8 + 16*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d
*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*sinh(d*x + c)^8 - 4*(3*a^3*b^2
+ 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^6 - 4*(3*a^3*b^
2 + 5*a^2*b^3 + 2*a*b^4 - 14*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^2 - 2*(a^5 + 5*a^4*b + 9*
a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*sinh(d*x + c)^6 + 8*(14*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x
+ c)^3 - 3*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d
*x + c))*sinh(d*x + c)^5 - 4*(6*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*
a^2*b^3 + 32*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^4 + 4*(35*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x +
c)^4 - 6*a^3*b^2 - 20*a^2*b^3 - 20*a*b^4 - 6*b^5 + (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*
b^5)*d*x - 15*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh
(d*x + c)^2)*sinh(d*x + c)^4 + 16*(7*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*cosh(d*x + c)^5 - 5*(3*a^3*b^2
+ 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^3 - (6*a^3*b^2
+ 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*d*x)*cosh(d*
x + c))*sinh(d*x + c)^3 + 2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x - 4*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2
*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^2 + 4*(14*(a^5 + 3*a^4*b + 3*a^3*b^2 + a
^2*b^3)*d*x*cosh(d*x + c)^6 - 3*a^3*b^2 - 5*a^2*b^3 - 2*a*b^4 - 15*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 +
5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x)*cosh(d*x + c)^4 + 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 +
2*a*b^4)*d*x - 6*(6*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3 + 32
*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^8 + 8*(
3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sinh(d*x + c)^8
+ 4*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^6 + 4*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^
4 + 7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*
cosh(d*x + c)^3 + 3*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^4*b + 3*a
^3*b^2 + a^2*b^3 + 2*(9*a^4*b + 33*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cosh(d*x + c)^4 + 2*(9*a^4*b + 33*
a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5 + 35*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^4 + 30*(3*a^4*b + 9
*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d
*x + c)^5 + 10*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^3 + (9*a^4*b + 33*a^3*b^2 + 51*a^2*b^
3 + 32*a*b^4 + 8*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x
+ c)^2 + 4*(7*(3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^6 + 3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4 + 15
*(3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^4 + 3*(9*a^4*b + 33*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4
+ 8*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cosh(d*x + c)^7 + 3*(3*a^4*b +
9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c)^5 + (9*a^4*b + 33*a^3*b^2 + 51*a^2*b^3 + 32*a*b^4 + 8*b^5)*cos
h(d*x + c)^3 + (3*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh(d*x + c
)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*(a
^5*cosh(d*x + c)^8 + 8*a^5*cosh(d*x + c)*sinh(d*x + c)^7 + a^5*sinh(d*x + c)^8 + 4*(a^5 + 2*a^4*b)*cosh(d*x +
c)^6 + 4*(7*a^5*cosh(d*x + c)^2 + a^5 + 2*a^4*b)*sinh(d*x + c)^6 + 8*(7*a^5*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b
)*cosh(d*x + c))*sinh(d*x + c)^5 + a^5 + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c)^4 + 2*(35*a^5*cosh(d*x
+ c)^4 + 3*a^5 + 8*a^4*b + 8*a^3*b^2 + 30*(a^5 + 2*a^4*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^5*cosh(d*x
+ c)^5 + 10*(a^5 + 2*a^4*b)*cosh(d*x + c)^3 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(a^5 + 2*a^4*b)*cosh(d*x + c)^2 + 4*(7*a^5*cosh(d*x + c)^6 + a^5 + 2*a^4*b + 15*(a^5 + 2*a^4*b)*cosh(d*x + c
)^4 + 3*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^5*cosh(d*x + c)^7 + 3*(a^5 + 2*a
^4*b)*cosh(d*x + c)^5 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*cosh(d*x + c)^3 + (a^5 + 2*a^4*b)*cosh(d*x + c))*sinh(d*
x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(2*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d*x*
cosh(d*x + c)^7 - 3*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4 - 2*(a^5 + 5*a^4*b + 9*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*d*x
)*cosh(d*x + c)^5 - 2*(6*a^3*b^2 + 20*a^2*b^3 + 20*a*b^4 + 6*b^5 - (3*a^5 + 17*a^4*b + 41*a^3*b^2 + 51*a^2*b^3
+ 32*a*b^4 + 8*b^5)*d*x)*cosh(d*x + c)^3 - (3*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral(coth(c + d*x)/(a + b*sech(c + d*x)**2)**3, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,\mathrm {coth}\left (c+d\,x\right )}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)/(a + b/cosh(c + d*x)^2)^3,x)

[Out]

int((cosh(c + d*x)^6*coth(c + d*x))/(b + a*cosh(c + d*x)^2)^3, x)

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