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

Optimal. Leaf size=39 \[ \frac {x}{a+b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {b} (a+b)} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {481, 207, 205} \[ \frac {x}{a+b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {b} (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a + b) - (Sqrt[a]*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a]])/(Sqrt[b]*(a + b))

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]

Rule 207

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

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 34, normalized size = 0.87 \[ \frac {x-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {b}}}{a+b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x - (Sqrt[a]*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a]])/Sqrt[b])/(a + b)

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fricas [A]  time = 0.47, size = 367, normalized size = 9.41 \[ \left [\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - 6 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 4 \, {\left ({\left (a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a b + b^{2}\right )} \sinh \relax (x)^{2} + a b - b^{2}\right )} \sqrt {-\frac {a}{b}}}{{\left (a + b\right )} \cosh \relax (x)^{4} + 4 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a + b\right )} \sinh \relax (x)^{4} + 2 \, {\left (a - b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \relax (x)^{2} + a - b\right )} \sinh \relax (x)^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \relax (x)^{3} + {\left (a - b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + a + b}\right ) + 2 \, x}{2 \, {\left (a + b\right )}}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + a - b\right )} \sqrt {\frac {a}{b}}}{2 \, a}\right ) - x}{a + b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 46, normalized size = 1.18 \[ -\frac {a \arctan \left (\frac {a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} {\left (a + b\right )}} + \frac {x}{a + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.20, size = 414, normalized size = 10.62 \[ -\frac {8 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a +8 b}+\frac {8 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a +8 b}+\frac {4 a^{2} \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{\left (4 a +4 b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {4 a \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{\left (4 a +4 b \right ) \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {4 a \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right ) b}{\left (4 a +4 b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {4 a^{2} \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (4 a +4 b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {4 a \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (4 a +4 b \right ) \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {4 a \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right ) b}{\left (4 a +4 b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/(8*a+8*b)*ln(tanh(1/2*x)-1)+8/(8*a+8*b)*ln(tanh(1/2*x)+1)+4*a^2/(4*a+4*b)/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/
2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*x)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-4*a/(4*a+4*b)/((2*(b*(a+b))^(1/2
)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*x)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+4*a/(4*a+4*b)/(b*(a+b))^(1/2)/((2
*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*x)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))*b+4*a^2/(4*a+4*b)/
(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*x)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+
4*a/(4*a+4*b)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*x)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+4*
a/(4*a+4*b)/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*x)/((2*(b*(a+b))^(1/2)+a+2*b
)*a)^(1/2))*b

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maxima [B]  time = 0.43, size = 74, normalized size = 1.90 \[ -\frac {{\left (a - b\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (2 \, x\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (a + b\right )}} + \frac {\arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, x\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {x}{a + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.29, size = 209, normalized size = 5.36 \[ \frac {x}{a+b}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{2\,x}\,\left (\frac {4\,a}{{\left (a+b\right )}^4}+\frac {\left (a^2-b^2\right )\,\left (a-b\right )}{{\left (a+b\right )}^3\,\sqrt {b\,{\left (a+b\right )}^2}\,\sqrt {a^2\,b+2\,a\,b^2+b^3}}\right )+\frac {\left (a-b\right )\,\left (a^2+2\,a\,b+b^2\right )}{{\left (a+b\right )}^3\,\sqrt {b\,{\left (a+b\right )}^2}\,\sqrt {a^2\,b+2\,a\,b^2+b^3}}\right )\,\left (a^2\,\sqrt {a^2\,b+2\,a\,b^2+b^3}+b^2\,\sqrt {a^2\,b+2\,a\,b^2+b^3}+2\,a\,b\,\sqrt {a^2\,b+2\,a\,b^2+b^3}\right )}{2\,\sqrt {a}}\right )}{\sqrt {a^2\,b+2\,a\,b^2+b^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(a + b) - (a^(1/2)*atan(((exp(2*x)*((4*a)/(a + b)^4 + ((a^2 - b^2)*(a - b))/((a + b)^3*(b*(a + b)^2)^(1/2)*(
2*a*b^2 + a^2*b + b^3)^(1/2))) + ((a - b)*(2*a*b + a^2 + b^2))/((a + b)^3*(b*(a + b)^2)^(1/2)*(2*a*b^2 + a^2*b
+ b^3)^(1/2)))*(a^2*(2*a*b^2 + a^2*b + b^3)^(1/2) + b^2*(2*a*b^2 + a^2*b + b^3)^(1/2) + 2*a*b*(2*a*b^2 + a^2*
b + b^3)^(1/2)))/(2*a^(1/2))))/(2*a*b^2 + a^2*b + b^3)^(1/2)

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sympy [A]  time = 1.54, size = 243, normalized size = 6.23 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{b} & \text {for}\: a = 0 \\- \frac {x \sinh ^{2}{\relax (x )}}{- 2 b \sinh ^{2}{\relax (x )} + 2 b \cosh ^{2}{\relax (x )}} + \frac {x \cosh ^{2}{\relax (x )}}{- 2 b \sinh ^{2}{\relax (x )} + 2 b \cosh ^{2}{\relax (x )}} - \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{- 2 b \sinh ^{2}{\relax (x )} + 2 b \cosh ^{2}{\relax (x )}} & \text {for}\: a = - b \\\frac {x - \frac {\sinh {\relax (x )}}{\cosh {\relax (x )}}}{a} & \text {for}\: b = 0 \\\frac {2 i \sqrt {b} x \sqrt {\frac {1}{a}}}{2 i a \sqrt {b} \sqrt {\frac {1}{a}} + 2 i b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} + \frac {\log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} \sinh {\relax (x )} + \cosh {\relax (x )} \right )}}{2 i a \sqrt {b} \sqrt {\frac {1}{a}} + 2 i b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} - \frac {\log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} \sinh {\relax (x )} + \cosh {\relax (x )} \right )}}{2 i a \sqrt {b} \sqrt {\frac {1}{a}} + 2 i b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (x/b, Eq(a, 0)), (-x*sinh(x)**2/(-2*b*sinh(x)**2 + 2*b*cosh(x)**2) + x
*cosh(x)**2/(-2*b*sinh(x)**2 + 2*b*cosh(x)**2) - sinh(x)*cosh(x)/(-2*b*sinh(x)**2 + 2*b*cosh(x)**2), Eq(a, -b)
), ((x - sinh(x)/cosh(x))/a, Eq(b, 0)), (2*I*sqrt(b)*x*sqrt(1/a)/(2*I*a*sqrt(b)*sqrt(1/a) + 2*I*b**(3/2)*sqrt(
1/a)) + log(-I*sqrt(b)*sqrt(1/a)*sinh(x) + cosh(x))/(2*I*a*sqrt(b)*sqrt(1/a) + 2*I*b**(3/2)*sqrt(1/a)) - log(I
*sqrt(b)*sqrt(1/a)*sinh(x) + cosh(x))/(2*I*a*sqrt(b)*sqrt(1/a) + 2*I*b**(3/2)*sqrt(1/a)), True))

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