3.2.89 \(\int \text {csch}(c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [189]

Optimal. Leaf size=42 \[ -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3294, 1167, 212} \begin {gather*} -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \cosh ^3(c+d x)}{3 d}-\frac {b \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

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

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 3294

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.67 \begin {gather*} -\frac {3 b \cosh (c+d x)}{4 d}+\frac {b \cosh (3 (c+d x))}{12 d}-\frac {a \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d) - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 +
(d*x)/2]])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(40)=80\).
time = 1.10, size = 88, normalized size = 2.10

method	result	size
risch	\(\frac {b \,{\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 b \,{\mathrm e}^{d x +c}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} b}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b}{24 d}+\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/24*b/d*exp(3*d*x+3*c)-3/8*b/d*exp(d*x+c)-3/8/d*exp(-d*x-c)*b+1/24/d*exp(-3*d*x-3*c)*b+a/d*ln(exp(d*x+c)-1)-a
/d*ln(exp(d*x+c)+1)

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Maxima [A]
time = 0.26, size = 71, normalized size = 1.69 \begin {gather*} \frac {1}{24} \, b {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {a \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + a*log(tanh(1/2*d*x + 1/
2*c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (40) = 80\).
time = 0.40, size = 395, normalized size = 9.40 \begin {gather*} \frac {b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 9 \, b \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{2} - 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{4} - 18 \, b \cosh \left (d x + c\right )^{2} - 3 \, b\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \, {\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 9*b*cosh(d*x + c)^4 + 3*(5*b
*cosh(d*x + c)^2 - 3*b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 9*b*cosh(d*x + c))*sinh(d*x + c)^3 - 9*b*co
sh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 18*b*cosh(d*x + c)^2 - 3*b)*sinh(d*x + c)^2 - 24*(a*cosh(d*x + c)^3 +
3*a*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3)*log(cosh(d*x + c)
+ sinh(d*x + c) + 1) + 24*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*cosh(d*x + c)*sinh(d*x
+ c)^2 + a*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(b*cosh(d*x + c)^5 - 6*b*cosh(d*x + c)^
3 - 3*b*cosh(d*x + c))*sinh(d*x + c) + b)/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*d*cosh(d*
x + c)*sinh(d*x + c)^2 + d*sinh(d*x + c)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 78, normalized size = 1.86 \begin {gather*} \frac {b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b e^{\left (d x + c\right )} - {\left (9 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 24 \, a \log \left (e^{\left (d x + c\right )} + 1\right ) + 24 \, a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b*e^(3*d*x + 3*c) - 9*b*e^(d*x + c) - (9*b*e^(2*d*x + 2*c) - b)*e^(-3*d*x - 3*c) - 24*a*log(e^(d*x + c)
+ 1) + 24*a*log(abs(e^(d*x + c) - 1)))/d

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Mupad [B]
time = 0.13, size = 96, normalized size = 2.29 \begin {gather*} \frac {b\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b\,{\mathrm {e}}^{-c-d\,x}}{8\,d}+\frac {b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {3\,b\,{\mathrm {e}}^{c+d\,x}}{8\,d}-\frac {2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*exp(- 3*c - 3*d*x))/(24*d) - (3*b*exp(- c - d*x))/(8*d) + (b*exp(3*c + 3*d*x))/(24*d) - (3*b*exp(c + d*x))/
(8*d) - (2*atan((a*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^2)^(1/2)))*(a^2)^(1/2))/(-d^2)^(1/2)

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