3.2.56 \(\int \frac {\coth ^3(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [156]

Optimal. Leaf size=110 \[ \frac {b^3}{2 a^2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {csch}^2(c+d x)}{2 (a+b)^2 d}+\frac {b^2 (3 a+b) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 (a+b)^3 d}+\frac {(a+3 b) \log (\sinh (c+d x))}{(a+b)^3 d} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 457, 90} \begin {gather*} \frac {b^3}{2 a^2 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b^2 (3 a+b) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d (a+b)^3}-\frac {\text {csch}^2(c+d x)}{2 d (a+b)^2}+\frac {(a+3 b) \log (\sinh (c+d x))}{d (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b^3/(2*a^2*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)) - Csch[c + d*x]^2/(2*(a + b)^2*d) + (b^2*(3*a + b)*Log[b + a*C
osh[c + d*x]^2])/(2*a^2*(a + b)^3*d) + ((a + 3*b)*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 ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^7}{\left (1-x^2\right )^2 \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x^3}{(1-x)^2 (b+a x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(a+b)^2 (-1+x)^2}+\frac {a+3 b}{(a+b)^3 (-1+x)}-\frac {b^3}{a (a+b)^2 (b+a x)^2}+\frac {b^2 (3 a+b)}{a (a+b)^3 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {b^3}{2 a^2 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {csch}^2(c+d x)}{2 (a+b)^2 d}+\frac {b^2 (3 a+b) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 (a+b)^3 d}+\frac {(a+3 b) \log (\sinh (c+d x))}{(a+b)^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.88, size = 130, normalized size = 1.18 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}^4(c+d x) \left (-\left ((a+b) \text {csch}^2(c+d x)\right )+2 (a+3 b) \log (\sinh (c+d x))+\frac {b^2 (3 a+b) \log \left (a+b+a \sinh ^2(c+d x)\right )}{a^2}+\frac {b^3 (a+b)}{a^2 \left (a+b+a \sinh ^2(c+d x)\right )}\right )}{8 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*(-((a + b)*Csch[c + d*x]^2) + 2*(a + 3*b)*Log[Sinh[c + d*x]
] + (b^2*(3*a + b)*Log[a + b + a*Sinh[c + d*x]^2])/a^2 + (b^3*(a + b))/(a^2*(a + b + a*Sinh[c + d*x]^2))))/(8*
(a + b)^3*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs. \(2(104)=208\).
time = 3.27, size = 260, normalized size = 2.36

