3.330 \(\int \frac {\cosh ^5(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=104 \[ -\frac {\left (3 a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac {(a-b)^2 \sinh (c+d x)}{2 a b^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x)}{b^2 d} \]

[Out]

-1/2*(3*a^2-2*a*b-b^2)*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/a^(3/2)/b^(5/2)/d+sinh(d*x+c)/b^2/d+1/2*(a-b)^2*sin
h(d*x+c)/a/b^2/d/(a+b*sinh(d*x+c)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3190, 390, 385, 205} \[ -\frac {\left (3 a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac {(a-b)^2 \sinh (c+d x)}{2 a b^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x)}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

-((3*a^2 - 2*a*b - b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d) + Sinh[c + d*x]/(b^2*d)
+ ((a - b)^2*Sinh[c + d*x])/(2*a*b^2*d*(a + b*Sinh[c + d*x]^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]

Rule 385

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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 ^5(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b^2}-\frac {a^2-b^2+2 (a-b) b x^2}{b^2 \left (a+b x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{b^2 d}-\frac {\operatorname {Subst}\left (\int \frac {a^2-b^2+2 (a-b) b x^2}{\left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{b^2 d}\\ &=\frac {\sinh (c+d x)}{b^2 d}+\frac {(a-b)^2 \sinh (c+d x)}{2 a b^2 d \left (a+b \sinh ^2(c+d x)\right )}-\frac {((a-b) (3 a+b)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{2 a b^2 d}\\ &=-\frac {(a-b) (3 a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac {\sinh (c+d x)}{b^2 d}+\frac {(a-b)^2 \sinh (c+d x)}{2 a b^2 d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.30, size = 106, normalized size = 1.02 \[ \frac {-\frac {\left (-3 a^2+2 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )}{a^{3/2}}+\frac {2 \sqrt {b} (a-b)^2 \sinh (c+d x)}{a (2 a+b \cosh (2 (c+d x))-b)}+2 \sqrt {b} \sinh (c+d x)}{2 b^{5/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

(-(((-3*a^2 + 2*a*b + b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/a^(3/2)) + 2*Sqrt[b]*Sinh[c + d*x] + (2*(a
- b)^2*Sqrt[b]*Sinh[c + d*x])/(a*(2*a - b + b*Cosh[2*(c + d*x)])))/(2*b^(5/2)*d)

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fricas [B]  time = 0.75, size = 2739, normalized size = 26.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(2*a^2*b^2*cosh(d*x + c)^6 + 12*a^2*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + 2*a^2*b^2*sinh(d*x + c)^6 + 2*(6*
a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 + 2*(15*a^2*b^2*cosh(d*x + c)^2 + 6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*
sinh(d*x + c)^4 - 2*a^2*b^2 + 8*(5*a^2*b^2*cosh(d*x + c)^3 + (6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*si
nh(d*x + c)^3 - 2*(6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 2*(15*a^2*b^2*cosh(d*x + c)^4 - 6*a^3*b +
7*a^2*b^2 - 2*a*b^3 + 6*(6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^2*b - 2*a*b^2
- b^3)*cosh(d*x + c)^5 + 5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^4 + (3*a^2*b - 2*a*b^2 - b^3
)*sinh(d*x + c)^5 + 2*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c)^3 + 2*(6*a^3 - 7*a^2*b + b^3 + 5*(3*a^2*b - 2*a*b^
2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^3 + 3*(6*a^3 - 7*a^2*
b + b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c) + (5*(3*a^2*b - 2*a*b^2 - b^
3)*cosh(d*x + c)^4 + 3*a^2*b - 2*a*b^2 - b^3 + 6*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*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))*sin
h(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^2*b^2*cosh(d*x + c)^5 + 2*(6*a^3*b - 7*a^2*b^2
+ 2*a*b^3)*cosh(d*x + c)^3 - (6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/(a^2*b^4*d*cosh(d*x
+ c)^5 + 5*a^2*b^4*d*cosh(d*x + c)*sinh(d*x + c)^4 + a^2*b^4*d*sinh(d*x + c)^5 + a^2*b^4*d*cosh(d*x + c) + 2*
(2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^3 + 2*(5*a^2*b^4*d*cosh(d*x + c)^2 + (2*a^3*b^3 - a^2*b^4)*d)*sinh(d*x +
c)^3 + 2*(5*a^2*b^4*d*cosh(d*x + c)^3 + 3*(2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*a^2*b^4
*d*cosh(d*x + c)^4 + a^2*b^4*d + 6*(2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^2)*sinh(d*x + c)), 1/2*(a^2*b^2*cosh(
d*x + c)^6 + 6*a^2*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*b^2*sinh(d*x + c)^6 + (6*a^3*b - 7*a^2*b^2 + 2*a*b^
3)*cosh(d*x + c)^4 + (15*a^2*b^2*cosh(d*x + c)^2 + 6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*sinh(d*x + c)^4 - a^2*b^2 +
4*(5*a^2*b^2*cosh(d*x + c)^3 + (6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (6*a^3*b - 7*a
^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + (15*a^2*b^2*cosh(d*x + c)^4 - 6*a^3*b + 7*a^2*b^2 - 2*a*b^3 + 6*(6*a^3*b -
7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(3*a^2
*b - 2*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^4 + (3*a^2*b - 2*a*b^2 - b^3)*sinh(d*x + c)^5 + 2*(6*a^3 - 7*a
^2*b + b^3)*cosh(d*x + c)^3 + 2*(6*a^3 - 7*a^2*b + b^3 + 5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + 2*(5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^3 + 3*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c))*sinh(d*x +
c)^2 + (3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c) + (5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^4 + 3*a^2*b - 2*a
*b^2 - b^3 + 6*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*
x + c) + sinh(d*x + c))/a) - ((3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x
+ c)*sinh(d*x + c)^4 + (3*a^2*b - 2*a*b^2 - b^3)*sinh(d*x + c)^5 + 2*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c)^3
+ 2*(6*a^3 - 7*a^2*b + b^3 + 5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(3*a^2*b - 2*
a*b^2 - b^3)*cosh(d*x + c)^3 + 3*(6*a^3 - 7*a^2*b + b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (3*a^2*b - 2*a*b^2 -
b^3)*cosh(d*x + c) + (5*(3*a^2*b - 2*a*b^2 - b^3)*cosh(d*x + c)^4 + 3*a^2*b - 2*a*b^2 - b^3 + 6*(6*a^3 - 7*a^
2*b + b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*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^2*b^2*cosh(d*x + c)^5 + 2*(6*a^3*b - 7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (6*a^3*b - 7*a
^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/(a^2*b^4*d*cosh(d*x + c)^5 + 5*a^2*b^4*d*cosh(d*x + c)*sinh(d*
x + c)^4 + a^2*b^4*d*sinh(d*x + c)^5 + a^2*b^4*d*cosh(d*x + c) + 2*(2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^3 + 2
*(5*a^2*b^4*d*cosh(d*x + c)^2 + (2*a^3*b^3 - a^2*b^4)*d)*sinh(d*x + c)^3 + 2*(5*a^2*b^4*d*cosh(d*x + c)^3 + 3*
(2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*a^2*b^4*d*cosh(d*x + c)^4 + a^2*b^4*d + 6*(2*a^3*b
^3 - a^2*b^4)*d*cosh(d*x + c)^2)*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)^5/(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,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]=[6,-20]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]=[66,-29]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]=[-21,2]Warning, need to cho
ose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a
,b]=[15,2]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.T
he choice was done assuming [a,b]=[-92,94]Warning, need to choose a branch for the root of a polynomial with p
arameters. 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. This might be wrong.The choice was done assuming [a,b]=[-90,-5]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]=[-94,-77]Warning, need to choose a branch for the root of a polynomial with parameters. Th
is might be wrong.The choice was done assuming [a,b]=[36,-73]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]=[91,55]Warning, need to c
hoose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming
[a,b]=[17,-27]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wro
ng.The choice was done assuming [a,b]=[24,-71]Warning, need to choose a branch for the root of a polynomial wi
th parameters. This might be wrong.The choice was done assuming [a,b]=[-39,-6]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]=[61,-49]
Undef/Unsigned Inf encountered in limitEvaluation time: 2.51Limit: Max order reached or unable to make series
expansion Error: Bad Argument Value

