3.4.41 \(\int \frac {\cosh ^4(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [341]

Optimal. Leaf size=114 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {a-b} d}+\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

3/8*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/a^(5/2)/d/(a-b)^(1/2)+1/4*tanh(d*x+c)/a/d/(a-(a-b)*tanh(d*x+c)^2)
^2+3/8*tanh(d*x+c)/a^2/d/(a-(a-b)*tanh(d*x+c)^2)

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Rubi [A]
time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3270, 205, 214} \begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d \sqrt {a-b}}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[a - b]*d) + Tanh[c + d*x]/(4*a*d*(a - (a - b)
*Tanh[c + d*x]^2)^2) + (3*Tanh[c + d*x])/(8*a^2*d*(a - (a - b)*Tanh[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 214

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

Rule 3270

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 d}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {a-b} d}+\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 102, normalized size = 0.89 \begin {gather*} \frac {\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a-b}}+\frac {\sqrt {a} (8 a-3 b+(2 a+3 b) \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(2 a-b+b \cosh (2 (c+d x)))^2}}{8 a^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a - b] + (Sqrt[a]*(8*a - 3*b + (2*a + 3*b)*Cosh[2*(c +
d*x)])*Sinh[2*(c + d*x)])/(2*a - b + b*Cosh[2*(c + d*x)])^2)/(8*a^(5/2)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(100)=200\).
time = 1.53, size = 300, normalized size = 2.63

