3.1.33 \(\int \frac {\sinh ^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [33]

Optimal. Leaf size=50 \[ \frac {x}{b}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a-b} b d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3250, 3260, 214} \begin {gather*} \frac {x}{b}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{b d \sqrt {a-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 214

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

Rule 3250

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[B*(x
/b), x] + Dist[(A*b - a*B)/b, Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]

Rule 3260

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 50, normalized size = 1.00 \begin {gather*} \frac {c+d x-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a-b}}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(42)=84\).
time = 1.01, size = 216, normalized size = 4.32

method	result	size
risch	\(\frac {x}{b}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 \left (a -b \right ) d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 \left (a -b \right ) d b}\)	\(120\)
derivativedivides	\(\frac {\frac {2 a^{2} \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\)	\(216\)
default	\(\frac {\frac {2 a^{2} \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b}}{d}\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(a+b*sinh(d*x+c)^2),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(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (42) = 84\).
time = 0.44, size = 464, normalized size = 9.28 \begin {gather*} \left [\frac {2 \, d x + \sqrt {\frac {a}{a - b}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left ({\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} - 3 \, a b + b^{2}\right )} \sqrt {\frac {a}{a - b}}}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right )}{2 \, b d}, \frac {d x - \sqrt {-\frac {a}{a - b}} \arctan \left (\frac {{\left (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 \, a - b\right )} \sqrt {-\frac {a}{a - b}}}{2 \, a}\right )}{b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*d*x + sqrt(a/(a - b))*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x +
c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a
*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*((a*b - b^2)*cosh(d*x + c)^
2 + 2*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b - b^2)*sinh(d*x + c)^2 + 2*a^2 - 3*a*b + b^2)*sqrt(a/(a -
b)))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2
+ 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*
x + c) + b)))/(b*d), (d*x - sqrt(-a/(a - b))*arctan(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 - b)*sqrt(-a/(a - b))/a))/(b*d)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.27, size = 64, normalized size = 1.28 \begin {gather*} -\frac {\frac {a \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b} - \frac {d x + c}{b}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.22, size = 473, normalized size = 9.46 \begin {gather*} \frac {x}{b}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\left (b^5\,\sqrt {b^3\,d^2-a\,b^2\,d^2}-a\,b^4\,\sqrt {b^3\,d^2-a\,b^2\,d^2}\right )\,\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {2\,\left (8\,a^2-8\,a\,b+b^2\right )\,\left (8\,a^{5/2}\,\sqrt {b^3\,d^2-a\,b^2\,d^2}-8\,a^{3/2}\,b\,\sqrt {b^3\,d^2-a\,b^2\,d^2}+\sqrt {a}\,b^2\,\sqrt {b^3\,d^2-a\,b^2\,d^2}\right )}{b^8\,d\,{\left (a-b\right )}^2\,\sqrt {b^3\,d^2-a\,b^2\,d^2}}+\frac {4\,\sqrt {a}\,\left (4\,a-2\,b\right )\,\left (8\,d\,a^3\,b-12\,d\,a^2\,b^2+4\,d\,a\,b^3\right )}{b^7\,\left (a-b\right )\,\sqrt {b^3\,d^2-a\,b^2\,d^2}\,\sqrt {-b^2\,d^2\,\left (a-b\right )}}\right )+\frac {2\,\left (2\,a^{3/2}\,b\,\sqrt {b^3\,d^2-a\,b^2\,d^2}-\sqrt {a}\,b^2\,\sqrt {b^3\,d^2-a\,b^2\,d^2}\right )\,\left (8\,a^2-8\,a\,b+b^2\right )}{b^8\,d\,{\left (a-b\right )}^2\,\sqrt {b^3\,d^2-a\,b^2\,d^2}}+\frac {4\,\sqrt {a}\,\left (2\,a^2\,b^2\,d-2\,a\,b^3\,d\right )\,\left (4\,a-2\,b\right )}{b^7\,\left (a-b\right )\,\sqrt {b^3\,d^2-a\,b^2\,d^2}\,\sqrt {-b^2\,d^2\,\left (a-b\right )}}\right )}{4\,a}\right )}{\sqrt {b^3\,d^2-a\,b^2\,d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b - (a^(1/2)*atan(((b^5*(b^3*d^2 - a*b^2*d^2)^(1/2) - a*b^4*(b^3*d^2 - a*b^2*d^2)^(1/2))*(exp(2*c)*exp(2*d*x
)*((2*(8*a^2 - 8*a*b + b^2)*(8*a^(5/2)*(b^3*d^2 - a*b^2*d^2)^(1/2) - 8*a^(3/2)*b*(b^3*d^2 - a*b^2*d^2)^(1/2) +
a^(1/2)*b^2*(b^3*d^2 - a*b^2*d^2)^(1/2)))/(b^8*d*(a - b)^2*(b^3*d^2 - a*b^2*d^2)^(1/2)) + (4*a^(1/2)*(4*a - 2
*b)*(4*a*b^3*d - 12*a^2*b^2*d + 8*a^3*b*d))/(b^7*(a - b)*(b^3*d^2 - a*b^2*d^2)^(1/2)*(-b^2*d^2*(a - b))^(1/2))
) + (2*(2*a^(3/2)*b*(b^3*d^2 - a*b^2*d^2)^(1/2) - a^(1/2)*b^2*(b^3*d^2 - a*b^2*d^2)^(1/2))*(8*a^2 - 8*a*b + b^
2))/(b^8*d*(a - b)^2*(b^3*d^2 - a*b^2*d^2)^(1/2)) + (4*a^(1/2)*(2*a^2*b^2*d - 2*a*b^3*d)*(4*a - 2*b))/(b^7*(a
- b)*(b^3*d^2 - a*b^2*d^2)^(1/2)*(-b^2*d^2*(a - b))^(1/2))))/(4*a)))/(b^3*d^2 - a*b^2*d^2)^(1/2)

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