3.1 \(\int (a+b \text{csch}^2(c+d x))^4 \, dx\)

Optimal. Leaf size=114 \[ -\frac{b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{b (2 a-b) \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}+a^4 x-\frac{b^3 (4 a-3 b) \coth ^5(c+d x)}{5 d}-\frac{b^4 \coth ^7(c+d x)}{7 d} \]

[Out]

a^4*x - ((2*a - b)*b*(2*a^2 - 2*a*b + b^2)*Coth[c + d*x])/d - (b^2*(6*a^2 - 8*a*b + 3*b^2)*Coth[c + d*x]^3)/(3
*d) - ((4*a - 3*b)*b^3*Coth[c + d*x]^5)/(5*d) - (b^4*Coth[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.0738219, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4128, 390, 206} \[ -\frac{b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{b (2 a-b) \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}+a^4 x-\frac{b^3 (4 a-3 b) \coth ^5(c+d x)}{5 d}-\frac{b^4 \coth ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csch[c + d*x]^2)^4,x]

[Out]

a^4*x - ((2*a - b)*b*(2*a^2 - 2*a*b + b^2)*Coth[c + d*x])/d - (b^2*(6*a^2 - 8*a*b + 3*b^2)*Coth[c + d*x]^3)/(3
*d) - ((4*a - 3*b)*b^3*Coth[c + d*x]^5)/(5*d) - (b^4*Coth[c + d*x]^7)/(7*d)

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 206

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

Rubi steps

\begin{align*} \int \left (a+b \text{csch}^2(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^4}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-(2 a-b) b \left (2 a^2-2 a b+b^2\right )-b^2 \left (6 a^2-8 a b+3 b^2\right ) x^2-(4 a-3 b) b^3 x^4-b^4 x^6+\frac{a^4}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac{(2 a-b) b \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}-\frac{b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{(4 a-3 b) b^3 \coth ^5(c+d x)}{5 d}-\frac{b^4 \coth ^7(c+d x)}{7 d}+\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=a^4 x-\frac{(2 a-b) b \left (2 a^2-2 a b+b^2\right ) \coth (c+d x)}{d}-\frac{b^2 \left (6 a^2-8 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{(4 a-3 b) b^3 \coth ^5(c+d x)}{5 d}-\frac{b^4 \coth ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 3.36043, size = 149, normalized size = 1.31 \[ \frac{16 \sinh ^8(c+d x) \left (a+b \text{csch}^2(c+d x)\right )^4 \left (105 a^4 (c+d x)-b \coth (c+d x) \left (2 b \left (105 a^2-56 a b+12 b^2\right ) \text{csch}^2(c+d x)-420 a^2 b+420 a^3+6 b^2 (14 a-3 b) \text{csch}^4(c+d x)+224 a b^2+15 b^3 \text{csch}^6(c+d x)-48 b^3\right )\right )}{105 d (a (-\cosh (2 (c+d x)))+a-2 b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(a + b*Csch[c + d*x]^2)^4*(105*a^4*(c + d*x) - b*Coth[c + d*x]*(420*a^3 - 420*a^2*b + 224*a*b^2 - 48*b^3 +
2*b*(105*a^2 - 56*a*b + 12*b^2)*Csch[c + d*x]^2 + 6*(14*a - 3*b)*b^2*Csch[c + d*x]^4 + 15*b^3*Csch[c + d*x]^6
))*Sinh[c + d*x]^8)/(105*d*(a - 2*b - a*Cosh[2*(c + d*x)])^4)

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Maple [A]  time = 0.033, size = 129, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( dx+c \right ) -4\,{a}^{3}b{\rm coth} \left (dx+c\right )+6\,{a}^{2}{b}^{2} \left ( 2/3-1/3\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ){\rm coth} \left (dx+c\right )+4\,a{b}^{3} \left ( -{\frac{8}{15}}-1/5\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{15}} \right ){\rm coth} \left (dx+c\right )+{b}^{4} \left ({\frac{16}{35}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{7}}+{\frac{6\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{35}}-{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{35}} \right ){\rm coth} \left (dx+c\right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*csch(d*x+c)^2)^4,x)

[Out]

1/d*(a^4*(d*x+c)-4*a^3*b*coth(d*x+c)+6*a^2*b^2*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+4*a*b^3*(-8/15-1/5*csch(d*x
+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)+b^4*(16/35-1/7*csch(d*x+c)^6+6/35*csch(d*x+c)^4-8/35*csch(d*x+c)^2)*coth
(d*x+c))

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Maxima [B]  time = 1.0314, size = 953, normalized size = 8.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*x + 32/35*b^4*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*
e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) - 21*e^(-4*d*x - 4*
c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x -
10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-
4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-
14*d*x - 14*c) - 1)) - 1/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8
*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1))) - 64/15*a*b^3*(5*e^(-2*d*x - 2*
c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c
) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x
- 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*
e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1))) + 8*a^2*b^2*(3*e^(-2*d*x - 2*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)))
+ 8*a^3*b/(d*(e^(-2*d*x - 2*c) - 1))

