3.147 \(\int \text {csch}^2(c+d x) (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=24 \[ \frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x)}{d} \]

[Out]

b*cosh(d*x+c)/d-a*coth(d*x+c)/d

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3220, 3767, 8, 2638} \[ \frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3),x]

[Out]

(b*Cosh[c + d*x])/d - (a*Coth[c + d*x])/d

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 35, normalized size = 1.46 \[ -\frac {a \coth (c+d x)}{d}+\frac {b \sinh (c) \sinh (d x)}{d}+\frac {b \cosh (c) \cosh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3),x]

[Out]

(b*Cosh[c]*Cosh[d*x])/d - (a*Coth[c + d*x])/d + (b*Sinh[c]*Sinh[d*x])/d

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fricas [A]  time = 0.93, size = 40, normalized size = 1.67 \[ -\frac {a \cosh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

-(a*cosh(d*x + c) - (b*cosh(d*x + c) + a)*sinh(d*x + c))/(d*sinh(d*x + c))

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giac [B]  time = 0.14, size = 59, normalized size = 2.46 \[ \frac {b e^{\left (d x + c\right )} + \frac {b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )} - b}{e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 23, normalized size = 0.96 \[ \frac {-\coth \left (d x +c \right ) a +b \cosh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2*(a+b*sinh(d*x+c)^3),x)

[Out]

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

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maxima [A]  time = 0.32, size = 47, normalized size = 1.96 \[ \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {2 \, a}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^2*(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 47, normalized size = 1.96 \[ \frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}-\frac {2\,a}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^3)/sinh(c + d*x)^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sinh ^{3}{\left (c + d x \right )}\right ) \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**2*(a+b*sinh(d*x+c)**3),x)

[Out]

Integral((a + b*sinh(c + d*x)**3)*csch(c + d*x)**2, x)

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