3.2.19 \(\int \coth ^3(c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [119]

Optimal. Leaf size=55 \[ -\frac {(a+b)^2 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 \log (\cosh (c+d x))}{d}+\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 55, 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 {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 \log (\cosh (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.14, size = 82, normalized size = 1.49 \begin {gather*} -\frac {2 \left (b+a \cosh ^2(c+d x)\right )^2 \left ((a+b)^2 \text {csch}^2(c+d x)-2 \left (b^2 \log (\cosh (c+d x))+\left (a^2-b^2\right ) \log (\sinh (c+d x))\right )\right )}{d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.71, size = 64, normalized size = 1.16

method	result	size
derivativedivides	\(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )-\frac {a b}{\sinh \left (d x +c \right )^{2}}+b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2}}-\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(64\)
default	\(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}\right )-\frac {a b}{\sinh \left (d x +c \right )^{2}}+b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2}}-\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(64\)
risch	\(-a^{2} x -\frac {2 a^{2} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{2}+2 a b +b^{2}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{d}+\frac {b^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(113\)

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*(a^2*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2)-a*b/sinh(d*x+c)^2+b^2*(-1/2/sinh(d*x+c)^2-ln(tanh(d*x+c))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (53) = 106\).
time = 0.48, size = 206, normalized size = 3.75 \begin {gather*} a^{2} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {4 \, a b}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \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]

a^2*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) -
e^(-4*d*x - 4*c) - 1))) - b^2*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1)/
d - 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) - 4*a*b/(d*(e^(d*x + c) - e^(-d*x - c)
)^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (53) = 106\).
time = 0.43, size = 637, normalized size = 11.58 \begin {gather*} -\frac {a^{2} d x \cosh \left (d x + c\right )^{4} + 4 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} d x \sinh \left (d x + c\right )^{4} + a^{2} d x - 2 \, {\left (a^{2} d x - a^{2} - 2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} d x + a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (a^{2} d x \cosh \left (d x + c\right )^{3} - {\left (a^{2} d x - a^{2} - 2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 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="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (53) = 106\).
time = 0.45, size = 148, normalized size = 2.69 \begin {gather*} \frac {b^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) + {\left (a^{2} - b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a^{2} + 8 \, a b + 6 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2}}{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="giac")

[Out]

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

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Mupad [B]
time = 0.22, size = 240, normalized size = 4.36 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (d\,\left (a^2-b^2\right )+b^2\,d\right )}{2\,d^2}-a^2\,x-\frac {2\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^4\,\sqrt {-d^2}+4\,b^4\,\sqrt {-d^2}-4\,a^2\,b^2\,\sqrt {-d^2}\right )}{a^2\,d\,\sqrt {a^4-4\,a^2\,b^2+4\,b^4}-2\,b^2\,d\,\sqrt {a^4-4\,a^2\,b^2+4\,b^4}}\right )\,\sqrt {a^4-4\,a^2\,b^2+4\,b^4}}{\sqrt {-d^2}}-\frac {2\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \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]

(log(exp(4*c + 4*d*x) - 1)*(d*(a^2 - b^2) + b^2*d))/(2*d^2) - a^2*x - (2*(2*a*b + a^2 + b^2))/(d*(exp(2*c + 2*
d*x) - 1)) - (atan((exp(2*c)*exp(2*d*x)*(a^4*(-d^2)^(1/2) + 4*b^4*(-d^2)^(1/2) - 4*a^2*b^2*(-d^2)^(1/2)))/(a^2
*d*(a^4 + 4*b^4 - 4*a^2*b^2)^(1/2) - 2*b^2*d*(a^4 + 4*b^4 - 4*a^2*b^2)^(1/2)))*(a^4 + 4*b^4 - 4*a^2*b^2)^(1/2)
)/(-d^2)^(1/2) - (2*(2*a*b + a^2 + b^2))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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