3.251 \(\int \frac {1}{(a-b \sinh ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=210 \[ \frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a d (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3209, 1205, 1166, 208} \[ \frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a d (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Sinh[c + d*x]^4)^(-2),x]

[Out]

((4*Sqrt[a] - 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[
b])^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(7/4)*(
Sqrt[a] + Sqrt[b])^(3/2)*d) - (b*Tanh[c + d*x]*(1 - 2*Tanh[c + d*x]^2))/(4*a*(a - b)*d*(a - 2*a*Tanh[c + d*x]^
2 + (a - b)*Tanh[c + d*x]^4))

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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1205

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(
2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {2 a (4 a-3 b) b}{a-b}+\frac {4 a (2 a-b) b x^2}{a-b}}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b d}\\ &=-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {\left (4 a-\sqrt {a} \sqrt {b}-3 b\right ) \operatorname {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right ) d}-\frac {\left (4 a+\sqrt {a} \sqrt {b}-3 b\right ) \operatorname {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right ) d}\\ &=\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 2.87, size = 230, normalized size = 1.10 \[ \frac {\frac {\left (-\sqrt {a} \sqrt {b}+4 a-3 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {\left (\sqrt {a} \sqrt {b}+4 a-3 b\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}-a}}+\frac {2 \sqrt {a} b (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b}}{8 a^{3/2} d (a-b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((4*a + Sqrt[a]*Sqrt[b] - 3*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt
[-a + Sqrt[a]*Sqrt[b]]) + ((4*a - Sqrt[a]*Sqrt[b] - 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a +
Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (2*Sqrt[a]*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(8*a -
3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)

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fricas [B]  time = 1.26, size = 6522, normalized size = 31.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/16*(8*b*cosh(d*x + c)^6 + 48*b*cosh(d*x + c)*sinh(d*x + c)^5 + 8*b*sinh(d*x + c)^6 - 8*(8*a - 3*b)*cosh(d*x
+ c)^4 + 8*(15*b*cosh(d*x + c)^2 - 8*a + 3*b)*sinh(d*x + c)^4 + 32*(5*b*cosh(d*x + c)^3 - (8*a - 3*b)*cosh(d*x
+ c))*sinh(d*x + c)^3 - 40*b*cosh(d*x + c)^2 + 8*(15*b*cosh(d*x + c)^4 - 6*(8*a - 3*b)*cosh(d*x + c)^2 - 5*b)
*sinh(d*x + c)^2 - ((a^2*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a
^2*b - a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^2
- (a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^2*b - a*b^2
)*d*cosh(d*x + c)^3 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b - a*b^2)*d*cosh(d*x +
c)^4 - 30*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh(d*x + c)^4 - 4*(a^2*b - a*b
^2)*d*cosh(d*x + c)^2 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - (8*a^3
- 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 - 15*(a^2*b -
a*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c
)^2 + (a^2*b - a*b^2)*d + 8*((a^2*b - a*b^2)*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - (8*a^3
- 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^3 - (a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^6 - 3*a^5*
b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^
12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 -
3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4 + 2*(16*a^8 - 57*a^7*b +
75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/
((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - (384*a^3*b - 680*a^2
*b^2 + 405*a*b^3 - 81*b^4)*cosh(d*x + c)^2 - 2*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cosh(d*x + c)*si
nh(d*x + c) - (384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*sinh(d*x + c)^2 + 2*(2*(2*a^10 - 7*a^9*b + 9*a^8*
b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*
a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + (120*a^5*b - 217*a^4*b^2 + 132*
a^3*b^3 - 27*a^2*b^4)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 127
3*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b
^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) + ((a^2*b - a*b^2)*d*cosh(d
*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b - a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b -
a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8
*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - 3*(a^2*b - a*b^2)*d*co
sh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 30*(a^2*b - a*b^2)*d*cosh(d*x + c)^2
- (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh(d*x + c)^4 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^2*b - a*b^2)
*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c))*sinh
(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 - 15*(a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^3 - 11*a^
2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^2 + (a^2*b - a*b^2)*d + 8*((a^2*b - a*b^2)
*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^3 - (a
^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b
- 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^
4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(384
*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4 + 2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*sq
rt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^
3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - (384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cosh(d*x + c)^2 -
2*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cosh(d*x + c)*sinh(d*x + c) - (384*a^3*b - 680*a^2*b^2 + 405
*a*b^3 - 81*b^4)*sinh(d*x + c)^2 - 2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4
*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*
b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d)*sqrt(-((a^6 - 3*a^5
*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a
^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 -
3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) + ((a^2*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x + c
)*sinh(d*x + c)^7 + (a^2*b - a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b - a*
b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^4
+ 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b
- a*b^2)*d*cosh(d*x + c)^4 - 30*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh(d*x +
c)^4 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2)*d*co
sh(d*x + c)^3 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*
x + c)^6 - 15*(a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^2*b -
a*b^2)*d)*sinh(d*x + c)^2 + (a^2*b - a*b^2)*d + 8*((a^2*b - a*b^2)*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)*d*cos
h(d*x + c)^5 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^3 - (a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)
)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 +
81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*
a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4 - 2
*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 -
522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) -
(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cosh(d*x + c)^2 - 2*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*
b^4)*cosh(d*x + c)*sinh(d*x + c) - (384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*sinh(d*x + c)^2 + 2*(2*(2*a^
10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4
+ 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - (120*a^5*b
- 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b
- 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4
- 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) - ((a^2
*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b - a*b^2)*d*sinh(d*x
+ c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*si
nh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - 3*
(a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 30*(a^2*b - a*b^2
)*d*cosh(d*x + c)^2 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh(d*x + c)^4 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 +
8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d
*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 - 15*(a^2*b - a*b^2)*d*cosh(d*x + c)^
4 - 3*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^2 + (a^2*b - a*b^2)*d
+ 8*((a^2*b - a*b^2)*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*
cosh(d*x + c)^3 - (a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*
d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a
^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3
*b^3)*d^2))*log(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4 - 2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3
+ 9*a^4*b^4)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^
11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - (384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4
)*cosh(d*x + c)^2 - 2*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cosh(d*x + c)*sinh(d*x + c) - (384*a^3*b
- 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*sinh(d*x + c)^2 - 2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4
)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20
*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d)*
sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81
*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*
b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) + 16*(3*b*cosh(d*x + c)^5 - 2*(8*a - 3*b)*cosh(d*x +
c)^3 - 5*b*cosh(d*x + c))*sinh(d*x + c) + 8*b)/((a^2*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d
*x + c)*sinh(d*x + c)^7 + (a^2*b - a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*
b - a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x
+ c)^4 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(
a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 30*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh
(d*x + c)^4 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2
)*d*cosh(d*x + c)^3 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*c
osh(d*x + c)^6 - 15*(a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^
2*b - a*b^2)*d)*sinh(d*x + c)^2 + (a^2*b - a*b^2)*d + 8*((a^2*b - a*b^2)*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)
*d*cosh(d*x + c)^5 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^3 - (a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*
x + c))