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {3 \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a}}{d}\)	\(300\)
default	\(\frac {-\frac {2 \left (-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {3 \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a}}{d}\)	\(300\)
risch	\(-\frac {8 a^{2} b \,{\mathrm e}^{6 d x +6 c}-3 b^{3} {\mathrm e}^{6 d x +6 c}+16 a^{3} {\mathrm e}^{4 d x +4 c}+8 a^{2} b \,{\mathrm e}^{4 d x +4 c}-18 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 b^{3} {\mathrm e}^{4 d x +4 c}+8 a^{2} b \,{\mathrm e}^{2 d x +2 c}+16 a \,b^{2} {\mathrm e}^{2 d x +2 c}-9 b^{3} {\mathrm e}^{2 d x +2 c}+2 a \,b^{2}+3 b^{3}}{4 b^{2} a^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{16 \sqrt {a^{2}-a b}\, d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{16 \sqrt {a^{2}-a b}\, d \,a^{2}}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2115 vs. \(2 (102) = 204\).
time = 0.43, size = 4486, normalized size = 39.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^6 + 24*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 +
3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^6 + 8*a^3
*b^2 + 4*a^2*b^3 - 12*a*b^4 + 4*(16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^4 + 4*(16
*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4 + 15*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 16*(5*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^3 + (16*a^5 - 8*a^4*b
- 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4*b + 8*a^3*b^2 - 25*a^2*b^3 + 9
*a*b^4)*cosh(d*x + c)^2 + 4*(8*a^4*b + 8*a^3*b^2 - 25*a^2*b^3 + 9*a*b^4 + 15*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3
+ 3*a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 - 3*(b^4*cosh(d*x + c)^8 + 8*b^4*cosh(d*x + c)*sinh(d*x + c)^7 + b^4*sinh(d*x + c)^8 + 4*(2*a*b^3 - b
^4)*cosh(d*x + c)^6 + 4*(7*b^4*cosh(d*x + c)^2 + 2*a*b^3 - b^4)*sinh(d*x + c)^6 + 8*(7*b^4*cosh(d*x + c)^3 + 3
*(2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*(35*b^4*
cosh(d*x + c)^4 + 8*a^2*b^2 - 8*a*b^3 + 3*b^4 + 30*(2*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + b^4 + 8*
(7*b^4*cosh(d*x + c)^5 + 10*(2*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c))*sin
h(d*x + c)^3 + 4*(2*a*b^3 - b^4)*cosh(d*x + c)^2 + 4*(7*b^4*cosh(d*x + c)^6 + 15*(2*a*b^3 - b^4)*cosh(d*x + c)
^4 + 2*a*b^3 - b^4 + 3*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(b^4*cosh(d*x + c)^7
+ 3*(2*a*b^3 - b^4)*cosh(d*x + c)^5 + (8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (2*a*b^3 - b^4)*cosh(d*
x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*si
nh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*
a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*(b*cosh(d*x + c)^2
+ 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4*b*co
sh(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)) + 8*(3*(8*a^4*b - 8*
a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^5 + 2*(16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4)*cosh
(d*x + c)^3 + (8*a^4*b + 8*a^3*b^2 - 25*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b^4 - a^3*b^5)*
d*cosh(d*x + c)^8 + 8*(a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b^4 - a^3*b^5)*d*sinh(d*x + c
)^8 + 4*(2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (2*
a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^6*b^2 - 16*a^5*b^3 + 11*a^4*b^4 - 3*a^3*b^5)*d*cosh
(d*x + c)^4 + 8*(7*(a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + 3*(2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)
)*sinh(d*x + c)^5 + 2*(35*(a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 30*(2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d*cosh(
d*x + c)^2 + (8*a^6*b^2 - 16*a^5*b^3 + 11*a^4*b^4 - 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(2*a^5*b^3 - 3*a^4*b^4 +
a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + 10*(2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5
)*d*cosh(d*x + c)^3 + (8*a^6*b^2 - 16*a^5*b^3 + 11*a^4*b^4 - 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(
7*(a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 15*(2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^6*b^
2 - 16*a^5*b^3 + 11*a^4*b^4 - 3*a^3*b^5)*d*cosh(d*x + c)^2 + (2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d)*sinh(d*x + c
)^2 + (a^4*b^4 - a^3*b^5)*d + 8*((a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^7 + 3*(2*a^5*b^3 - 3*a^4*b^4 + a^3*b^5)*d
*cosh(d*x + c)^5 + (8*a^6*b^2 - 16*a^5*b^3 + 11*a^4*b^4 - 3*a^3*b^5)*d*cosh(d*x + c)^3 + (2*a^5*b^3 - 3*a^4*b^
4 + a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c
)^6 + 12*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(8*a^4*b - 8*a^3*b^2 -
3*a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^6 + 4*a^3*b^2 + 2*a^2*b^3 - 6*a*b^4 + 2*(16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27
*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^4 + 2*(16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4 + 15*(8*a^4*b -
8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*
a*b^4)*cosh(d*x + c)^3 + (16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3
+ 2*(8*a^4*b + 8*a^3*b^2 - 25*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2 + 2*(8*a^4*b + 8*a^3*b^2 - 25*a^2*b^3 + 9*a*
b^4 + 15*(8*a^4*b - 8*a^3*b^2 - 3*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 8*a^4*b - 26*a^3*b^2 + 27*a
^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*(b^4*cosh(d*x + c)^8 + 8*b^4*cosh(d*x + c)*sinh(d*x + c
)^7 + b^4*sinh(d*x + c)^8 + 4*(2*a*b^3 - b^4)*c...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**4/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (102) = 204\).
time = 1.75, size = 244, normalized size = 2.14 \begin {gather*} \frac {\frac {3 \, \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a^{2}} - \frac {2 \, {\left (8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} + 3 \, b^{3}\right )}}{{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2} a^{2} b^{2}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*(3*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a^2) - 2*(8*a^2*b*e^(6*d*x
+ 6*c) - 3*b^3*e^(6*d*x + 6*c) + 16*a^3*e^(4*d*x + 4*c) + 8*a^2*b*e^(4*d*x + 4*c) - 18*a*b^2*e^(4*d*x + 4*c)
+ 9*b^3*e^(4*d*x + 4*c) + 8*a^2*b*e^(2*d*x + 2*c) + 16*a*b^2*e^(2*d*x + 2*c) - 9*b^3*e^(2*d*x + 2*c) + 2*a*b^2
+ 3*b^3)/((b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2*a^2*b^2))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^4/(a + b*sinh(c + d*x)^2)^3,x)

[Out]

int(cosh(c + d*x)^4/(a + b*sinh(c + d*x)^2)^3, x)

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