3.231 \(\int \frac{\sinh ^2(c+d x)}{a+b \cosh (c+d x)} \, dx\)

Optimal. Leaf size=73 \[ \frac{2 \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d}-\frac{a x}{b^2}+\frac{\sinh (c+d x)}{b d} \]

[Out]

-((a*x)/b^2) + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/(b^2*d) + Sinh
[c + d*x]/(b*d)

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Rubi [A]  time = 0.123418, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2695, 2735, 2659, 205} \[ \frac{2 \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d}-\frac{a x}{b^2}+\frac{\sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*x)/b^2) + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tanh[(c + d*x)/2])/Sqrt[a + b]])/(b^2*d) + Sinh
[c + d*x]/(b*d)

Rule 2695

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(b*(m + p)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.180409, size = 69, normalized size = 0.95 \[ \frac{2 \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )-a (c+d x)+b \sinh (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a*(c + d*x)) + 2*Sqrt[-a^2 + b^2]*ArcTan[((a - b)*Tanh[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + b*Sinh[c + d*x])/(
b^2*d)

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Maple [B]  time = 0.009, size = 177, normalized size = 2.4 \begin{align*} 2\,{\frac{{a}^{2}}{d{b}^{2}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-2\,{\frac{1}{d\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{1}{bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{a}{d{b}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{1}{bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{d{b}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/b^2/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^2-2/d/((a+b)*(a-b))^(1/2)
*arctanh((a-b)*tanh(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))-1/d/b/(tanh(1/2*d*x+1/2*c)+1)-1/d*a/b^2*ln(tanh(1/2*d*
x+1/2*c)+1)-1/d/b/(tanh(1/2*d*x+1/2*c)-1)+1/d*a/b^2*ln(tanh(1/2*d*x+1/2*c)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.03002, size = 1085, normalized size = 14.86 \begin{align*} \left [-\frac{2 \, a d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{2} - b \sinh \left (d x + c\right )^{2} - 2 \, \sqrt{a^{2} - b^{2}}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \log \left (\frac{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \,{\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) + b}\right ) + 2 \,{\left (a d x - b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{2 \,{\left (b^{2} d \cosh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )\right )}}, -\frac{2 \, a d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{2} - b \sinh \left (d x + c\right )^{2} + 4 \, \sqrt{-a^{2} + b^{2}}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{a^{2} - b^{2}}\right ) + 2 \,{\left (a d x - b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{2 \,{\left (b^{2} d \cosh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*a*d*x*cosh(d*x + c) - b*cosh(d*x + c)^2 - b*sinh(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(cosh(d*x + c) + sinh
(d*x + c))*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + 2*a^2 - b^2 + 2*(b^2*cosh(d*
x + c) + a*b)*sinh(d*x + c) - 2*sqrt(a^2 - b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/(b*cosh(d*x + c)^2 +
b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) + b)) + 2*(a*d*x - b*cosh(d*x +
c))*sinh(d*x + c) + b)/(b^2*d*cosh(d*x + c) + b^2*d*sinh(d*x + c)), -1/2*(2*a*d*x*cosh(d*x + c) - b*cosh(d*x +
c)^2 - b*sinh(d*x + c)^2 + 4*sqrt(-a^2 + b^2)*(cosh(d*x + c) + sinh(d*x + c))*arctan(-sqrt(-a^2 + b^2)*(b*cos
h(d*x + c) + b*sinh(d*x + c) + a)/(a^2 - b^2)) + 2*(a*d*x - b*cosh(d*x + c))*sinh(d*x + c) + b)/(b^2*d*cosh(d*
x + c) + b^2*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(sinh(d*x+c)**2/(a+b*cosh(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.24338, size = 131, normalized size = 1.79 \begin{align*} -\frac{{\left (d x + c\right )} a}{b^{2} d} + \frac{e^{\left (d x + c\right )}}{2 \, b d} - \frac{e^{\left (-d x - c\right )}}{2 \, b d} + \frac{2 \,{\left (a^{2} - b^{2}\right )} \arctan \left (\frac{b e^{\left (d x + c\right )} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}} b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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