3.1.6 \(\int \text {csch}(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [6]

Optimal. Leaf size=25 \[ -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \cosh (c+d x)}{d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3093, 3855} \begin {gather*} \frac {b \cosh (c+d x)}{d}-\frac {a \tanh ^{-1}(\cosh (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3093

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

Rule 3855

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(25)=50\).
time = 0.03, size = 62, normalized size = 2.48 \begin {gather*} \frac {b \cosh (c) \cosh (d x)}{d}-\frac {a \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \sinh (c) \sinh (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Cosh[c]*Cosh[d*x])/d - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 + (d*x)/2]])/d + (b*Sinh[c]*Sinh[d*
x])/d

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Maple [A]
time = 0.77, size = 24, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {-2 a \arctanh \left ({\mathrm e}^{d x +c}\right )+b \cosh \left (d x +c \right )}{d}\)	\(24\)
default	\(\frac {-2 a \arctanh \left ({\mathrm e}^{d x +c}\right )+b \cosh \left (d x +c \right )}{d}\)	\(24\)
risch	\(\frac {b \,{\mathrm e}^{d x +c}}{2 d}+\frac {{\mathrm e}^{-d x -c} b}{2 d}+\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 43, normalized size = 1.72 \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 {a \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (25) = 50\).
time = 0.41, size = 126, normalized size = 5.04 \begin {gather*} \frac {b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a \cosh \left (d x + c\right ) + a \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 (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + b}{2 \, {\left (d \cosh \left (d x + c\right ) + d \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)*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - 2*(a*cosh(d*x + c) + a*sinh(d*x
+ c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*(a*cosh(d*x + c) + a*sinh(d*x + c))*log(cosh(d*x + c) + sinh
(d*x + c) - 1) + b)/(d*cosh(d*x + c) + d*sinh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 50, normalized size = 2.00 \begin {gather*} \frac {b e^{\left (d x + c\right )} + b e^{\left (-d x - c\right )} - 2 \, a \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.14, size = 66, normalized size = 2.64 \begin {gather*} \frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {2\,\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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