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Fricas [B]  time = 1.58094, size = 2356, normalized size = 20.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(4*(105*a^3*b - 105*a^2*b^2 + 56*a*b^3 - 12*b^4)*cosh(d*x + c)^7 + 28*(105*a^3*b - 105*a^2*b^2 + 56*a*b
^3 - 12*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 - (105*a^4*d*x + 420*a^3*b - 420*a^2*b^2 + 224*a*b^3 - 48*b^4)*sinh
(d*x + c)^7 - 28*(75*a^3*b - 105*a^2*b^2 + 56*a*b^3 - 12*b^4)*cosh(d*x + c)^5 + 7*(105*a^4*d*x + 420*a^3*b - 4
20*a^2*b^2 + 224*a*b^3 - 48*b^4 - 3*(105*a^4*d*x + 420*a^3*b - 420*a^2*b^2 + 224*a*b^3 - 48*b^4)*cosh(d*x + c)
^2)*sinh(d*x + c)^5 + 140*((105*a^3*b - 105*a^2*b^2 + 56*a*b^3 - 12*b^4)*cosh(d*x + c)^3 - (75*a^3*b - 105*a^2
*b^2 + 56*a*b^3 - 12*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(45*a^3*b - 75*a^2*b^2 + 56*a*b^3 - 12*b^4)*cosh
(d*x + c)^3 - 7*(315*a^4*d*x + 5*(105*a^4*d*x + 420*a^3*b - 420*a^2*b^2 + 224*a*b^3 - 48*b^4)*cosh(d*x + c)^4
+ 1260*a^3*b - 1260*a^2*b^2 + 672*a*b^3 - 144*b^4 - 10*(105*a^4*d*x + 420*a^3*b - 420*a^2*b^2 + 224*a*b^3 - 48
*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 28*(3*(105*a^3*b - 105*a^2*b^2 + 56*a*b^3 - 12*b^4)*cosh(d*x + c)^5 -
10*(75*a^3*b - 105*a^2*b^2 + 56*a*b^3 - 12*b^4)*cosh(d*x + c)^3 + 9*(45*a^3*b - 75*a^2*b^2 + 56*a*b^3 - 12*b^
4)*cosh(d*x + c))*sinh(d*x + c)^2 - 420*(5*a^3*b - 9*a^2*b^2 + 8*a*b^3 - 4*b^4)*cosh(d*x + c) - 7*((105*a^4*d*
x + 420*a^3*b - 420*a^2*b^2 + 224*a*b^3 - 48*b^4)*cosh(d*x + c)^6 - 525*a^4*d*x - 5*(105*a^4*d*x + 420*a^3*b -
420*a^2*b^2 + 224*a*b^3 - 48*b^4)*cosh(d*x + c)^4 - 2100*a^3*b + 2100*a^2*b^2 - 1120*a*b^3 + 240*b^4 + 9*(105
*a^4*d*x + 420*a^3*b - 420*a^2*b^2 + 224*a*b^3 - 48*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^7 +
7*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^5 + 7*(5*d*cosh(d*x + c)^4 - 10*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x +
c)^3 + 7*(d*cosh(d*x + c)^6 - 5*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20137, size = 451, normalized size = 3.96 \begin{align*} \frac{{\left (d x + c\right )} a^{4}}{d} - \frac{8 \,{\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} - 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} - 1365 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 2310 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 1400 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 420 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 1890 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 252 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 735 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 392 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 84 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b - 105 \, a^{2} b^{2} + 56 \, a b^{3} - 12 \, b^{4}\right )}}{105 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)*a^4/d - 8/105*(105*a^3*b*e^(12*d*x + 12*c) - 630*a^3*b*e^(10*d*x + 10*c) + 315*a^2*b^2*e^(10*d*x + 1
0*c) + 1575*a^3*b*e^(8*d*x + 8*c) - 1365*a^2*b^2*e^(8*d*x + 8*c) + 560*a*b^3*e^(8*d*x + 8*c) - 2100*a^3*b*e^(6
*d*x + 6*c) + 2310*a^2*b^2*e^(6*d*x + 6*c) - 1400*a*b^3*e^(6*d*x + 6*c) + 420*b^4*e^(6*d*x + 6*c) + 1575*a^3*b
*e^(4*d*x + 4*c) - 1890*a^2*b^2*e^(4*d*x + 4*c) + 1176*a*b^3*e^(4*d*x + 4*c) - 252*b^4*e^(4*d*x + 4*c) - 630*a
^3*b*e^(2*d*x + 2*c) + 735*a^2*b^2*e^(2*d*x + 2*c) - 392*a*b^3*e^(2*d*x + 2*c) + 84*b^4*e^(2*d*x + 2*c) + 105*
a^3*b - 105*a^2*b^2 + 56*a*b^3 - 12*b^4)/(d*(e^(2*d*x + 2*c) - 1)^7)