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maple [B]  time = 0.12, size = 1403, normalized size = 13.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x)

[Out]

-1/d/b^2/(tanh(1/2*d*x+1/2*c)-1)-1/d/b^2/(tanh(1/2*d*x+1/2*c)+1)-1/d/b^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d
*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*a*tanh(1/2*d*x+1/2*c)^3+2/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d
*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)^3-1/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1
/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)^3+1/d/b^2/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x
+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*a*tanh(1/2*d*x+1/2*c)-2/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1
/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)+1/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^
2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)+3/2/d/b^2*a^2/(-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))-3/2/d/b^2*a/((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))-5/2/d/b*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))+3/2
/d/b^2*a^2/(-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))+3/2/d/b^2*a/((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))-5/2/d/b*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/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))+1/2/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))-1/d/b/((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))+1/2/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))+1/2/d/a/((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))+1/2/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))*b
-1/2/d/a/((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))+1/2/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))*b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a b e^{\left (6 \, d x + 6 \, c\right )} - a b + {\left (6 \, a^{2} e^{\left (4 \, c\right )} - 7 \, a b e^{\left (4 \, c\right )} + 2 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - {\left (6 \, a^{2} e^{\left (2 \, c\right )} - 7 \, a b e^{\left (2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{2 \, {\left (a b^{3} d e^{\left (5 \, d x + 5 \, c\right )} + a b^{3} d e^{\left (d x + c\right )} + 2 \, {\left (2 \, a^{2} b^{2} d e^{\left (3 \, c\right )} - a b^{3} d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )}\right )}} - \frac {1}{32} \, \int \frac {32 \, {\left ({\left (3 \, a^{2} e^{\left (3 \, c\right )} - 2 \, a b e^{\left (3 \, c\right )} - b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (3 \, a^{2} e^{c} - 2 \, a b e^{c} - b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + a b^{3} + 2 \, {\left (2 \, a^{2} b^{2} e^{\left (2 \, c\right )} - a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/2*(a*b*e^(6*d*x + 6*c) - a*b + (6*a^2*e^(4*c) - 7*a*b*e^(4*c) + 2*b^2*e^(4*c))*e^(4*d*x) - (6*a^2*e^(2*c) -
7*a*b*e^(2*c) + 2*b^2*e^(2*c))*e^(2*d*x))/(a*b^3*d*e^(5*d*x + 5*c) + a*b^3*d*e^(d*x + c) + 2*(2*a^2*b^2*d*e^(3
*c) - a*b^3*d*e^(3*c))*e^(3*d*x)) - 1/32*integrate(32*((3*a^2*e^(3*c) - 2*a*b*e^(3*c) - b^2*e^(3*c))*e^(3*d*x)
+ (3*a^2*e^c - 2*a*b*e^c - b^2*e^c)*e^(d*x))/(a*b^3*e^(4*d*x + 4*c) + a*b^3 + 2*(2*a^2*b^2*e^(2*c) - a*b^3*e^
(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^5}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2)^2,x)

[Out]

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2)^2, x)

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

[Out]

Timed out

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