3.1.42 \(\int \frac {\sinh ^4(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [42]

Optimal. Leaf size=102 \[ \frac {x}{b^2}-\frac {\sqrt {a} (2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 (a-b)^{3/2} b^2 d}-\frac {a \tanh (c+d x)}{2 (a-b) b d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3266, 481, 536, 212, 214} \begin {gather*} -\frac {\sqrt {a} (2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 b^2 d (a-b)^{3/2}}-\frac {a \tanh (c+d x)}{2 b d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {x}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 3266

Int[sin[(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^(m + 1)/f, Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p +
1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.58, size = 99, normalized size = 0.97 \begin {gather*} -\frac {-2 (c+d x)+\frac {\sqrt {a} (2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{(a-b)^{3/2}}+\frac {a b \sinh (2 (c+d x))}{(a-b) (2 a-b+b \cosh (2 (c+d x)))}}{2 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(-2*(c + d*x) + (Sqrt[a]*(2*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(3/2) + (a*b*S
inh[2*(c + d*x)])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])))/(b^2*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(318\) vs. \(2(90)=180\).
time = 1.09, size = 319, normalized size = 3.13

method	result	size
risch	\(\frac {x}{b^{2}}+\frac {a \left (2 a \,{\mathrm e}^{2 d x +2 c}-b \,{\mathrm e}^{2 d x +2 c}+b \right )}{b^{2} \left (a -b \right ) d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right ) a}{2 \left (a -b \right )^{2} d \,b^{2}}-\frac {3 \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{4 \left (a -b \right )^{2} d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right ) a}{2 \left (a -b \right )^{2} d \,b^{2}}+\frac {3 \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{4 \left (a -b \right )^{2} d b}\)	\(315\)
derivativedivides	\(\frac {\frac {2 a \left (\frac {-\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a -b \right )}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (2 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a -2 b}\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}}{d}\)	\(319\)
default	\(\frac {\frac {2 a \left (\frac {-\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right )}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a -b \right )}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (2 a -3 b \right ) a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a -2 b}\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{2}}}{d}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*a/b^2*((-1/2*b/(a-b)*tanh(1/2*d*x+1/2*c)^3-1/2*b/(a-b)*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)+1/2*(2*a-3*b)/(a-b)*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)*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))))+1/b^2*ln(tanh(1/2*d*x+1/2*c)+1)-1/b^2*ln(tanh(1/2*d*x+1/2*c)-1))

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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(sinh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^2,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. 740 vs. \(2 (91) = 182\).
time = 0.49, size = 1772, normalized size = 17.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a*b - b^2)*d*x*cosh(d*x + c)^4 + 16*(a*b - b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 4*(a*b - b^2)*d*x
*sinh(d*x + c)^4 + 4*(a*b - b^2)*d*x + 4*(2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^2 - a*b)*cosh(d*x + c)^2 + 4*(6*(a
*b - b^2)*d*x*cosh(d*x + c)^2 + 2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^2 - a*b)*sinh(d*x + c)^2 + ((2*a*b - 3*b^2)*
cosh(d*x + c)^4 + 4*(2*a*b - 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b - 3*b^2)*sinh(d*x + c)^4 + 2*(4*a^2
- 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b - 3*b^2)*cosh(d*x + c)^2 + 4*a^2 - 8*a*b + 3*b^2)*sinh(d*x + c
)^2 + 2*a*b - 3*b^2 + 4*((2*a*b - 3*b^2)*cosh(d*x + c)^3 + (4*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c
))*sqrt(a/(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*((a*b - b^2)*cosh(d*x + c)^2 + 2*(a*b
- b^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b - b^2)*sinh(d*x + c)^2 + 2*a^2 - 3*a*b + b^2)*sqrt(a/(a - b)))/(b*co
sh(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*c
osh(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*b + 8*(2*(a*b - b^2)*d*x*cosh(d*x + c)^3 + (2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^2 - a*b)*cosh(d*x + c))*
sinh(d*x + c))/((a*b^3 - b^4)*d*cosh(d*x + c)^4 + 4*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^3 - b
^4)*d*sinh(d*x + c)^4 + 2*(2*a^2*b^2 - 3*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 2*(3*(a*b^3 - b^4)*d*cosh(d*x + c)^2
+ (2*a^2*b^2 - 3*a*b^3 + b^4)*d)*sinh(d*x + c)^2 + (a*b^3 - b^4)*d + 4*((a*b^3 - b^4)*d*cosh(d*x + c)^3 + (2*
a^2*b^2 - 3*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*(a*b - b^2)*d*x*cosh(d*x + c)^4 + 8*(a*b - b^
2)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(a*b - b^2)*d*x*sinh(d*x + c)^4 + 2*(a*b - b^2)*d*x + 2*(2*(2*a^2 - 3
*a*b + b^2)*d*x + 2*a^2 - a*b)*cosh(d*x + c)^2 + 2*(6*(a*b - b^2)*d*x*cosh(d*x + c)^2 + 2*(2*a^2 - 3*a*b + b^2
)*d*x + 2*a^2 - a*b)*sinh(d*x + c)^2 - ((2*a*b - 3*b^2)*cosh(d*x + c)^4 + 4*(2*a*b - 3*b^2)*cosh(d*x + c)*sinh
(d*x + c)^3 + (2*a*b - 3*b^2)*sinh(d*x + c)^4 + 2*(4*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b - 3*b^
2)*cosh(d*x + c)^2 + 4*a^2 - 8*a*b + 3*b^2)*sinh(d*x + c)^2 + 2*a*b - 3*b^2 + 4*((2*a*b - 3*b^2)*cosh(d*x + c)
^3 + (4*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a/(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)*sqrt(-a/(a - b))/a) + 2*a*b + 4*(2*(a*b - b^2)*d*
x*cosh(d*x + c)^3 + (2*(2*a^2 - 3*a*b + b^2)*d*x + 2*a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/((a*b^3 - b^4)*d
*cosh(d*x + c)^4 + 4*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^3 - b^4)*d*sinh(d*x + c)^4 + 2*(2*a^
2*b^2 - 3*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 2*(3*(a*b^3 - b^4)*d*cosh(d*x + c)^2 + (2*a^2*b^2 - 3*a*b^3 + b^4)*
d)*sinh(d*x + c)^2 + (a*b^3 - b^4)*d + 4*((a*b^3 - b^4)*d*cosh(d*x + c)^3 + (2*a^2*b^2 - 3*a*b^3 + b^4)*d*cosh
(d*x + c))*sinh(d*x + 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(sinh(d*x+c)**4/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [A]
time = 1.36, size = 168, normalized size = 1.65 \begin {gather*} -\frac {\frac {{\left (2 \, a^{2} - 3 \, 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 b^{2} - b^{3}\right )} \sqrt {-a^{2} + a b}} - \frac {2 \, {\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b e^{\left (2 \, d x + 2 \, c\right )} + a b\right )}}{{\left (a b^{2} - 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 )}} - \frac {2 \, {\left (d x + c\right )}}{b^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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