3.1.33 \(\int \frac {\sinh ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [33]

Optimal. Leaf size=194 \[ \frac {3 \left (a^2+8 a b+8 b^2\right ) x}{8 a^4}-\frac {3 \sqrt {b} \sqrt {a+b} (a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^4 d}-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

3/8*(a^2+8*a*b+8*b^2)*x/a^4-3/2*(a+2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*b^(1/2)*(a+b)^(1/2)/a^4/d-1/8
*(5*a+6*b)*cosh(d*x+c)*sinh(d*x+c)/a^2/d/(a+b-b*tanh(d*x+c)^2)+1/4*cosh(d*x+c)^3*sinh(d*x+c)/a/d/(a+b-b*tanh(d
*x+c)^2)-3/8*b*(3*a+4*b)*tanh(d*x+c)/a^3/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]
time = 0.20, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4217, 481, 541, 536, 212, 214} \begin {gather*} -\frac {3 \sqrt {b} \sqrt {a+b} (a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^4 d}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {(5 a+6 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {3 x \left (a^2+8 a b+8 b^2\right )}{8 a^4}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 a d \left (a-b \tanh ^2(c+d x)+b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a^2 + 8*a*b + 8*b^2)*x)/(8*a^4) - (3*Sqrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a
+ b]])/(2*a^4*d) - ((5*a + 6*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d*(a + b - b*Tanh[c + d*x]^2)) + (Cosh[c +
d*x]^3*Sinh[c + d*x])/(4*a*d*(a + b - b*Tanh[c + d*x]^2)) - (3*b*(3*a + 4*b)*Tanh[c + d*x])/(8*a^3*d*(a + b -
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 541

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

Rule 4217

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1
+ ff^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rubi steps

