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

Optimal. Leaf size=72 \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {b (2 a-b) \tanh (c+d x)}{d}+\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]

[Out]

a*(a-2*b)*coth(d*x+c)/d-1/3*a^2*coth(d*x+c)^3/d-(2*a-b)*b*tanh(d*x+c)/d-1/3*b^2*tanh(d*x+c)^3/d

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3663, 448} \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {b (2 a-b) \tanh (c+d x)}{d}+\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 448

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

Rule 3663

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.52, size = 59, normalized size = 0.82 \[ \frac {b \tanh (c+d x) \left (-6 a+b \text {sech}^2(c+d x)+2 b\right )-a \coth (c+d x) \left (a \text {csch}^2(c+d x)-2 a+6 b\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a*Coth[c + d*x]*(-2*a + 6*b + a*Csch[c + d*x]^2)) + b*(-6*a + 2*b + b*Sech[c + d*x]^2)*Tanh[c + d*x])/(3*d)

________________________________________________________________________________________

fricas [B]  time = 0.71, size = 393, normalized size = 5.46 \[ -\frac {8 \, {\left ({\left (a^{2} + 6 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 8 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 6 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 6 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} - 6 \, a b + 3 \, b^{2} + 8 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{8} + 56 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{6} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} - 4 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} - 6 \, d \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} - d \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3*((a^2 + 6*a*b + b^2)*cosh(d*x + c)^4 + 8*(a^2 + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 6*a*b + b^2)*
sinh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 6*a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 - 2*b^2)*si
nh(d*x + c)^2 + 3*a^2 - 6*a*b + 3*b^2 + 8*((a^2 + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x +
c))/(d*cosh(d*x + c)^8 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*d*co
sh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 4*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - 2*d)*sinh(d*
x + c)^4 + 8*(7*d*cosh(d*x + c)^5 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*d*cosh(d*x + c)^6 - 6*d*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - d*cosh(d*x + c)^3)*sinh(d*x + c) + 3*d)

________________________________________________________________________________________

giac [B]  time = 0.26, size = 143, normalized size = 1.99 \[ -\frac {4 \, {\left (3 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 6 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 3 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 8 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 12 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} + 6 \, a b - b^{2}\right )}}{3 \, d {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*a^2*e^(8*d*x + 8*c) + 6*a*b*e^(8*d*x + 8*c) + 3*b^2*e^(8*d*x + 8*c) + 8*a^2*e^(6*d*x + 6*c) - 8*b^2*e^
(6*d*x + 6*c) + 6*a^2*e^(4*d*x + 4*c) - 12*a*b*e^(4*d*x + 4*c) + 6*b^2*e^(4*d*x + 4*c) - a^2 + 6*a*b - b^2)/(d
*(e^(4*d*x + 4*c) - 1)^3)

________________________________________________________________________________________

maple [A]  time = 0.46, size = 81, normalized size = 1.12 \[ \frac {a^{2} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+2 a b \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )+b^{2} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^2*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+2*a*b*(-1/sinh(d*x+c)/cosh(d*x+c)-2*tanh(d*x+c))+b^2*(2/3+1/3*sec
h(d*x+c)^2)*tanh(d*x+c))

________________________________________________________________________________________

maxima [B]  time = 0.32, size = 210, normalized size = 2.92 \[ \frac {4}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {8 \, a b}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.08, size = 143, normalized size = 1.99 \[ -\frac {4\,\left (6\,a\,b-a^2-b^2+6\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}+8\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}+3\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}+3\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}-12\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}+6\,a\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}\right )}{3\,d\,{\left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tanh(c + d*x)^2)^2/sinh(c + d*x)^4,x)

[Out]

-(4*(6*a*b - a^2 - b^2 + 6*a^2*exp(4*c + 4*d*x) + 8*a^2*exp(6*c + 6*d*x) + 3*a^2*exp(8*c + 8*d*x) + 6*b^2*exp(
4*c + 4*d*x) - 8*b^2*exp(6*c + 6*d*x) + 3*b^2*exp(8*c + 8*d*x) - 12*a*b*exp(4*c + 4*d*x) + 6*a*b*exp(8*c + 8*d
*x)))/(3*d*(exp(4*c + 4*d*x) - 1)^3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________