3.1.14 \(\int \text {csch}^2(c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [14]

Optimal. Leaf size=50 \[ -\frac {(a+b)^2 \coth (c+d x)}{d}-\frac {2 b (a+b) \tanh (c+d x)}{d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 50, 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 = {4217, 276} \begin {gather*} -\frac {2 b (a+b) \tanh (c+d x)}{d}-\frac {(a+b)^2 \coth (c+d x)}{d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 4217

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(50)=100\).
time = 1.21, size = 109, normalized size = 2.18 \begin {gather*} -\frac {4 \left (b+a \cosh ^2(c+d x)\right )^2 \text {sech}^3(c+d x) \left (b^2 \text {sech}(c) \sinh (d x)+\cosh ^2(c+d x) \left (-3 (a+b)^2 \coth (c+d x) \text {csch}(c)+b (6 a+5 b) \text {sech}(c)\right ) \sinh (d x)+b^2 \cosh (c+d x) \tanh (c)\right )}{3 d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(b + a*Cosh[c + d*x]^2)^2*Sech[c + d*x]^3*(b^2*Sech[c]*Sinh[d*x] + Cosh[c + d*x]^2*(-3*(a + b)^2*Coth[c +
d*x]*Csch[c] + b*(6*a + 5*b)*Sech[c])*Sinh[d*x] + b^2*Cosh[c + d*x]*Tanh[c]))/(3*d*(a + 2*b + a*Cosh[2*(c + d*
x)])^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(48)=96\).
time = 2.10, size = 129, normalized size = 2.58

method	result	size
risch	\(-\frac {2 \left (3 a^{2} {\mathrm e}^{6 d x +6 c}+9 a^{2} {\mathrm e}^{4 d x +4 c}+12 a b \,{\mathrm e}^{4 d x +4 c}+9 a^{2} {\mathrm e}^{2 d x +2 c}+24 a b \,{\mathrm e}^{2 d x +2 c}+16 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}+12 a b +8 b^{2}\right )}{3 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*a^2*exp(6*d*x+6*c)+9*a^2*exp(4*d*x+4*c)+12*a*b*exp(4*d*x+4*c)+9*a^2*exp(2*d*x+2*c)+24*a*b*exp(2*d*x+2*
c)+16*b^2*exp(2*d*x+2*c)+3*a^2+12*a*b+8*b^2)/d/(1+exp(2*d*x+2*c))^3/(exp(2*d*x+2*c)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (48) = 96\).
time = 0.30, size = 140, normalized size = 2.80 \begin {gather*} -\frac {16}{3} \, b^{2} {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}} + \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + \frac {2 \, a^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac {8 \, a b}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/3*b^2*(2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) + 1)) + 1/(d*(2*e
^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) + 1))) + 2*a^2/(d*(e^(-2*d*x - 2*c) - 1)) + 8*a*b/(d*(
e^(-4*d*x - 4*c) - 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (48) = 96\).
time = 0.37, size = 284, normalized size = 5.68 \begin {gather*} -\frac {4 \, {\left ({\left (3 \, a^{2} + 6 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{2} + 6 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - 2 \, {\left (3 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + {\left (9 \, a^{2} + 18 \, a b + 8 \, b^{2}\right )} \cosh \left (d x + c\right ) - 2 \, {\left (3 \, {\left (3 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, a b + 4 \, b^{2}\right )} \sinh \left (d x + c\right )\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5} + d \cosh \left (d x + c\right )^{3} + {\left (10 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{3} + {\left (10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 2 \, d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*((3*a^2 + 6*a*b + 4*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 6*a*b + 4*b^2)*cosh(d*x + c)*sinh(d*x + c)^2 - 2*(3
*a*b + 2*b^2)*sinh(d*x + c)^3 + (9*a^2 + 18*a*b + 8*b^2)*cosh(d*x + c) - 2*(3*(3*a*b + 2*b^2)*cosh(d*x + c)^2
+ 3*a*b + 4*b^2)*sinh(d*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5 + d
*cosh(d*x + c)^3 + (10*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^3 + (10*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*s
inh(d*x + c)^2 - 2*d*cosh(d*x + c) + (5*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 + 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 )^{2} \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (48) = 96\).
time = 0.43, size = 111, normalized size = 2.22 \begin {gather*} -\frac {2 \, {\left (\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} - \frac {6 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b + 5 \, b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}\right )}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*(a^2 + 2*a*b + b^2)/(e^(2*d*x + 2*c) - 1) - (6*a*b*e^(4*d*x + 4*c) + 3*b^2*e^(4*d*x + 4*c) + 12*a*b*e^
(2*d*x + 2*c) + 12*b^2*e^(2*d*x + 2*c) + 6*a*b + 5*b^2)/(e^(2*d*x + 2*c) + 1)^3)/d

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Mupad [B]
time = 1.49, size = 215, normalized size = 4.30 \begin {gather*} \frac {\frac {2\,\left (3\,b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+2\,a\,b\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\frac {2\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,b^2+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 (a^2+2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {2\,\left (b^2+2\,a\,b\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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