3.193 \(\int \text{csch}^5(c+d x) (a+b \sinh ^4(c+d x)) \, dx\)

Optimal. Leaf size=64 \[ -\frac{(3 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a \coth (c+d x) \text{csch}(c+d x)}{8 d} \]

[Out]

-((3*a + 8*b)*ArcTanh[Cosh[c + d*x]])/(8*d) + (3*a*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a*Coth[c + d*x]*Csch[
c + d*x]^3)/(4*d)

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Rubi [A]  time = 0.0627252, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3215, 1157, 385, 206} \[ -\frac{(3 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac{a \coth (c+d x) \text{csch}^3(c+d x)}{4 d}+\frac{3 a \coth (c+d x) \text{csch}(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4),x]

[Out]

-((3*a + 8*b)*ArcTanh[Cosh[c + d*x]])/(8*d) + (3*a*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a*Coth[c + d*x]*Csch[
c + d*x]^3)/(4*d)

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

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

Mathematica [B]  time = 0.0291577, size = 139, normalized size = 2.17 \[ -\frac{a \text{csch}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{a \text{sech}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{3 a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{32 d}+\frac{3 a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{8 d}+\frac{b \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4),x]

[Out]

(3*a*Csch[(c + d*x)/2]^2)/(32*d) - (a*Csch[(c + d*x)/2]^4)/(64*d) - (b*Log[Cosh[c/2 + (d*x)/2]])/d + (b*Log[Si
nh[c/2 + (d*x)/2]])/d + (3*a*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a*Sech[(c + d*x)/2]^2)/(32*d) + (a*Sech[(c + d
*x)/2]^4)/(64*d)

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Maple [A]  time = 0.039, size = 54, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( a \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (dx+c\right )}{8}} \right ){\rm coth} \left (dx+c\right )-{\frac{3\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{4}} \right ) -2\,b{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4),x)

[Out]

1/d*(a*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp(d*x+c)))-2*b*arctanh(exp(d*x+c)))

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Maxima [B]  time = 1.04026, size = 235, normalized size = 3.67 \begin{align*} -\frac{1}{8} \, a{\left (\frac{3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - b{\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}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/8*a*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e
^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8
*d*x - 8*c) - 1))) - b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d)

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Fricas [B]  time = 1.86017, size = 3906, normalized size = 61.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

1/8*(6*a*cosh(d*x + c)^7 + 42*a*cosh(d*x + c)*sinh(d*x + c)^6 + 6*a*sinh(d*x + c)^7 - 22*a*cosh(d*x + c)^5 + 2
*(63*a*cosh(d*x + c)^2 - 11*a)*sinh(d*x + c)^5 + 10*(21*a*cosh(d*x + c)^3 - 11*a*cosh(d*x + c))*sinh(d*x + c)^
4 - 22*a*cosh(d*x + c)^3 + 2*(105*a*cosh(d*x + c)^4 - 110*a*cosh(d*x + c)^2 - 11*a)*sinh(d*x + c)^3 + 2*(63*a*
cosh(d*x + c)^5 - 110*a*cosh(d*x + c)^3 - 33*a*cosh(d*x + c))*sinh(d*x + c)^2 + 6*a*cosh(d*x + c) - ((3*a + 8*
b)*cosh(d*x + c)^8 + 8*(3*a + 8*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a + 8*b)*sinh(d*x + c)^8 - 4*(3*a + 8*b)
*cosh(d*x + c)^6 + 4*(7*(3*a + 8*b)*cosh(d*x + c)^2 - 3*a - 8*b)*sinh(d*x + c)^6 + 8*(7*(3*a + 8*b)*cosh(d*x +
c)^3 - 3*(3*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a + 8*b)*cosh(d*x + c)^4 + 2*(35*(3*a + 8*b)*cosh(
d*x + c)^4 - 30*(3*a + 8*b)*cosh(d*x + c)^2 + 9*a + 24*b)*sinh(d*x + c)^4 + 8*(7*(3*a + 8*b)*cosh(d*x + c)^5 -
10*(3*a + 8*b)*cosh(d*x + c)^3 + 3*(3*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a + 8*b)*cosh(d*x + c)^2
+ 4*(7*(3*a + 8*b)*cosh(d*x + c)^6 - 15*(3*a + 8*b)*cosh(d*x + c)^4 + 9*(3*a + 8*b)*cosh(d*x + c)^2 - 3*a - 8
*b)*sinh(d*x + c)^2 + 8*((3*a + 8*b)*cosh(d*x + c)^7 - 3*(3*a + 8*b)*cosh(d*x + c)^5 + 3*(3*a + 8*b)*cosh(d*x
+ c)^3 - (3*a + 8*b)*cosh(d*x + c))*sinh(d*x + c) + 3*a + 8*b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((3*a
+ 8*b)*cosh(d*x + c)^8 + 8*(3*a + 8*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a + 8*b)*sinh(d*x + c)^8 - 4*(3*a +
8*b)*cosh(d*x + c)^6 + 4*(7*(3*a + 8*b)*cosh(d*x + c)^2 - 3*a - 8*b)*sinh(d*x + c)^6 + 8*(7*(3*a + 8*b)*cosh(d
*x + c)^3 - 3*(3*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a + 8*b)*cosh(d*x + c)^4 + 2*(35*(3*a + 8*b)*c
osh(d*x + c)^4 - 30*(3*a + 8*b)*cosh(d*x + c)^2 + 9*a + 24*b)*sinh(d*x + c)^4 + 8*(7*(3*a + 8*b)*cosh(d*x + c)
^5 - 10*(3*a + 8*b)*cosh(d*x + c)^3 + 3*(3*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a + 8*b)*cosh(d*x +
c)^2 + 4*(7*(3*a + 8*b)*cosh(d*x + c)^6 - 15*(3*a + 8*b)*cosh(d*x + c)^4 + 9*(3*a + 8*b)*cosh(d*x + c)^2 - 3*a
- 8*b)*sinh(d*x + c)^2 + 8*((3*a + 8*b)*cosh(d*x + c)^7 - 3*(3*a + 8*b)*cosh(d*x + c)^5 + 3*(3*a + 8*b)*cosh(
d*x + c)^3 - (3*a + 8*b)*cosh(d*x + c))*sinh(d*x + c) + 3*a + 8*b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*
(21*a*cosh(d*x + c)^6 - 55*a*cosh(d*x + c)^4 - 33*a*cosh(d*x + c)^2 + 3*a)*sinh(d*x + c))/(d*cosh(d*x + c)^8 +
8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 - 4*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 - d)*sin
h(d*x + c)^6 + 8*(7*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cos
h(d*x + c)^4 - 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 - 10*d*cosh(d*x + c)^3 + 3
*d*cosh(d*x + c))*sinh(d*x + c)^3 - 4*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 - 15*d*cosh(d*x + c)^4 + 9*d*
cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - 3*d*cosh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 - d*co
sh(d*x + c))*sinh(d*x + c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**5*(a+b*sinh(d*x+c)**4),x)

[Out]

Timed out

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Giac [B]  time = 1.16487, size = 174, normalized size = 2.72 \begin{align*} -\frac{{\left (3 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{16 \, d} + \frac{{\left (3 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{16 \, d} + \frac{3 \, a{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 20 \, a{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{4 \,{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

-1/16*(3*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d + 1/16*(3*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) - 2)
/d + 1/4*(3*a*(e^(d*x + c) + e^(-d*x - c))^3 - 20*a*(e^(d*x + c) + e^(-d*x - c)))/(((e^(d*x + c) + e^(-d*x - c
))^2 - 4)^2*d)