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

Optimal. Leaf size=32 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]

[Out]

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

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Rubi [A]  time = 0.0365203, antiderivative size = 32, normalized size of antiderivative = 1., 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 = {3190, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3190

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

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

Mathematica [A]  time = 0.0099342, size = 32, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{d}\arctan \left ({b\sinh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.57284, size = 1207, normalized size = 37.72 \begin{align*} \left [-\frac{\sqrt{-a b} \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 + 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 ) - 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}{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 \, a b d}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, a}\right ) + \sqrt{a b} \arctan \left (\frac{{\left (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 - b\right )} \cosh \left (d x + c\right ) +{\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt{a b}}{2 \, a b}\right )}{a b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.01131, size = 128, normalized size = 4. \begin{align*} \begin{cases} \frac{\tilde{\infty } x \cosh{\left (c \right )}}{\sinh ^{2}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{\sinh{\left (c + d x \right )}}{a d} & \text{for}\: b = 0 \\- \frac{1}{b d \sinh{\left (c + d x \right )}} & \text{for}\: a = 0 \\\frac{x \cosh{\left (c \right )}}{a + b \sinh ^{2}{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sinh{\left (c + d x \right )} \right )}}{2 \sqrt{a} b d \sqrt{\frac{1}{b}}} + \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sinh{\left (c + d x \right )} \right )}}{2 \sqrt{a} b d \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError