3.1.55 \(\int \frac {1}{(a+b \sinh ^2(c+d x))^3} \, dx\) [55]

Optimal. Leaf size=154 \[ \frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a-b)^{5/2} d}-\frac {b \cosh (c+d x) \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {3 (2 a-b) b \cosh (c+d x) \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3263, 3252, 12, 3260, 214} \begin {gather*} -\frac {3 b (2 a-b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a-b)^{5/2}}-\frac {b \sinh (c+d x) \cosh (c+d x)}{4 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^2)^(-3),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 3252

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp
[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Dist[
1/(2*a*(a + b)*(p + 1)), Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*(p + 1) + b*(2*p + 3)) + 2*(A*b
- a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]

Rule 3260

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 3263

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^
(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && N
eQ[a + b, 0] && LtQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.81, size = 132, normalized size = 0.86 \begin {gather*} \frac {\frac {\left (8 a^2-8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{(a-b)^{5/2}}+\frac {\sqrt {a} b \left (-16 a^2+16 a b-3 b^2+3 b (-2 a+b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}}{8 a^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(5/2) + (Sqrt[a]*b*(-16*a^2 +
16*a*b - 3*b^2 + 3*b*(-2*a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)^2*(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. \(417\) vs. \(2(140)=280\).
time = 1.43, size = 418, normalized size = 2.71

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {b \left (8 a -5 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (8 a^{2}-29 a b +12 b^{2}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (8 a^{2}-29 a b +12 b^{2}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {b \left (8 a -5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (8 a^{2}-8 a b +3 b^{2}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(418\)
default	\(\frac {-\frac {2 \left (\frac {b \left (8 a -5 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (8 a^{2}-29 a b +12 b^{2}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (8 a^{2}-29 a b +12 b^{2}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {b \left (8 a -5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (8 a^{2}-8 a b +3 b^{2}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(418\)
risch	\(\frac {8 a^{2} b \,{\mathrm e}^{6 d x +6 c}-8 a \,b^{2} {\mathrm e}^{6 d x +6 c}+3 b^{3} {\mathrm e}^{6 d x +6 c}+48 a^{3} {\mathrm e}^{4 d x +4 c}-72 a^{2} b \,{\mathrm e}^{4 d x +4 c}+42 a \,b^{2} {\mathrm e}^{4 d x +4 c}-9 b^{3} {\mathrm e}^{4 d x +4 c}+40 a^{2} b \,{\mathrm e}^{2 d x +2 c}-40 a \,b^{2} {\mathrm e}^{2 d x +2 c}+9 b^{3} {\mathrm e}^{2 d x +2 c}+6 a \,b^{2}-3 b^{3}}{4 d \,a^{2} \left (a -b \right )^{2} \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d a}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b^{2}}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{2 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d a}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b^{2}}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d \,a^{2}}\)	\(732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*(1/8*b*(8*a-5*b)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7-1/8*(8*a^2-29*a*b+12*b^2)/a^2*b/(a^2-2*a*b+b^
2)*tanh(1/2*d*x+1/2*c)^5-1/8*(8*a^2-29*a*b+12*b^2)/a^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3+1/8*b*(8*a-5*b)
/a/(a^2-2*a*b+b^2)*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)^2-1/4/a*(8*a^2-8*a*b+3*b^2)/(a^2-2*a*b+b^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))-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))))

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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(1/(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. 2835 vs. \(2 (140) = 280\).
time = 0.52, size = 5925, normalized size = 38.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(8*a^4*b - 16*a^3*b^2 + 11*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 + 24*(8*a^4*b - 16*a^3*b^2 + 11*a^2*b^3
- 3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(8*a^4*b - 16*a^3*b^2 + 11*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^6 +
24*a^3*b^2 - 36*a^2*b^3 + 12*a*b^4 + 12*(16*a^5 - 40*a^4*b + 38*a^3*b^2 - 17*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^
4 + 12*(16*a^5 - 40*a^4*b + 38*a^3*b^2 - 17*a^2*b^3 + 3*a*b^4 + 5*(8*a^4*b - 16*a^3*b^2 + 11*a^2*b^3 - 3*a*b^4
)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(8*a^4*b - 16*a^3*b^2 + 11*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + 3*(
16*a^5 - 40*a^4*b + 38*a^3*b^2 - 17*a^2*b^3 + 3*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(40*a^4*b - 80*a^3*b
^2 + 49*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 + 4*(40*a^4*b - 80*a^3*b^2 + 49*a^2*b^3 - 9*a*b^4 + 15*(8*a^4*b - 1
6*a^3*b^2 + 11*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 18*(16*a^5 - 40*a^4*b + 38*a^3*b^2 - 17*a^2*b^3 + 3*a*b^4)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(8*a^2*b^2 - 8*a*b^3 +
3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*a^2*b^2 - 8*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(16*a^3*b - 24*a^2*b^
2 + 14*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(16*a^3*b - 24*a^2*b^2 + 14*a*b^3 - 3*b^4 + 7*(8*a^2*b^2 - 8*a*b^3 +
3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(16*a^3*b -
24*a^2*b^2 + 14*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(64*a^4 - 128*a^3*b + 112*a^2*b^2 - 48*a*b^3
+ 9*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 64*a^4 - 128*a^3*b + 112*a^2
*b^2 - 48*a*b^3 + 9*b^4 + 30*(16*a^3*b - 24*a^2*b^2 + 14*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*a
^2*b^2 - 8*a*b^3 + 3*b^4 + 8*(7*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(16*a^3*b - 24*a^2*b^2 + 14
*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (64*a^4 - 128*a^3*b + 112*a^2*b^2 - 48*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*
x + c)^3 + 4*(16*a^3*b - 24*a^2*b^2 + 14*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*c
osh(d*x + c)^6 + 15*(16*a^3*b - 24*a^2*b^2 + 14*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 16*a^3*b - 24*a^2*b^2 + 14*a*
b^3 - 3*b^4 + 3*(64*a^4 - 128*a^3*b + 112*a^2*b^2 - 48*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((8
*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(16*a^3*b - 24*a^2*b^2 + 14*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (
64*a^4 - 128*a^3*b + 112*a^2*b^2 - 48*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (16*a^3*b - 24*a^2*b^2 + 14*a*b^3 - 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*sinh(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*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)) + 8*(3*(8
*a^4*b - 16*a^3*b^2 + 11*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 6*(16*a^5 - 40*a^4*b + 38*a^3*b^2 - 17*a^2*b^3 +
3*a*b^4)*cosh(d*x + c)^3 + (40*a^4*b - 80*a^3*b^2 + 49*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6
*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d*co
sh(d*x + c)*sinh(d*x + c)^7 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d*sinh(d*x + c)^8 + 4*(2*a^7*b - 7*a
^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)
*d*cosh(d*x + c)^2 + (2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^8 - 3
2*a^7*b + 51*a^6*b^2 - 41*a^5*b^3 + 17*a^4*b^4 - 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^6*b^2 - 3*a^5*b^3 + 3*
a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + 3*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 2*(35*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 30*(2*a^7*b - 7*a^
6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (8*a^8 - 32*a^7*b + 51*a^6*b^2 - 41*a^5*b^3 + 17*
a^4*b^4 - 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x
+ c)^2 + 8*(7*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + 10*(2*a^7*b - 7*a^6*b^2 + 9*a^5
*b^3 - 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + (8*a^8 - 32*a^7*b + 51*a^6*b^2 - 41*a^5*b^3 + 17*a^4*b^4 - 3*a
^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6
+ 15*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^8 - 32*a^7*b + 51*a^6*
b^2 - 41*a^5*b^3 + 17*a^4*b^4 - 3*a^3*b^5)*d*cosh(d*x + c)^2 + (2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 +
a^3*b^5)*d)*sinh(d*x + c)^2 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*d + 8*((a^6*b^2 - 3*a^5*b^3 + 3*a^4*
b^4 - a^3*b^5)*d*cosh(d*x + c)^7 + 3*(2*a^7*b -...

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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(1/(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. 302 vs. \(2 (140) = 280\).
time = 0.81, size = 302, normalized size = 1.96 \begin {gather*} \frac {\frac {{\left (8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 72 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 42 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 40 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 40 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b^{2} - 3 \, b^{3}\right )}}{{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\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}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*((8*a^2 - 8*a*b + 3*b^2)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^4 - 2*a^3*b + a^2*
b^2)*sqrt(-a^2 + a*b)) + 2*(8*a^2*b*e^(6*d*x + 6*c) - 8*a*b^2*e^(6*d*x + 6*c) + 3*b^3*e^(6*d*x + 6*c) + 48*a^3
*e^(4*d*x + 4*c) - 72*a^2*b*e^(4*d*x + 4*c) + 42*a*b^2*e^(4*d*x + 4*c) - 9*b^3*e^(4*d*x + 4*c) + 40*a^2*b*e^(2
*d*x + 2*c) - 40*a*b^2*e^(2*d*x + 2*c) + 9*b^3*e^(2*d*x + 2*c) + 6*a*b^2 - 3*b^3)/((a^4 - 2*a^3*b + a^2*b^2)*(
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))/d

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

[Out]

int(1/(a + b*sinh(c + d*x)^2)^3, x)

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