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

Optimal. Leaf size=40 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {b} d \sqrt {a-b}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3186, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {b} d \sqrt {a-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [C]  time = 0.13, size = 91, normalized size = 2.28 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{\sqrt {b} d \sqrt {a-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.72, size = 502, normalized size = 12.55 \[ \left [-\frac {\sqrt {-a b + b^{2}} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right )\right )} \sqrt {-a b + b^{2}} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right )}{2 \, {\left (a b - b^{2}\right )} d}, \frac {\sqrt {a b - b^{2}} \arctan \left (-\frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a b - b^{2}}}\right ) - \sqrt {a b - b^{2}} \arctan \left (-\frac {\sqrt {a b - b^{2}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, {\left (a - b\right )}}\right )}{{\left (a b - b^{2}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[31,78]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[85,31]Warning, need to choose a branch for the root of a p
olynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[46,18]Warning, need to choo
se a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,
b]=[-27,57]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [a,b]=[-18,-81]Warning, need to choose a branch for the root of a polynomial with
parameters. This might be wrong.The choice was done assuming [a,b]=[-10,75]Undef/Unsigned Inf encountered in
limitEvaluation time: 0.8Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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maple [A]  time = 0.04, size = 51, normalized size = 1.28 \[ \frac {\arctan \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{d \sqrt {a b -b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.99, size = 116, normalized size = 2.90 \[ \frac {\ln \left (-\frac {4\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^2\,\left (a-b\right )}-\frac {8\,a\,{\mathrm {e}}^{c+d\,x}}{{\left (-b\right )}^{5/2}\,\sqrt {a-b}}\right )-\ln \left (\frac {8\,a\,{\mathrm {e}}^{c+d\,x}}{{\left (-b\right )}^{5/2}\,\sqrt {a-b}}-\frac {4\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^2\,\left (a-b\right )}\right )}{2\,\sqrt {-b}\,d\,\sqrt {a-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(- (4*(a + a*exp(2*c + 2*d*x)))/(b^2*(a - b)) - (8*a*exp(c + d*x))/((-b)^(5/2)*(a - b)^(1/2))) - log((8*a*
exp(c + d*x))/((-b)^(5/2)*(a - b)^(1/2)) - (4*(a + a*exp(2*c + 2*d*x)))/(b^2*(a - b))))/(2*(-b)^(1/2)*d*(a - b
)^(1/2))

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

[Out]

Timed out

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