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

Optimal. Leaf size=74 \[ \frac {a \sinh (x)}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}-\frac {a^2 \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]

[Out]

-a^2*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)-b*cosh(x)/(a^2-b^2)+a*sinh(x)/(a^2-b^2)

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Rubi [A]  time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3099, 3074, 206, 2638} \[ \frac {a \sinh (x)}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}-\frac {a^2 \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh
[x])/(a^2 - b^2)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2638

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

Rule 3074

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

Rule 3099

Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(a*Sin[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2)/
(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a,
b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]

Rubi steps

\begin {align*} \int \frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx &=\frac {a \sinh (x)}{a^2-b^2}-\frac {a^2 \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac {b \int \sinh (x) \, dx}{a^2-b^2}\\ &=-\frac {b \cosh (x)}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2}-\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a^2-b^2}\\ &=-\frac {a^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {b \cosh (x)}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 89, normalized size = 1.20 \[ \frac {a \left (\sqrt {a-b} (a+b) \sinh (x)-2 a \sqrt {a+b} \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )\right )-b \sqrt {a-b} (a+b) \cosh (x)}{(a-b)^{3/2} (a+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[a - b]*b*(a + b)*Cosh[x]) + a*(-2*a*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])] +
Sqrt[a - b]*(a + b)*Sinh[x]))/((a - b)^(3/2)*(a + b)^2)

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fricas [B]  time = 0.45, size = 435, normalized size = 5.88 \[ \left [-\frac {a^{3} + a^{2} b - a b^{2} - b^{3} - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \relax (x)^{2} - 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \relax (x) \sinh \relax (x) - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \relax (x)^{2} - 2 \, {\left (a^{2} \cosh \relax (x) + a^{2} \sinh \relax (x)\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} - a + b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + a - b}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)\right )}}, -\frac {a^{3} + a^{2} b - a b^{2} - b^{3} - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \relax (x)^{2} - 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \relax (x) \sinh \relax (x) - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \relax (x)^{2} - 4 \, {\left (a^{2} \cosh \relax (x) + a^{2} \sinh \relax (x)\right )} \sqrt {a^{2} - b^{2}} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(a^3 + a^2*b - a*b^2 - b^3 - (a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2 - 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(
x)*sinh(x) - (a^3 - a^2*b - a*b^2 + b^3)*sinh(x)^2 - 2*(a^2*cosh(x) + a^2*sinh(x))*sqrt(-a^2 + b^2)*log(((a +
b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)
/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x
) + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)), -1/2*(a^3 + a^2*b - a*b^2 - b^3 - (a^3 - a^2*b - a*b^2 + b^3)*cosh(x)^2
- 2*(a^3 - a^2*b - a*b^2 + b^3)*cosh(x)*sinh(x) - (a^3 - a^2*b - a*b^2 + b^3)*sinh(x)^2 - 4*(a^2*cosh(x) + a^2
*sinh(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))))/((a^4 - 2*a^2*b^2 + b^4
)*cosh(x) + (a^4 - 2*a^2*b^2 + b^4)*sinh(x))]

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giac [A]  time = 0.12, size = 61, normalized size = 0.82 \[ -\frac {2 \, a^{2} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} - \frac {e^{\left (-x\right )}}{2 \, {\left (a - b\right )}} + \frac {e^{x}}{2 \, {\left (a + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^2*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(a^2 - b^2)^(3/2) - 1/2*e^(-x)/(a - b) + 1/2*e^x/(a + b)

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maple [A]  time = 0.20, size = 93, normalized size = 1.26 \[ -\frac {8}{\left (8 a -8 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 a^{2} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {a^{2}-b^{2}}}-\frac {8}{\left (8 a +8 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/(8*a-8*b)/(tanh(1/2*x)+1)-2*a^2/(a+b)/(a-b)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2
))-8/(8*a+8*b)/(tanh(1/2*x)-1)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 1.72, size = 157, normalized size = 2.12 \[ \frac {{\mathrm {e}}^x}{2\,a+2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,a-2\,b}-\frac {a^2\,\ln \left (-\frac {2\,a^2}{{\left (a+b\right )}^{5/2}\,\sqrt {b-a}}-\frac {2\,a^2\,{\mathrm {e}}^x}{-a^3-a^2\,b+a\,b^2+b^3}\right )}{{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}+\frac {a^2\,\ln \left (\frac {2\,a^2}{{\left (a+b\right )}^{5/2}\,\sqrt {b-a}}-\frac {2\,a^2\,{\mathrm {e}}^x}{-a^3-a^2\,b+a\,b^2+b^3}\right )}{{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)/(2*a + 2*b) - exp(-x)/(2*a - 2*b) - (a^2*log(- (2*a^2)/((a + b)^(5/2)*(b - a)^(1/2)) - (2*a^2*exp(x))/(
a*b^2 - a^2*b - a^3 + b^3)))/((a + b)^(3/2)*(b - a)^(3/2)) + (a^2*log((2*a^2)/((a + b)^(5/2)*(b - a)^(1/2)) -
(2*a^2*exp(x))/(a*b^2 - a^2*b - a^3 + b^3)))/((a + b)^(3/2)*(b - a)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**2/(a*cosh(x)+b*sinh(x)),x)

[Out]

Timed out

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