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

Optimal. Leaf size=139 \[ -\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} d (a-b)^{5/2}}+\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{8 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 139, 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 = {3173, 12, 3181, 208} \[ -\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} d (a-b)^{5/2}}+\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{8 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(3/2)*(a - b)^(5/2)*d) + (Cosh[c + d*x]*Sinh[c
+ d*x])/(4*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) + ((2*a + b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a*(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 208

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

Rule 3173

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Sim
p[((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 3181

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]

Rubi steps

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

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Mathematica [A]  time = 1.36, size = 121, normalized size = 0.87 \[ \frac {\frac {\sinh (2 (c+d x)) \left (8 a^2+b (2 a+b) \cosh (2 (c+d x))-4 a b-b^2\right )}{a (a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}-\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} (a-b)^{5/2}}}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.82, size = 5519, normalized size = 39.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^6 + 24*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)*s
inh(d*x + c)^5 + 4*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*sinh(d*x + c)^6 + 8*a^3*b^2 - 4*a^2*b^3 - 4*a*b^4 + 4*(16*a
^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 4*(16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^
3 - 3*a*b^4 + 15*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(4*a^3*b^2 - 5*a^2*b
^3 + a*b^4)*cosh(d*x + c)^3 + (16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c))*sinh(d*x +
c)^3 + 4*(16*a^4*b - 20*a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2 + 4*(16*a^4*b - 20*a^3*b^2 + a^2*b^3 + 3*
a*b^4 + 15*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*
a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((4*a*b^3 - b^4)*cosh(d*x + c)^8 + 8*(4*a*b^3 - b^4)*cosh(d*x + c)*s
inh(d*x + c)^7 + (4*a*b^3 - b^4)*sinh(d*x + c)^8 + 4*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^6 + 4*(8*a^2*b^
2 - 6*a*b^3 + b^4 + 7*(4*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a*b^3 - b^4)*cosh(d*x + c)^3
+ 3*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4)*
cosh(d*x + c)^4 + 2*(35*(4*a*b^3 - b^4)*cosh(d*x + c)^4 + 32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4 + 30*(8*a^2
*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*a*b^3 - b^4 + 8*(7*(4*a*b^3 - b^4)*cosh(d*x + c)^5
+ 10*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^3 + (32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4)*cosh(d*x + c))*s
inh(d*x + c)^3 + 4*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^2 + 4*(7*(4*a*b^3 - b^4)*cosh(d*x + c)^6 + 15*(8*
a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^4 + 8*a^2*b^2 - 6*a*b^3 + b^4 + 3*(32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3
*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a*b^3 - b^4)*cosh(d*x + c)^7 + 3*(8*a^2*b^2 - 6*a*b^3 + b^4)*co
sh(d*x + c)^5 + (32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (8*a^2*b^2 - 6*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*
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*(4*a^3*b^2
- 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^5 + 2*(16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3
+ (16*a^4*b - 20*a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5
- a^2*b^6)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*sinh(d*x + c)^8 + 4*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3
*b^5 + a^2*b^6)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^2 + (2*a^
6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d)*sinh(d*x + c)^6 + 2*(8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3
- 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d
*cosh(d*x + c)^3 + 3*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^
5 + 2*(35*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^4 + 30*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^
4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 + (8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*
a^2*b^6)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 +
8*(7*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^5 + 10*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5
*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^3 + (8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b
^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^6 + 15
*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^4 + 3*(8*a^7*b - 32*a^6*b^2 + 51*a^
5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c)^2 + (2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^
5 + a^2*b^6)*d)*sinh(d*x + c)^2 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d + 8*((a^5*b^3 - 3*a^4*b^4 + 3*
a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^7 + 3*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x
+ c)^5 + (8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c)^3 + (2*a^6*
b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(4*a^3*b^2 - 5*a^2
*b^3 + a*b^4)*cosh(d*x + c)^6 + 12*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(4*a^3*b^
2 - 5*a^2*b^3 + a*b^4)*sinh(d*x + c)^6 + 4*a^3*b^2 - 2*a^2*b^3 - 2*a*b^4 + 2*(16*a^5 - 24*a^4*b + 6*a^3*b^2 +
5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 2*(16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4 + 15*(4*a^3*b^2
- 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^3 +
