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]

1/2*(a-2*b)*arctanh(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 = 40, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.03, 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]

-1/8*(a*Csch[(c + d*x)/2]^2)/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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fricas [B]  time = 0.59, size = 484, normalized size = 12.10 \[ -\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 )}} \]

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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giac [B]  time = 0.14, size = 96, normalized size = 2.40 \[ \frac {{\left (a - 2 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - {\left (a - 2 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4}}{4 \, d} \]

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) - (a - 2*b)*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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maple [A]  time = 0.07, size = 40, normalized size = 1.00 \[ \frac {a \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )-2 b \arctanh \left ({\mathrm e}^{d x +c}\right )}{d} \]

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 = 0.33, size = 125, normalized size = 3.12 \[ \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} \]

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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mupad [B]  time = 0.13, size = 131, normalized size = 3.28 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {-d^2}-2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^2-4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2-4\,a\,b+4\,b^2}}{\sqrt {-d^2}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((exp(d*x)*exp(c)*(a*(-d^2)^(1/2) - 2*b*(-d^2)^(1/2)))/(d*(a^2 - 4*a*b + 4*b^2)^(1/2)))*(a^2 - 4*a*b + 4*
b^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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right ) \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \]

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]

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

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