method	result	size
derivativedivides	\(\frac {-\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {1}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a +12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {b^{2} \left (-\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a +b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\)	\(260\)
default	\(\frac {-\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right )}-\frac {1}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a +12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {b^{2} \left (-\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a +b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\)	\(260\)
risch	\(\frac {x}{a^{2}}-\frac {2 a x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 a c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {6 b c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{2} x}{a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 b^{2} c}{a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 b^{3} x}{a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 b^{3} c}{a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}-b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+4 a^{2} b \,{\mathrm e}^{2 d x +2 c}+2 b^{3} {\mathrm e}^{2 d x +2 c}+a^{3}-b^{3}\right )}{a^{2} \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\)	\(595\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/8*tanh(1/2*d*x+1/2*c)^2/(a^2+2*a*b+b^2)-1/8/(a+b)^2/tanh(1/2*d*x+1/2*c)^2+1/4/(a+b)^3*(4*a+12*b)*ln(ta
nh(1/2*d*x+1/2*c))-1/a^2*ln(tanh(1/2*d*x+1/2*c)+1)+b^2/a^2/(a+b)^3*(-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)+1/2*(3*a+b)*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))-1/a^2*
ln(tanh(1/2*d*x+1/2*c)-1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (104) = 208\).
time = 0.30, size = 384, normalized size = 3.49 \begin {gather*} \frac {{\left (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^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d} + \frac {{\left (a + 3 \, b\right )} \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 {{\left (a + 3 \, b\right )} \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 {2 \, {\left ({\left (a^{3} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{3} + 2 \, a^{2} b + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{3} - b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} + 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{5} + 6 \, a^{4} b + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(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^5 + 3*a^4*b + 3*a^3*b^2 + a
^2*b^3)*d) + (a + 3*b)*log(e^(-d*x - c) + 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + (a + 3*b)*log(e^(-d*x - c)
- 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 2*((a^3 - b^3)*e^(-2*d*x - 2*c) + 2*(a^3 + 2*a^2*b + b^3)*e^(-4*d*x
- 4*c) + (a^3 - b^3)*e^(-6*d*x - 6*c))/((a^5 + 2*a^4*b + a^3*b^2 + 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*e^(-2*d*x
- 2*c) - 2*(a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*e^(-4*d*x - 4*c) + 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*e^(-6*d*
x - 6*c) + (a^5 + 2*a^4*b + a^3*b^2)*e^(-8*d*x - 8*c))*d) + (d*x + c)/(a^2*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3624 vs. \(2 (104) = 208\).
time = 0.59, size = 3624, normalized size = 32.95 \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)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/2*(2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^8 + 16*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*c
osh(d*x + c)*sinh(d*x + c)^7 + 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*sinh(d*x + c)^8 + 4*(a^4 + a^3*b - a*
b^3 - b^4 + 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x + c)^6 + 4*(14*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*
b^3)*d*x*cosh(d*x + c)^2 + a^4 + a^3*b - a*b^3 - b^4 + 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*sinh(d*x + c
)^6 + 8*(14*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^3 + 3*(a^4 + a^3*b - a*b^3 - b^4 + 2*(a^3*b
+ 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(2*a^4 + 6*a^3*b + 4*a^2*b^2 + 2*a*b^3 +
2*b^4 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*x)*cosh(d*x + c)^4 + 4*(35*(a^4 + 3*a^3*b + 3*a^2*b^
2 + a*b^3)*d*x*cosh(d*x + c)^4 + 2*a^4 + 6*a^3*b + 4*a^2*b^2 + 2*a*b^3 + 2*b^4 - (a^4 + 7*a^3*b + 15*a^2*b^2 +
13*a*b^3 + 4*b^4)*d*x + 15*(a^4 + a^3*b - a*b^3 - b^4 + 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 16*(7*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^5 + 5*(a^4 + a^3*b - a*b^
3 - b^4 + 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x + c)^3 + (2*a^4 + 6*a^3*b + 4*a^2*b^2 + 2*a*b^3
+ 2*b^4 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^4 + 3*a^3
*b + 3*a^2*b^2 + a*b^3)*d*x + 4*(a^4 + a^3*b - a*b^3 - b^4 + 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d
*x + c)^2 + 4*(14*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^6 + 15*(a^4 + a^3*b - a*b^3 - b^4 + 2*
(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x + c)^4 + a^4 + a^3*b - a*b^3 - b^4 + 2*(a^3*b + 3*a^2*b^2 +
3*a*b^3 + b^4)*d*x + 6*(2*a^4 + 6*a^3*b + 4*a^2*b^2 + 2*a*b^3 + 2*b^4 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3
+ 4*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^2*b^2 + a*b^3)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + a*b^3)
*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b^2 + a*b^3)*sinh(d*x + c)^8 + 4*(3*a*b^3 + b^4)*cosh(d*x + c)^6 + 4*(
3*a*b^3 + b^4 + 7*(3*a^2*b^2 + a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b^2 + a*b^3)*cosh(d*x + c
)^3 + 3*(3*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2*b^2 + 13*a*b^3 + 4*b^4)*cosh(d*x + c)^4 + 2*
(35*(3*a^2*b^2 + a*b^3)*cosh(d*x + c)^4 - 3*a^2*b^2 - 13*a*b^3 - 4*b^4 + 30*(3*a*b^3 + b^4)*cosh(d*x + c)^2)*s
inh(d*x + c)^4 + 3*a^2*b^2 + a*b^3 + 8*(7*(3*a^2*b^2 + a*b^3)*cosh(d*x + c)^5 + 10*(3*a*b^3 + b^4)*cosh(d*x +
c)^3 - (3*a^2*b^2 + 13*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a*b^3 + b^4)*cosh(d*x + c)^2 + 4*(
7*(3*a^2*b^2 + a*b^3)*cosh(d*x + c)^6 + 15*(3*a*b^3 + b^4)*cosh(d*x + c)^4 + 3*a*b^3 + b^4 - 3*(3*a^2*b^2 + 13
*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^2*b^2 + a*b^3)*cosh(d*x + c)^7 + 3*(3*a*b^3 + b^4)*
cosh(d*x + c)^5 - (3*a^2*b^2 + 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 + (3*a*b^3 + 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^4 + 3*a^3*b)*cosh(d*x + c)^8 + 8*(a^4 + 3*a^3*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^
4 + 3*a^3*b)*sinh(d*x + c)^8 + 4*(a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 4*(a^3*b + 3*a^2*b^2 + 7*(a^4 + 3*a^3*b
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 + 3*a^3*b)*cosh(d*x + c)^3 + 3*(a^3*b + 3*a^2*b^2)*cosh(d*x + c
))*sinh(d*x + c)^5 - 2*(a^4 + 7*a^3*b + 12*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*(a^4 + 3*a^3*b)*cosh(d*x + c)^4 -
a^4 - 7*a^3*b - 12*a^2*b^2 + 30*(a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 3*a^3*b + 8*(7*(a
^4 + 3*a^3*b)*cosh(d*x + c)^5 + 10*(a^3*b + 3*a^2*b^2)*cosh(d*x + c)^3 - (a^4 + 7*a^3*b + 12*a^2*b^2)*cosh(d*x
+ c))*sinh(d*x + c)^3 + 4*(a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2 + 4*(7*(a^4 + 3*a^3*b)*cosh(d*x + c)^6 + 15*(a^
3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + a^3*b + 3*a^2*b^2 - 3*(a^4 + 7*a^3*b + 12*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + 8*((a^4 + 3*a^3*b)*cosh(d*x + c)^7 + 3*(a^3*b + 3*a^2*b^2)*cosh(d*x + c)^5 - (a^4 + 7*a^3*b + 12*a^
2*b^2)*cosh(d*x + c)^3 + (a^3*b + 3*a^2*b^2)*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^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^7 + 3*(a^4 + a^3*b - a*b^3 - b^
4 + 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x + c)^5 + 2*(2*a^4 + 6*a^3*b + 4*a^2*b^2 + 2*a*b^3 + 2*
b^4 - (a^4 + 7*a^3*b + 15*a^2*b^2 + 13*a*b^3 + 4*b^4)*d*x)*cosh(d*x + c)^3 + (a^4 + a^3*b - a*b^3 - b^4 + 2*(a
^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*
cosh(d*x + c)^8 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^6 + 3*a^5*b + 3
*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^8 + 4*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^6 + 4*(7*(
a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2 + (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*d)*sinh(d*x
+ c)^6 - 2*(a^6 + 7*a^5*b + 15*a^4*b^2 + 13*a^3*b^3 + 4*a^2*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a^6 + 3*a^5*b + 3*
a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 + 3*(a^5*b...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(c + d*x)**3/(a + b*sech(c + d*x)**2)**2, 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)^3/(a+b*sech(d*x+c)^2)^2,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 )}^4\,{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cosh(c + d*x)^4*coth(c + d*x)^3)/(b + a*cosh(c + d*x)^2)^2, x)

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