3.320 \(\int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=52 \[ \frac {\sinh (c+d x)}{b d}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 52, 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 = {3190, 388, 205} \[ \frac {\sinh (c+d x)}{b d}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]

[Out]

-(((a - b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d)) + Sinh[c + d*x]/(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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 50, normalized size = 0.96 \[ \frac {\frac {\sinh (c+d x)}{b}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

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fricas [B]  time = 0.59, size = 659, normalized size = 12.67 \[ \left [\frac {a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + \sqrt {-a b} {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \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 ) - a b}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )\right )}}, \frac {a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} - 2 \, \sqrt {a b} {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\frac {\sqrt {a b} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, a}\right ) - 2 \, \sqrt {a b} {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \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}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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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(cosh(d*x+c)^3/(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]Warning, need to choose a branch f
or the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[4,51]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]=[44,-86]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[34,-93]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]=[80,-1]Undef/Unsigned Inf
encountered in limitEvaluation time: 1.54Limit: Max order reached or unable to make series expansion Error: Ba
d Argument Value

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maple [B]  time = 0.10, size = 732, normalized size = 14.08 \[ \frac {a^{2} \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 )}{d b \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {a \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 )}{d b \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {2 a \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 )}{d \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}+\frac {a^{2} \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 )}{d b \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {a \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 )}{d b \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}-\frac {2 a \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 )}{d \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\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 )}{d \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}+\frac {\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 ) b}{d \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\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 )}{d \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\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 ) b}{d \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}-\frac {1}{d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}}{2 \, b d} - \frac {1}{8} \, \int \frac {16 \, {\left ({\left (a e^{\left (3 \, c\right )} - b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a e^{c} - b e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{2} e^{\left (4 \, d x + 4 \, c\right )} + b^{2} + 2 \, {\left (2 \, a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)/(b*d) - 1/8*integrate(16*((a*e^(3*c) - b*e^(3*c))*e^(3*d*x) + (a*e^c -
b*e^c)*e^(d*x))/(b^2*e^(4*d*x + 4*c) + b^2 + 2*(2*a*b*e^(2*c) - b^2*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 1.14, size = 426, normalized size = 8.19 \[ \frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a\,b^3\,d\,\sqrt {a^2-2\,a\,b+b^2}+2\,a^3\,b\,d\,\sqrt {a^2-2\,a\,b+b^2}-4\,a^2\,b^2\,d\,\sqrt {a^2-2\,a\,b+b^2}\right )}{a^2\,b^7\,d^2\,\left (a-b\right )}-\frac {2\,\left (a^3\,\sqrt {a\,b^3\,d^2}-b^3\,\sqrt {a\,b^3\,d^2}+3\,a\,b^2\,\sqrt {a\,b^3\,d^2}-3\,a^2\,b\,\sqrt {a\,b^3\,d^2}\right )}{a^2\,b^5\,d\,\sqrt {{\left (a-b\right )}^2}\,\sqrt {a\,b^3\,d^2}}\right )\,\sqrt {a\,b^3\,d^2}}{4\,a^2-8\,a\,b+4\,b^2}+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (a^3\,\sqrt {a\,b^3\,d^2}-b^3\,\sqrt {a\,b^3\,d^2}+3\,a\,b^2\,\sqrt {a\,b^3\,d^2}-3\,a^2\,b\,\sqrt {a\,b^3\,d^2}\right )}{a\,b\,d\,\sqrt {{\left (a-b\right )}^2}\,\left (4\,a^2-8\,a\,b+4\,b^2\right )}\right )+2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a-b\right )\,\sqrt {a\,b^3\,d^2}}{2\,a\,b\,d\,\sqrt {{\left (a-b\right )}^2}}\right )\right )\,\sqrt {a^2-2\,a\,b+b^2}}{2\,\sqrt {a\,b^3\,d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^3/(a + b*sinh(c + d*x)^2),x)

[Out]

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

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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(cosh(d*x+c)**3/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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