\begin {align*} \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^3 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {a+b+(4 a+5 b) x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 (a+b) (a+2 b)-3 b (5 a+6 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 d}\\ &=-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {6 (a+b)^2 (a+4 b)+6 b (a+b) (3 a+4 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{16 a^3 (a+b) d}\\ &=-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {(3 b (a+b) (a+2 b)) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^4 d}+\frac {\left (3 \left (a^2+8 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 d}\\ &=\frac {3 \left (a^2+8 a b+8 b^2\right ) x}{8 a^4}-\frac {3 \sqrt {b} \sqrt {a+b} (a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^4 d}-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1080\) vs. \(2(194)=388\).
time = 11.81, size = 1080, normalized size = 5.57 \begin {gather*} -\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (16 x+\frac {\left (a^3-6 a^2 b-24 a b^2-16 b^3\right ) \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 c)-\sinh (2 c))}{b (a+b)^{3/2} d \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\left (a^2+8 a b+8 b^2\right ) \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b (a+b) d (a+2 b+a \cosh (2 (c+d x)))}\right )}{256 a^2 \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {3 (a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (\frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{3/2} (a+b)^{3/2} d}-\frac {a \sinh (2 (c+d x))}{8 b (a+b) d (a+2 b+a \cosh (2 (c+d x)))}\right )}{128 \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (\frac {\left (a^5-30 a^4 b-480 a^3 b^2-1600 a^2 b^3-1920 a b^4-768 b^5\right ) \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\text {sech}(2 c) \left (32 b \left (5 a^4+39 a^3 b+106 a^2 b^2+120 a b^3+48 b^4\right ) d x \cosh (2 c)+16 a b \left (5 a^3+29 a^2 b+48 a b^2+24 b^3\right ) d x \cosh (2 d x)+80 a^4 b d x \cosh (4 c+2 d x)+464 a^3 b^2 d x \cosh (4 c+2 d x)+768 a^2 b^3 d x \cosh (4 c+2 d x)+384 a b^4 d x \cosh (4 c+2 d x)+a^5 \sinh (2 c)+34 a^4 b \sinh (2 c)+224 a^3 b^2 \sinh (2 c)+576 a^2 b^3 \sinh (2 c)+640 a b^4 \sinh (2 c)+256 b^5 \sinh (2 c)-a^5 \sinh (2 d x)-62 a^4 b \sinh (2 d x)-318 a^3 b^2 \sinh (2 d x)-512 a^2 b^3 \sinh (2 d x)-256 a b^4 \sinh (2 d x)-12 a^4 b \sinh (2 (c+2 d x))-36 a^3 b^2 \sinh (2 (c+2 d x))-24 a^2 b^3 \sinh (2 (c+2 d x))-30 a^4 b \sinh (4 c+2 d x)-158 a^3 b^2 \sinh (4 c+2 d x)-256 a^2 b^3 \sinh (4 c+2 d x)-128 a b^4 \sinh (4 c+2 d x)-12 a^4 b \sinh (6 c+4 d x)-36 a^3 b^2 \sinh (6 c+4 d x)-24 a^2 b^3 \sinh (6 c+4 d x)+2 a^4 b \sinh (4 c+6 d x)+2 a^3 b^2 \sinh (4 c+6 d x)+2 a^4 b \sinh (8 c+6 d x)+2 a^3 b^2 \sinh (8 c+6 d x)\right )}{a+2 b+a \cosh (2 (c+d x))}\right )}{1024 a^4 b (a+b) d \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/256*((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(16*x + ((a^3 - 6*a^2*b - 24*a*b^2 - 16*b^3)*ArcTanh
[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c]
- Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a^2 + 8*a*b + 8
*b^2)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(a^2*(a
+ b*Sech[c + d*x]^2)^2) + (3*(a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((a + 2*b)*ArcTanh[(Sqrt[b]*T
anh[c + d*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(3/2)*d) - (a*Sinh[2*(c + d*x)])/(8*b*(a + b)*d*(a + 2*b + a*Co
sh[2*(c + d*x)]))))/(128*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(((a^
5 - 30*a^4*b - 480*a^3*b^2 - 1600*a^2*b^3 - 1920*a*b^4 - 768*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*(
(a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c
]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(32*b*(5*a^4 + 39*a^3*b + 106*a^2*b^2 + 120*a*b^3
+ 48*b^4)*d*x*Cosh[2*c] + 16*a*b*(5*a^3 + 29*a^2*b + 48*a*b^2 + 24*b^3)*d*x*Cosh[2*d*x] + 80*a^4*b*d*x*Cosh[4
*c + 2*d*x] + 464*a^3*b^2*d*x*Cosh[4*c + 2*d*x] + 768*a^2*b^3*d*x*Cosh[4*c + 2*d*x] + 384*a*b^4*d*x*Cosh[4*c +
2*d*x] + a^5*Sinh[2*c] + 34*a^4*b*Sinh[2*c] + 224*a^3*b^2*Sinh[2*c] + 576*a^2*b^3*Sinh[2*c] + 640*a*b^4*Sinh[
2*c] + 256*b^5*Sinh[2*c] - a^5*Sinh[2*d*x] - 62*a^4*b*Sinh[2*d*x] - 318*a^3*b^2*Sinh[2*d*x] - 512*a^2*b^3*Sinh
[2*d*x] - 256*a*b^4*Sinh[2*d*x] - 12*a^4*b*Sinh[2*(c + 2*d*x)] - 36*a^3*b^2*Sinh[2*(c + 2*d*x)] - 24*a^2*b^3*S
inh[2*(c + 2*d*x)] - 30*a^4*b*Sinh[4*c + 2*d*x] - 158*a^3*b^2*Sinh[4*c + 2*d*x] - 256*a^2*b^3*Sinh[4*c + 2*d*x
] - 128*a*b^4*Sinh[4*c + 2*d*x] - 12*a^4*b*Sinh[6*c + 4*d*x] - 36*a^3*b^2*Sinh[6*c + 4*d*x] - 24*a^2*b^3*Sinh[
6*c + 4*d*x] + 2*a^4*b*Sinh[4*c + 6*d*x] + 2*a^3*b^2*Sinh[4*c + 6*d*x] + 2*a^4*b*Sinh[8*c + 6*d*x] + 2*a^3*b^2
*Sinh[8*c + 6*d*x]))/(a + 2*b + a*Cosh[2*(c + d*x)])))/(1024*a^4*b*(a + b)*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(176)=352\).
time = 2.60, size = 459, normalized size = 2.37

