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

Optimal. Leaf size=40 \[ \frac{(a-2 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.04142, antiderivative size = 40, 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, 3770} \[ \frac{(a-2 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - 2*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - (a*Coth[c + d*x]*Csch[c + d*x])/(2*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 3770

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

Rubi steps

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

Mathematica [B]  time = 0.0320482, size = 99, normalized size = 2.48 \[ -\frac{a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{b \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.034, size = 40, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \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)^3*(a+b*sinh(d*x+c)^2),x)

[Out]

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

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Maxima [B]  time = 1.05464, size = 169, normalized size = 4.22 \begin{align*} \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 )} - 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)^3*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.9364, size = 1323, normalized size = 33.08 \begin{align*} -\frac{2 \, a \cosh \left (d x + c\right )^{3} + 6 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, a \sinh \left (d x + c\right )^{3} + 2 \, a \cosh \left (d x + c\right ) -{\left ({\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{4} + 4 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a - 2 \, b\right )} \sinh \left (d x + c\right )^{4} - 2 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left ({\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{3} -{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a - 2 \, b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) +{\left ({\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{4} + 4 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a - 2 \, b\right )} \sinh \left (d x + c\right )^{4} - 2 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left ({\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{3} -{\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a - 2 \, b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )}{2 \,{\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*cosh(d*x + c)^3 + 6*a*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*sinh(d*x + c)^3 + 2*a*cosh(d*x + c) - ((a
- 2*b)*cosh(d*x + c)^4 + 4*(a - 2*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - 2*b)*sinh(d*x + c)^4 - 2*(a - 2*b)*c
osh(d*x + c)^2 + 2*(3*(a - 2*b)*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^2 + 4*((a - 2*b)*cosh(d*x + c)^3 - (a
- 2*b)*cosh(d*x + c))*sinh(d*x + c) + a - 2*b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a - 2*b)*cosh(d*x +
c)^4 + 4*(a - 2*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - 2*b)*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 +
2*(3*(a - 2*b)*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^2 + 4*((a - 2*b)*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x
+ c))*sinh(d*x + c) + a - 2*b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(3*a*cosh(d*x + c)^2 + a)*sinh(d*x +
c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d
*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 - d*cosh(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)**3*(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.25302, size = 136, normalized size = 3.4 \begin{align*} \frac{{\left (a - 2 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{4 \, d} - \frac{{\left (a - 2 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{4 \, d} - \frac{a{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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