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

Optimal. Leaf size=77 \[ \frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} d}-\frac {(a-b) \sinh (c+d x)}{2 a b d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 77, 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, 385, 205} \[ \frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} d}-\frac {(a-b) \sinh (c+d x)}{2 a b d \left (a+b \sinh ^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.35, size = 75, normalized size = 0.97 \[ \frac {\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}-\frac {(a-b) \sinh (c+d x)}{2 a b \left (a+b \sinh ^2(c+d x)\right )}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 1615, normalized size = 20.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(a^2*b - a*b^2)*cosh(d*x + c)^3 + 12*(a^2*b - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + 4*(a^2*b - a*b^2
)*sinh(d*x + c)^3 + ((a*b + b^2)*cosh(d*x + c)^4 + 4*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b + b^2)*s
inh(d*x + c)^4 + 2*(2*a^2 + a*b - b^2)*cosh(d*x + c)^2 + 2*(3*(a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 + a*b - b^2)
*sinh(d*x + c)^2 + a*b + b^2 + 4*((a*b + b^2)*cosh(d*x + c)^3 + (2*a^2 + a*b - b^2)*cosh(d*x + c))*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)*co
sh(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)) - 4*(a^2*b - a*b^2)*cosh(d*x + c) - 4*(a^2*b -
a*b^2 - 3*(a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^2*b^3*d*cosh(d*x + c)^4 + 4*a^2*b^3*d*cosh(d*x +
c)*sinh(d*x + c)^3 + a^2*b^3*d*sinh(d*x + c)^4 + a^2*b^3*d + 2*(2*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + 2*(3*
a^2*b^3*d*cosh(d*x + c)^2 + (2*a^3*b^2 - a^2*b^3)*d)*sinh(d*x + c)^2 + 4*(a^2*b^3*d*cosh(d*x + c)^3 + (2*a^3*b
^2 - a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(a^2*b - a*b^2)*cosh(d*x + c)^3 + 6*(a^2*b - a*b^2)*cos
h(d*x + c)*sinh(d*x + c)^2 + 2*(a^2*b - a*b^2)*sinh(d*x + c)^3 - ((a*b + b^2)*cosh(d*x + c)^4 + 4*(a*b + b^2)*
cosh(d*x + c)*sinh(d*x + c)^3 + (a*b + b^2)*sinh(d*x + c)^4 + 2*(2*a^2 + a*b - b^2)*cosh(d*x + c)^2 + 2*(3*(a*
b + b^2)*cosh(d*x + c)^2 + 2*a^2 + a*b - b^2)*sinh(d*x + c)^2 + a*b + b^2 + 4*((a*b + b^2)*cosh(d*x + c)^3 + (
2*a^2 + a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*x + c)
)/a) - ((a*b + b^2)*cosh(d*x + c)^4 + 4*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b + b^2)*sinh(d*x + c)^
4 + 2*(2*a^2 + a*b - b^2)*cosh(d*x + c)^2 + 2*(3*(a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 + a*b - b^2)*sinh(d*x + c
)^2 + a*b + b^2 + 4*((a*b + b^2)*cosh(d*x + c)^3 + (2*a^2 + a*b - b^2)*cosh(d*x + c))*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*(a^2*b - a*b^2)*cosh(d*x + c) - 2*(a^
2*b - a*b^2 - 3*(a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^2*b^3*d*cosh(d*x + c)^4 + 4*a^2*b^3*d*cosh(
d*x + c)*sinh(d*x + c)^3 + a^2*b^3*d*sinh(d*x + c)^4 + a^2*b^3*d + 2*(2*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 +
2*(3*a^2*b^3*d*cosh(d*x + c)^2 + (2*a^3*b^2 - a^2*b^3)*d)*sinh(d*x + c)^2 + 4*(a^2*b^3*d*cosh(d*x + c)^3 + (2
*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c))*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)^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]=[-27,-68]War
ning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[-70,50]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]=[-63,-1]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,-7]Undef/Unsigned Inf
encountered in limitEvaluation time: 1.8Limit: Max order reached or unable to make series expansion Error: Ba
d Argument Value

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maple [B]  time = 0.12, size = 808, normalized size = 10.49 \[ \frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right )}-\frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) a}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right )}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \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 )}{2 d b \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 )}{2 d b \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 )}{2 d b \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 )}{2 d b \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 )}{2 d a \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}{2 d a \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 )}{2 d a \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}{2 d a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/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/(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/(tan
h(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)-1/2/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))+1/2/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*(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/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/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*t
anh(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)*arctan
h(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 \[ -\frac {{\left (a e^{\left (3 \, c\right )} - b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (a e^{c} - b e^{c}\right )} e^{\left (d x\right )}}{a b^{2} d e^{\left (4 \, d x + 4 \, c\right )} + a b^{2} d + 2 \, {\left (2 \, a^{2} b d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \frac {1}{8} \, \int \frac {8 \, {\left ({\left (a e^{\left (3 \, c\right )} + b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a e^{c} + b e^{c}\right )} e^{\left (d x\right )}\right )}}{a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + a b^{2} + 2 \, {\left (2 \, a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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