method	result	size
risch	\(\frac {3 x}{8 a^{2}}+\frac {3 x b}{a^{3}}+\frac {3 x \,b^{2}}{a^{4}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 a^{2} d}-\frac {{\mathrm e}^{2 d x +2 c} b}{4 a^{3} d}-\frac {{\mathrm e}^{2 d x +2 c}}{8 a^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c} b}{4 a^{3} d}+\frac {{\mathrm e}^{-2 d x -2 c}}{8 a^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 a^{2} d}+\frac {b \left (a +b \right ) \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{2 d x +2 c}+a \right )}{a^{4} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {a b +b^{2}}+2 b}{a}\right ) b}{2 d \,a^{4}}+\frac {3 \sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {a b +b^{2}}+2 b}{a}\right )}{4 d \,a^{3}}-\frac {3 \sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {a b +b^{2}}-2 b}{a}\right ) b}{2 d \,a^{4}}-\frac {3 \sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {a b +b^{2}}-2 b}{a}\right )}{4 d \,a^{3}}\)	\(403\)
derivativedivides	\(\frac {\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-24 a b -24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{4}}+\frac {2 b \left (\frac {\left (-\frac {1}{2} a^{2}-\frac {1}{2} a b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \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 )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a^{2}+9 a b +6 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{4}}-\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a -8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (3 a^{2}+24 a b +24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{4}}}{d}\)	\(459\)
default	\(\frac {\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-24 a b -24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{4}}+\frac {2 b \left (\frac {\left (-\frac {1}{2} a^{2}-\frac {1}{2} a b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \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 )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a^{2}+9 a b +6 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{4}}-\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a -8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (3 a^{2}+24 a b +24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{4}}}{d}\)	\(459\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/4/a^2/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(a+8*b)/a^3/(tanh(1/2*d*x+1/2*c)-
1)^2-1/8*(3*a+8*b)/a^3/(tanh(1/2*d*x+1/2*c)-1)+1/8/a^4*(-3*a^2-24*a*b-24*b^2)*ln(tanh(1/2*d*x+1/2*c)-1)+2*b/a^
4*(((-1/2*a^2-1/2*a*b)*tanh(1/2*d*x+1/2*c)^3-1/2*a*(a+b)*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(
1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)+1/2*(3*a^2+9*a*b+6*b^2)*(-1/4/b^(1/2
)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b
)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))))-1/4/a^2/(tanh(1/2*d*
x+1/2*c)+1)^4+1/2/a^2/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-a-8*b)/a^3/(tanh(1/2*d*x+1/2*c)+1)^2-1/8*(3*a+8*b)/a^3/(
tanh(1/2*d*x+1/2*c)+1)+1/8/a^4*(3*a^2+24*a*b+24*b^2)*ln(tanh(1/2*d*x+1/2*c)+1))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (185) = 370\).
time = 0.55, size = 1299, normalized size = 6.70 \begin {gather*} -\frac {{\left (3 \, a^{3} b + 42 \, a^{2} b^{2} + 88 \, a b^{3} + 48 \, b^{4}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{64 \, {\left (a^{5} + a^{4} b\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {{\left (3 \, a^{2} b + 12 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {{\left (3 \, a^{3} b + 42 \, a^{2} b^{2} + 88 \, a b^{3} + 48 \, b^{4}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{64 \, {\left (a^{5} + a^{4} b\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {{\left (3 \, a^{2} b + 12 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {3 \, {\left (3 \, a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{32 \, {\left (a^{3} + a^{2} b\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {a^{3} b + 8 \, a^{2} b^{2} + 8 \, a b^{3} + {\left (a^{3} b + 18 \, a^{2} b^{2} + 48 \, a b^{3} + 32 \, b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{16 \, {\left (a^{6} + a^{5} b + {\left (a^{6} + a^{5} b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} d} - \frac {a^{3} b + 8 \, a^{2} b^{2} + 8 \, a b^{3} + {\left (a^{3} b + 18 \, a^{2} b^{2} + 48 \, a b^{3} + 32 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, {\left (a^{6} + a^{5} b + 2 \, {\left (a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{6} + a^{5} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {a^{2} b + 2 \, a b^{2} + {\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{4 \, {\left (a^{5} + a^{4} b + {\left (a^{5} + a^{4} b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} d} - \frac {a^{2} b + 2 \, a b^{2} + {\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, {\left (a^{5} + a^{4} b + 2 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{5} + a^{4} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac {3 \, {\left (a b + {\left (a b + 2 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{8 \, {\left (a^{4} + a^{3} b + 2 \, {\left (a^{4} + 3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{4} + a^{3} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {3 \, {\left (d x + c\right )}}{8 \, a^{2} d} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a^{2} d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a^{2} d} + \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{2 \, a^{3} d} - \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{3} d} + \frac {a e^{\left (4 \, d x + 4 \, c\right )} - 16 \, b e^{\left (2 \, d x + 2 \, c\right )}}{64 \, a^{3} d} + \frac {16 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, a^{3} d} + \frac {{\left (a b + 3 \, b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{4} d} - \frac {{\left (a b + 3 \, b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{4} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(3*a^3*b + 42*a^2*b^2 + 88*a*b^3 + 48*b^4)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2
*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5 + a^4*b)*sqrt((a + b)*b)*d) - 1/16*(3*a^2*b + 12*a*b^2 + 8*b
^3)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/(
(a^4 + a^3*b)*sqrt((a + b)*b)*d) + 1/64*(3*a^3*b + 42*a^2*b^2 + 88*a*b^3 + 48*b^4)*log((a*e^(-2*d*x - 2*c) + a
+ 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5 + a^4*b)*sqrt((a + b)*b)
*d) + 1/16*(3*a^2*b + 12*a*b^2 + 8*b^3)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x -
2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^4 + a^3*b)*sqrt((a + b)*b)*d) + 3/32*(3*a*b + 2*b^2)*log((a*e^(-2*d*x
- 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^3 + a^2*b)*sqrt
((a + b)*b)*d) + 1/16*(a^3*b + 8*a^2*b^2 + 8*a*b^3 + (a^3*b + 18*a^2*b^2 + 48*a*b^3 + 32*b^4)*e^(2*d*x + 2*c))
/((a^6 + a^5*b + (a^6 + a^5*b)*e^(4*d*x + 4*c) + 2*(a^6 + 3*a^5*b + 2*a^4*b^2)*e^(2*d*x + 2*c))*d) - 1/16*(a^3
*b + 8*a^2*b^2 + 8*a*b^3 + (a^3*b + 18*a^2*b^2 + 48*a*b^3 + 32*b^4)*e^(-2*d*x - 2*c))/((a^6 + a^5*b + 2*(a^6 +
3*a^5*b + 2*a^4*b^2)*e^(-2*d*x - 2*c) + (a^6 + a^5*b)*e^(-4*d*x - 4*c))*d) + 1/4*(a^2*b + 2*a*b^2 + (a^2*b +
8*a*b^2 + 8*b^3)*e^(2*d*x + 2*c))/((a^5 + a^4*b + (a^5 + a^4*b)*e^(4*d*x + 4*c) + 2*(a^5 + 3*a^4*b + 2*a^3*b^2
)*e^(2*d*x + 2*c))*d) - 1/4*(a^2*b + 2*a*b^2 + (a^2*b + 8*a*b^2 + 8*b^3)*e^(-2*d*x - 2*c))/((a^5 + a^4*b + 2*(
a^5 + 3*a^4*b + 2*a^3*b^2)*e^(-2*d*x - 2*c) + (a^5 + a^4*b)*e^(-4*d*x - 4*c))*d) - 3/8*(a*b + (a*b + 2*b^2)*e^
(-2*d*x - 2*c))/((a^4 + a^3*b + 2*(a^4 + 3*a^3*b + 2*a^2*b^2)*e^(-2*d*x - 2*c) + (a^4 + a^3*b)*e^(-4*d*x - 4*c
))*d) + 3/8*(d*x + c)/(a^2*d) - 1/8*e^(2*d*x + 2*c)/(a^2*d) + 1/8*e^(-2*d*x - 2*c)/(a^2*d) + 1/2*b*log(a*e^(4*
d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^3*d) - 1/2*b*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x -
4*c) + a)/(a^3*d) + 1/64*(a*e^(4*d*x + 4*c) - 16*b*e^(2*d*x + 2*c))/(a^3*d) + 1/64*(16*b*e^(-2*d*x - 2*c) - a
*e^(-4*d*x - 4*c))/(a^3*d) + 1/4*(a*b + 3*b^2)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^4*d
) - 1/4*(a*b + 3*b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^4*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2464 vs. \(2 (185) = 370\).
time = 0.44, size = 5169, normalized size = 26.64 \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*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/64*(a^3*cosh(d*x + c)^12 + 12*a^3*cosh(d*x + c)*sinh(d*x + c)^11 + a^3*sinh(d*x + c)^12 - 6*(a^3 + 2*a^2*b)
*cosh(d*x + c)^10 + 6*(11*a^3*cosh(d*x + c)^2 - a^3 - 2*a^2*b)*sinh(d*x + c)^10 + 20*(11*a^3*cosh(d*x + c)^3 -
3*(a^3 + 2*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^9 - (15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^
2)*d*x)*cosh(d*x + c)^8 + (495*a^3*cosh(d*x + c)^4 - 15*a^3 - 64*a^2*b - 64*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^