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giac [A]  time = 0.23, size = 127, normalized size = 0.60 \[ \frac {b e^{\left (6 \, d x + 6 \, c\right )} - 8 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 5 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{2 \, {\left (a^{2} - a b\right )} {\left (b e^{\left (8 \, d x + 8 \, c\right )} - 4 \, b e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/2*(b*e^(6*d*x + 6*c) - 8*a*e^(4*d*x + 4*c) + 3*b*e^(4*d*x + 4*c) - 5*b*e^(2*d*x + 2*c) + b)/((a^2 - a*b)*(b*
e^(8*d*x + 8*c) - 4*b*e^(6*d*x + 6*c) - 16*a*e^(4*d*x + 4*c) + 6*b*e^(4*d*x + 4*c) - 4*b*e^(2*d*x + 2*c) + b)*
d)

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maple [C]  time = 0.11, size = 534, normalized size = 2.54 \[ -\frac {b \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a -b \right )}+\frac {5 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{2 d \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a -b \right )}+\frac {5 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{2 d \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a -b \right )}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -3 b \right ) \textit {\_R}^{6}+\left (-12 a +5 b \right ) \textit {\_R}^{4}+\left (-5 b +12 a \right ) \textit {\_R}^{2}-4 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 d a \left (a -b \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*sinh(d*x+c)^4)^2,x)

[Out]

-1/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4
-4*tanh(1/2*d*x+1/2*c)^2*a+a)*b/a/(a-b)*tanh(1/2*d*x+1/2*c)^7+5/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/
2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/a/(a-b)*tanh(1/2*d*
x+1/2*c)^5*b+5/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*
d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/a/(a-b)*tanh(1/2*d*x+1/2*c)^3*b-1/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tan
h(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)*b/a/(a-
b)*tanh(1/2*d*x+1/2*c)-1/16/d/a/(a-b)*sum(((4*a-3*b)*_R^6+(-12*a+5*b)*_R^4+(-5*b+12*a)*_R^2-4*a+3*b)/(_R^7*a-3
*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+
a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (8 \, a e^{\left (4 \, c\right )} - 3 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - b e^{\left (6 \, d x + 6 \, c\right )} + 5 \, b e^{\left (2 \, d x + 2 \, c\right )} - b}{2 \, {\left (a^{2} b d - a b^{2} d + {\left (a^{2} b d e^{\left (8 \, c\right )} - a b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 4 \, {\left (a^{2} b d e^{\left (6 \, c\right )} - a b^{2} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \, {\left (8 \, a^{3} d e^{\left (4 \, c\right )} - 11 \, a^{2} b d e^{\left (4 \, c\right )} + 3 \, a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 4 \, {\left (a^{2} b d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + \int -\frac {2 \, {\left (8 \, a e^{\left (4 \, c\right )} - 5 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - b e^{\left (6 \, d x + 6 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )}}{a^{2} b - a b^{2} + {\left (a^{2} b e^{\left (8 \, c\right )} - a b^{2} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 4 \, {\left (a^{2} b e^{\left (6 \, c\right )} - a b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \, {\left (8 \, a^{3} e^{\left (4 \, c\right )} - 11 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 4 \, {\left (a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/2*((8*a*e^(4*c) - 3*b*e^(4*c))*e^(4*d*x) - b*e^(6*d*x + 6*c) + 5*b*e^(2*d*x + 2*c) - b)/(a^2*b*d - a*b^2*d
+ (a^2*b*d*e^(8*c) - a*b^2*d*e^(8*c))*e^(8*d*x) - 4*(a^2*b*d*e^(6*c) - a*b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^3*d
*e^(4*c) - 11*a^2*b*d*e^(4*c) + 3*a*b^2*d*e^(4*c))*e^(4*d*x) - 4*(a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c))*e^(2*d*x)
) + integrate(-(2*(8*a*e^(4*c) - 5*b*e^(4*c))*e^(4*d*x) - b*e^(6*d*x + 6*c) - b*e^(2*d*x + 2*c))/(a^2*b - a*b^
2 + (a^2*b*e^(8*c) - a*b^2*e^(8*c))*e^(8*d*x) - 4*(a^2*b*e^(6*c) - a*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^3*e^(4*c)
- 11*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c))*e^(4*d*x) - 4*(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 = -0.00 \[ \int \frac {1}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*sinh(c + d*x)^4)^2,x)

[Out]

int(1/(a - b*sinh(c + d*x)^4)^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(1/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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