3.2.91 \(\int \text {csch}^3(c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [191]

Optimal. Leaf size=47 \[ \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d} \]

[Out]

1/2*a*arctanh(cosh(d*x+c))/d+b*cosh(d*x+c)/d-1/2*a*coth(d*x+c)*csch(d*x+c)/d

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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3294, 1171, 396, 212} \begin {gather*} \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4),x]

[Out]

(a*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*Cosh[c + d*x])/d - (a*Coth[c + d*x]*Csch[c + d*x])/(2*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 396

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

Rule 1171

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] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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

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Mathematica [A]
time = 0.02, size = 82, normalized size = 1.74 \begin {gather*} \frac {b \cosh (c) \cosh (d x)}{d}-\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {b \sinh (c) \sinh (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4),x]

[Out]

(b*Cosh[c]*Cosh[d*x])/d - (a*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)
/2]^2)/(8*d) + (b*Sinh[c]*Sinh[d*x])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(43)=86\).
time = 1.25, size = 95, normalized size = 2.02

method	result	size
risch	\(\frac {b \,{\mathrm e}^{d x +c}}{2 d}+\frac {{\mathrm e}^{-d x -c} b}{2 d}-\frac {a \,{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (43) = 86\).
time = 0.27, size = 115, normalized size = 2.45 \begin {gather*} \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a {\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} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

1/2*b*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/2*a*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x
- c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - 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. 690 vs. \(2 (43) = 86\).
time = 0.41, size = 690, normalized size = 14.68 \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} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, b \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, b \cosh \left (d x + c\right )^{4} - 6 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + {\left (a \cosh \left (d x + c\right )^{5} + 5 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{5} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{3} - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + a \cosh \left (d x + c\right ) + {\left (5 \, a \cosh \left (d x + c\right )^{4} - 6 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (a \cosh \left (d x + c\right )^{5} + 5 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{5} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, a \cosh \left (d x + c\right )^{3} - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + a \cosh \left (d x + c\right ) + {\left (5 \, a \cosh \left (d x + c\right )^{4} - 6 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{5} - 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{2 \, {\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} - 2 \, d \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{4} - 6 \, d \cosh \left (d x + c\right )^{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)^3*(a+b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

1/2*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - (2*a + b)*cosh(d*x + c)^4 + (
15*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x +
c)^3 - (2*a + b)*cosh(d*x + c)^2 + (15*b*cosh(d*x + c)^4 - 6*(2*a + b)*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x +
c)^2 + (a*cosh(d*x + c)^5 + 5*a*cosh(d*x + c)*sinh(d*x + c)^4 + a*sinh(d*x + c)^5 - 2*a*cosh(d*x + c)^3 + 2*(5
*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^3 + 2*(5*a*cosh(d*x + c)^3 - 3*a*cosh(d*x + c))*sinh(d*x + c)^2 + a*cosh
(d*x + c) + (5*a*cosh(d*x + c)^4 - 6*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) +
1) - (a*cosh(d*x + c)^5 + 5*a*cosh(d*x + c)*sinh(d*x + c)^4 + a*sinh(d*x + c)^5 - 2*a*cosh(d*x + c)^3 + 2*(5*
a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^3 + 2*(5*a*cosh(d*x + c)^3 - 3*a*cosh(d*x + c))*sinh(d*x + c)^2 + a*cosh(
d*x + c) + (5*a*cosh(d*x + c)^4 - 6*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) -
1) + 2*(3*b*cosh(d*x + c)^5 - 2*(2*a + b)*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) + b)/(d*cos
h(d*x + c)^5 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5 - 2*d*cosh(d*x + c)^3 + 2*(5*d*cosh(d*x +
c)^2 - d)*sinh(d*x + c)^3 + 2*(5*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + d*cosh(d*x + c) + (
5*d*cosh(d*x + c)^4 - 6*d*cosh(d*x + c)^2 + d)*sinh(d*x + c))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**3*(a+b*sinh(d*x+c)**4),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (43) = 86\).
time = 0.44, size = 107, normalized size = 2.28 \begin {gather*} \frac {2 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + a \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - a \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.72, size = 126, normalized size = 2.68 \begin {gather*} \frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {\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}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\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*sinh(c + d*x)^4)/sinh(c + d*x)^3,x)

[Out]

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

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