2)*d*x - 270*(a^3 + 2*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*a^3*cosh(d*x + c)^5 - 90*(a^3 + 2*a^2*b)
*cosh(d*x + c)^3 - (15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x +
c)^7 + 16*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^6 + 4*(231*
a^3*cosh(d*x + c)^6 - 315*(a^3 + 2*a^2*b)*cosh(d*x + c)^4 + 16*a^2*b + 48*a*b^2 + 32*b^3 + 12*(a^3 + 10*a^2*b
+ 24*a*b^2 + 16*b^3)*d*x - 7*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^2
)*sinh(d*x + c)^6 + 8*(99*a^3*cosh(d*x + c)^7 - 189*(a^3 + 2*a^2*b)*cosh(d*x + c)^5 - 7*(15*a^3 + 64*a^2*b + 6
4*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^3 + 12*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2
*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^
2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^4 + (495*a^3*cosh(d*x + c)^8 - 1260*(a^3 + 2*a^2*b)*cosh(d*x + c)^6 - 70*(15
*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^4 + 15*a^3 + 128*a^2*b + 128*a*b^
2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x + 240*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3
)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(55*a^3*cosh(d*x + c)^9 - 180*(a^3 + 2*a^2*b)*cosh(d*x + c)^7 - 14
*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^5 + 80*(4*a^2*b + 12*a*b^2 +
8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^3 + (15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^
3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 + 6*(a^3 + 2*a^2*b)*cosh(d*x + c)^2 + 2*(33*a
^3*cosh(d*x + c)^10 - 135*(a^3 + 2*a^2*b)*cosh(d*x + c)^8 - 14*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2
*b + 8*a*b^2)*d*x)*cosh(d*x + c)^6 + 120*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*
d*x)*cosh(d*x + c)^4 + 3*a^3 + 6*a^2*b + 3*(15*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 48*((a^2 + 2*a*b)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x
+ c)^7 + (a^2 + 2*a*b)*sinh(d*x + c)^8 + 2*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^6 + 2*(14*(a^2 + 2*a*b)*cosh(d
*x + c)^2 + a^2 + 4*a*b + 4*b^2)*sinh(d*x + c)^6 + 4*(14*(a^2 + 2*a*b)*cosh(d*x + c)^3 + 3*(a^2 + 4*a*b + 4*b^
2)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^2 + 2*a*b)*cosh(d*x + c)^4 + (70*(a^2 + 2*a*b)*cosh(d*x + c)^4 + 30*(a^
2 + 4*a*b + 4*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^4 + 4*(14*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 10*(
a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(14*(a^2 + 2*a*b)*cosh
(d*x + c)^6 + 15*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^4 + 3*(a^2 + 2*a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*
(2*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 3*(a^2 + 4*a*b + 4*b^2)*cosh(d*x + c)^5 + (a^2 + 2*a*b)*cosh(d*x + c)^3)*si
nh(d*x + c))*sqrt(a*b + b^2)*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c
)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b
+ 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^2 + 2*a*cos
h(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(a*b + b^2))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c
)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d
*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 4*(3*a^3*cosh(d*x + c)^11 -
15*(a^3 + 2*a^2*b)*cosh(d*x + c)^9 - 2*(15*a^3 + 64*a^2*b + 64*a*b^2 - 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(
d*x + c)^7 + 24*(4*a^2*b + 12*a*b^2 + 8*b^3 + 3*(a^3 + 10*a^2*b + 24*a*b^2 + 16*b^3)*d*x)*cosh(d*x + c)^5 + (1
5*a^3 + 128*a^2*b + 128*a*b^2 + 24*(a^3 + 8*a^2*b + 8*a*b^2)*d*x)*cosh(d*x + c)^3 + 3*(a^3 + 2*a^2*b)*cosh(d*x
+ c))*sinh(d*x + c))/(a^5*d*cosh(d*x + c)^8 + 8*a^5*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^5*d*sinh(d*x + c)^8 +
a^5*d*cosh(d*x + c)^4 + 2*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^6 + 2*(14*a^5*d*cosh(d*x + c)^2 + (a^5 + 2*a^4*b)*d
)*sinh(d*x + c)^6 + 4*(14*a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)^5 + (70*a^5
*d*cosh(d*x + c)^4 + a^5*d + 30*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a^5*d*cosh(d*x + c)
^5 + a^5*d*cosh(d*x + c) + 10*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^3 + 2*(14*a^5*d*cosh(d*x + c)^6
+ 3*a^5*d*cosh(d*x + c)^2 + 15*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^4)*sinh(d*x + c)^2 + 4*(2*a^5*d*cosh(d*x + c)^
7 + a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*d...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(c + d*x)**4/(a + b*sech(c + d*x)**2)**2, x)

