3.4.22 \(\int \frac {\cosh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [322]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 32, 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 = {3269, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \end {gather*}

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 211

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

Rule 3269

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]

Rubi steps

\begin {align*} \int \frac {\cosh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {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*}

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Mathematica [A]
time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \end {gather*}

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 = 0.53, size = 24, normalized size = 0.75

method	result	size
derivativedivides	\(\frac {\arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{d \sqrt {a b}}\)	\(24\)
default	\(\frac {\arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{d \sqrt {a b}}\)	\(24\)
risch	\(-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (24) = 48\).
time = 0.48, size = 459, normalized size = 14.34 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (29) = 58\).
time = 0.86, size = 107, normalized size = 3.34 \begin {gather*} \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 {x \cosh {\left (c \right )}}{a + b \sinh ^{2}{\left (c \right )}} & \text {for}\: d = 0 \\- \frac {1}{b d \sinh {\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {\sinh {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {\log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{2 b d \sqrt {- \frac {a}{b}}} - \frac {\log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{2 b d \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

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

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

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 0.87, size = 23, normalized size = 0.72 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{\sqrt {a\,b}}\right )}{d\,\sqrt {a\,b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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