∫ coth4(c + dx)(a + bsech2(c + dx)) dx [110]

Optimal. Leaf size=34 \[ a x-\frac {a \coth (c+d x)}{d}-\frac {(a+b) \coth ^3(c+d x)}{3 d} \]

a*x-a*coth(d*x+c)/d-1/3*(a+b)*coth(d*x+c)^3/d

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Rubi [A]
time = 0.05, antiderivative size = 34, 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 = {4226, 1816, 213} \begin {gather*} -\frac {(a+b) \coth ^3(c+d x)}{3 d}-\frac {a \coth (c+d x)}{d}+a x \end {gather*}

Int[Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2),x]

a*x - (a*Coth[c + d*x])/d - ((a + b)*Coth[c + d*x]^3)/(3*d)

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])

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x

^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[

{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2

*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 49, normalized size = 1.44 \begin {gather*} -\frac {b \coth ^3(c+d x)}{3 d}-\frac {a \coth ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\tanh ^2(c+d x)\right )}{3 d} \end {gather*}

Integrate[Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2),x]

-1/3*(b*Coth[c + d*x]^3)/d - (a*Coth[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[c + d*x]^2])/(3*d)

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method	result	size

risch	\(a x -\frac {2 \left (6 a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}-6 a \,{\mathrm e}^{2 d x +2 c}+4 a +b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(64\)

int(coth(d*x+c)^4*(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (32) = 64\).
time = 0.27, size = 170, normalized size = 5.00 \begin {gather*} \frac {1}{3} \, a {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {2}{3} \, b {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \end {gather*}

integrate(coth(d*x+c)^4*(a+b*sech(d*x+c)^2),x, algorithm="maxima")

1/3*a*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) - 2)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*

c) + e^(-6*d*x - 6*c) - 1))) + 2/3*b*(3*e^(-4*d*x - 4*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d

*x - 6*c) - 1)) + 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (32) = 64\).
time = 0.38, size = 140, normalized size = 4.12 \begin {gather*} -\frac {{\left (4 \, a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a d x + 4 \, a + b\right )} \sinh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) + 3 \, {\left (3 \, a d x - {\left (3 \, a d x + 4 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a + b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

integrate(coth(d*x+c)^4*(a+b*sech(d*x+c)^2),x, algorithm="fricas")

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

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

*x + c)^3 + 3*(d*cosh(d*x + c)^2 - d)*sinh(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth ^{4}{\left (c + d x \right )}\, dx \end {gather*}

integrate(coth(d*x+c)**4*(a+b*sech(d*x+c)**2),x)

Integral((a + b*sech(c + d*x)**2)*coth(c + d*x)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (32) = 64\).
time = 0.43, size = 70, normalized size = 2.06 \begin {gather*} \frac {3 \, {\left (d x + c\right )} a - \frac {2 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \end {gather*}

integrate(coth(d*x+c)^4*(a+b*sech(d*x+c)^2),x, algorithm="giac")

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

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Mupad [B]
time = 1.49, size = 161, normalized size = 4.74 \begin {gather*} a\,x-\frac {\frac {2\,b}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a+b\right )}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,\left (2\,a+b\right )}{3\,d}+\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a+b\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {2\,\left (2\,a+b\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}

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

a*x - ((2*b)/(3*d) + (2*exp(2*c + 2*d*x)*(2*a + b))/(3*d))/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - ((2*(

2*a + b))/(3*d) + (4*b*exp(2*c + 2*d*x))/(3*d) + (2*exp(4*c + 4*d*x)*(2*a + b))/(3*d))/(3*exp(2*c + 2*d*x) - 3

*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - (2*(2*a + b))/(3*d*(exp(2*c + 2*d*x) - 1))

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3.2.10 \(\int \coth ^4(c+d x) (a+b \text {sech}^2(c+d x)) \, dx\) [110]