3.4.34 \(\int \frac {\cosh (c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [334]

Optimal. Leaf size=66 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

1/2*sinh(d*x+c)/a/d/(a+b*sinh(d*x+c)^2)+1/2*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/a^(3/2)/d/b^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3269, 205, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Sinh[c + d*x]/(2*a*d*(a + b*Sinh[c + d*x]^2))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{2 a d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 64, normalized size = 0.97 \begin {gather*} \frac {\frac {\text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sinh[c + d*x]/(2*a*(a + b*Sinh[c + d*x]^2)))/d

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Maple [A]
time = 0.55, size = 55, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\frac {\sinh \left (d x +c \right )}{2 a \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {\arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{d}\)	\(55\)
default	\(\frac {\frac {\sinh \left (d x +c \right )}{2 a \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {\arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{d}\)	\(55\)
risch	\(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d a}\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/2*sinh(d*x+c)/a/(a+b*sinh(d*x+c)^2)+1/2/a/(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)^2,x, algorithm="maxima")

[Out]

(e^(3*d*x + 3*c) - e^(d*x + c))/(a*b*d*e^(4*d*x + 4*c) + a*b*d + 2*(2*a^2*d*e^(2*c) - a*b*d*e^(2*c))*e^(2*d*x)
) + 1/2*integrate(2*(e^(3*d*x + 3*c) + e^(d*x + c))/(a*b*e^(4*d*x + 4*c) + a*b + 2*(2*a^2*e^(2*c) - a*b*e^(2*c
))*e^(2*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (54) = 108\).
time = 0.44, size = 1320, normalized size = 20.00 \begin {gather*} \left [\frac {4 \, a b \cosh \left (d x + c\right )^{3} + 12 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 4 \, a b \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) - {\left (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 )} \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 ) + 4 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{4 \, {\left (a^{2} b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a^{2} b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} b^{2} d \sinh \left (d x + c\right )^{4} + a^{2} b^{2} d + 2 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} b^{2} d \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{2} b^{2} d \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}, \frac {2 \, a b \cosh \left (d x + c\right )^{3} + 6 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, a b \sinh \left (d x + c\right )^{3} - 2 \, a b \cosh \left (d x + c\right ) + {\left (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 )} \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 ) + {\left (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 )} \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 ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a^{2} b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} b^{2} d \sinh \left (d x + c\right )^{4} + a^{2} b^{2} d + 2 \, {\left (2 \, a^{3} b - a^{2} b^{2}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} b^{2} d \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{2} b^{2} d \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{3} b - a^{2} b^{2}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}\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)^2,x, algorithm="fricas")

[Out]

[1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b*sinh(d*x + c)^3 - 4*a*b*cosh(d*x +
c) - (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)*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)*co
sh(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)) + 4*(3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d
*x + c))/(a^2*b^2*d*cosh(d*x + c)^4 + 4*a^2*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*b^2*d*sinh(d*x + c)^4 +
a^2*b^2*d + 2*(2*a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*a^2*b^2*d*cosh(d*x + c)^2 + (2*a^3*b - a^2*b^2)*d)*
sinh(d*x + c)^2 + 4*(a^2*b^2*d*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*a
*b*cosh(d*x + c)^3 + 6*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*b*sinh(d*x + c)^3 - 2*a*b*cosh(d*x + c) + (b*co
sh(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*c
osh(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)
*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) + (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*si
nh(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)*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)) + 2*(3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c))/(a^2*b^2*d*cosh(d*
x + c)^4 + 4*a^2*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*b^2*d*sinh(d*x + c)^4 + a^2*b^2*d + 2*(2*a^3*b - a^
2*b^2)*d*cosh(d*x + c)^2 + 2*(3*a^2*b^2*d*cosh(d*x + c)^2 + (2*a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 + 4*(a^2*b^
2*d*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (54) = 108\).
time = 6.52, size = 377, normalized size = 5.71 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x \cosh {\left (c \right )}}{\sinh ^{4}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {1}{3 b^{2} d \sinh ^{3}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {x \cosh {\left (c \right )}}{\left (a + b \sinh ^{2}{\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\\frac {a \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} - \frac {a \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} + \frac {2 b \sqrt {- \frac {a}{b}} \sinh {\left (c + d x \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} + \frac {b \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} - \frac {b \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} & \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)**2,x)

[Out]

Piecewise((zoo*x*cosh(c)/sinh(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (-1/(3*b**2*d*sinh(c + d*x)**3), Eq(a, 0
)), (x*cosh(c)/(a + b*sinh(c)**2)**2, Eq(d, 0)), (sinh(c + d*x)/(a**2*d), Eq(b, 0)), (a*log(-sqrt(-a/b) + sinh
(c + d*x))/(4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2) - a*log(sqrt(-a/b) + sinh(c + d*x)
)/(4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2) + 2*b*sqrt(-a/b)*sinh(c + d*x)/(4*a**2*b*d*
sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2) + b*log(-sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**2/(4*
a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2) - b*log(sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x
)**2/(4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2), 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)^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 = 0.91, size = 54, normalized size = 0.82 \begin {gather*} \frac {\mathrm {sinh}\left (c+d\,x\right )}{2\,a\,\left (b\,d\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\,d\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {sinh}\left (c+d\,x\right )}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}\,d} \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)^2,x)

[Out]

sinh(c + d*x)/(2*a*(a*d + b*d*sinh(c + d*x)^2)) + atan((b^(1/2)*sinh(c + d*x))/a^(1/2))/(2*a^(3/2)*b^(1/2)*d)

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