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

Optimal. Leaf size=16 \[ b x-\frac{a \coth (c+d x)}{d} \]

[Out]

b*x - (a*Coth[c + d*x])/d

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Rubi [A]  time = 0.0266243, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3012, 8} \[ b x-\frac{a \coth (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (a*Coth[c + d*x])/d

Rule 3012

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0195822, size = 16, normalized size = 1. \[ b x-\frac{a \coth (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (a*Coth[c + d*x])/d

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Maple [A]  time = 0.027, size = 22, normalized size = 1.4 \begin{align*}{\frac{-{\rm coth} \left (dx+c\right )a+ \left ( dx+c \right ) b}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02413, size = 31, normalized size = 1.94 \begin{align*} b x + \frac{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + 2*a/(d*(e^(-2*d*x - 2*c) - 1))

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Fricas [B]  time = 1.8503, size = 89, normalized size = 5.56 \begin{align*} -\frac{a \cosh \left (d x + c\right ) -{\left (b d x + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.29289, size = 41, normalized size = 2.56 \begin{align*} \frac{{\left (d x + c\right )} b}{d} - \frac{2 \, a}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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