(16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(16*a^4*b - 20*a^3*b
^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2 + 2*(16*a^4*b - 20*a^3*b^2 + a^2*b^3 + 3*a*b^4 + 15*(4*a^3*b^2 - 5*a^2
*b^3 + a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 24*a^4*b + 6*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(
d*x + c)^2 - ((4*a*b^3 - b^4)*cosh(d*x + c)^8 + 8*(4*a*b^3 - b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a*b^3 - b
^4)*sinh(d*x + c)^8 + 4*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^6 + 4*(8*a^2*b^2 - 6*a*b^3 + b^4 + 7*(4*a*b^
3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a*b^3 - b^4)*cosh(d*x + c)^3 + 3*(8*a^2*b^2 - 6*a*b^3 + b^
4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 2*(35*(4*a*
b^3 - b^4)*cosh(d*x + c)^4 + 32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4 + 30*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*
x + c)^2)*sinh(d*x + c)^4 + 4*a*b^3 - b^4 + 8*(7*(4*a*b^3 - b^4)*cosh(d*x + c)^5 + 10*(8*a^2*b^2 - 6*a*b^3 + b
^4)*cosh(d*x + c)^3 + (32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^2*b^2
- 6*a*b^3 + b^4)*cosh(d*x + c)^2 + 4*(7*(4*a*b^3 - b^4)*cosh(d*x + c)^6 + 15*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh
(d*x + c)^4 + 8*a^2*b^2 - 6*a*b^3 + b^4 + 3*(32*a^3*b - 40*a^2*b^2 + 20*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d
*x + c)^2 + 8*((4*a*b^3 - b^4)*cosh(d*x + c)^7 + 3*(8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c)^5 + (32*a^3*b - 4
0*a^2*b^2 + 20*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (8*a^2*b^2 - 6*a*b^3 + b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt
(-a^2 + a*b)*arctan(-1/2*(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)*s
qrt(-a^2 + a*b)/(a^2 - a*b)) + 4*(3*(4*a^3*b^2 - 5*a^2*b^3 + a*b^4)*cosh(d*x + c)^5 + 2*(16*a^5 - 24*a^4*b + 6
*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (16*a^4*b - 20*a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c))*s
inh(d*x + c))/((a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*
b^5 - a^2*b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*sinh(d*x + c)^8
+ 4*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3 - 3*a^4*b^4 +
3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^2 + (2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d)*sinh(d*
x + c)^6 + 2*(8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c)^4 + 8*(
7*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^3 + 3*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3
*b^5 + a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*
x + c)^4 + 30*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 + (8*a^7*b - 32*a^6*
b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*
b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c
)^5 + 10*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^3 + (8*a^7*b - 32*a^6*b^2 +
51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 - 3*a^4*b^
4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^6 + 15*(2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*co
sh(d*x + c)^4 + 3*(8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c)^2
+ (2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d)*sinh(d*x + c)^2 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*
b^5 - a^2*b^6)*d + 8*((a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d*cosh(d*x + c)^7 + 3*(2*a^6*b^2 - 7*a^5*b^3
+ 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^5 + (8*a^7*b - 32*a^6*b^2 + 51*a^5*b^3 - 41*a^4*b^4 + 17*a
^3*b^5 - 3*a^2*b^6)*d*cosh(d*x + c)^3 + (2*a^6*b^2 - 7*a^5*b^3 + 9*a^4*b^4 - 5*a^3*b^5 + a^2*b^6)*d*cosh(d*x +
c))*sinh(d*x + c))]

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giac [B]  time = 3.55, size = 277, normalized size = 1.99 \[ -\frac {\frac {{\left (4 \, a - b\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (4 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} + b^{3}\right )}}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 1408, normalized size = 10.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)^2*a/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^7-1/4/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)^2/(a^2-2
*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7*b-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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5*a+9/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*t
anh(1/2*d*x+1/2*c)^2*b+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5*b+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)^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5*b^2-1/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3*a+9/4/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x
+1/2*c)^3*b+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)^2/a/(a^2-2*a*b
+b^2)*tanh(1/2*d*x+1/2*c)^3*b^2+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)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-1/4/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)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)*b+1/2/d/(a^2-2*a*b+b^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/(a^2-2*a*b+b^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))
*b-1/2/d/(a^2-2*a*b+b^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^2-2*a*b+b^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))*b-1/8/d/(a^2-2*a*b+b^2)*b/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))-1/8/d/(a^2-2*a*b+b^2)/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^2+1/8/d/(a^2-2*a*b+b^2)*b/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))-1/8/d/(a^2-2*a*b+b^2)/a/(-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))*b^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2/(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 b-a positive or negative?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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