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Giac [A]
time = 1.27, size = 323, normalized size = 1.66 \begin {gather*} \frac {\frac {24 \, {\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (d x + c\right )}}{a^{4}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 144 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{4}} - \frac {96 \, {\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{4}} + \frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )}}{a^{4}} + \frac {64 \, {\left (a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} b + a b^{2}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} a^{4}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(24*(a^2 + 8*a*b + 8*b^2)*(d*x + c)/a^4 - (18*a^2*e^(4*d*x + 4*c) + 144*a*b*e^(4*d*x + 4*c) + 144*b^2*e^(
4*d*x + 4*c) - 8*a^2*e^(2*d*x + 2*c) - 16*a*b*e^(2*d*x + 2*c) + a^2)*e^(-4*d*x - 4*c)/a^4 - 96*(a^2*b + 3*a*b^
2 + 2*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^4) + (a^2*e^(4*d*x +
4*c) - 8*a^2*e^(2*d*x + 2*c) - 16*a*b*e^(2*d*x + 2*c))/a^4 + 64*(a^2*b*e^(2*d*x + 2*c) + 3*a*b^2*e^(2*d*x + 2
*c) + 2*b^3*e^(2*d*x + 2*c) + a^2*b + a*b^2)/((a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) +
a)*a^4))/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\,{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\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/cosh(c + d*x)^2)^2,x)

[Out]

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

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