3.48 \(\int \frac {\text {csch}^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=165 \[ -\frac {5 \sqrt {b} (3 a-4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 d (a+b)^{9/2}}-\frac {b (7 a-4 b) \tanh (c+d x)}{8 d (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a b \tanh (c+d x)}{4 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\coth ^3(c+d x)}{3 d (a+b)^3}+\frac {(a-2 b) \coth (c+d x)}{d (a+b)^4} \]

[Out]

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

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Rubi [A]  time = 0.27, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4132, 456, 1259, 1261, 208} \[ -\frac {5 \sqrt {b} (3 a-4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 d (a+b)^{9/2}}-\frac {b (7 a-4 b) \tanh (c+d x)}{8 d (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a b \tanh (c+d x)}{4 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\coth ^3(c+d x)}{3 d (a+b)^3}+\frac {(a-2 b) \coth (c+d x)}{d (a+b)^4} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(-5*(3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*d) + ((a - 2*b)*Coth[c
+ d*x])/((a + b)^4*d) - Coth[c + d*x]^3/(3*(a + b)^3*d) - (a*b*Tanh[c + d*x])/(4*(a + b)^3*d*(a + b - b*Tanh[c
+ d*x]^2)^2) - ((7*a - 4*b)*b*Tanh[c + d*x])/(8*(a + b)^4*d*(a + b - b*Tanh[c + d*x]^2))

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 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 4132

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 {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{x^4 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b \operatorname {Subst}\left (\int \frac {\frac {4}{b (a+b)}-\frac {4 a x^2}{b (a+b)^2}-\frac {3 a x^4}{(a+b)^3}}{x^4 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {8 b (a+b)-8 (a-b) b x^2-\frac {(7 a-4 b) b^2 x^4}{a+b}}{x^4 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 b (a+b)^3 d}\\ &=-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {8 b}{x^4}+\frac {8 b (-a+2 b)}{(a+b) x^2}-\frac {5 b^2 (-3 a+4 b)}{(a+b) \left (-a-b+b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{8 b (a+b)^3 d}\\ &=\frac {(a-2 b) \coth (c+d x)}{(a+b)^4 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^3 d}-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {(5 (3 a-4 b) b) \operatorname {Subst}\left (\int \frac {1}{-a-b+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^4 d}\\ &=-\frac {5 (3 a-4 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} d}+\frac {(a-2 b) \coth (c+d x)}{(a+b)^4 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^3 d}-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B]  time = 5.53, size = 985, normalized size = 5.97 \[ -\frac {(\cosh (2 (c+d x)) a+a+2 b) \text {sech}^6(c+d x) \left (\frac {\text {csch}(c) \text {sech}(2 c) \left (224 \sinh (2 c-d x) a^4-224 \sinh (2 c+d x) a^4+176 \sinh (4 c+d x) a^4+48 \sinh (2 c+3 d x) a^4-96 \sinh (4 c+3 d x) a^4+48 \sinh (6 c+3 d x) a^4+16 \sinh (2 c+5 d x) a^4+16 \sinh (6 c+5 d x) a^4+16 \sinh (4 c+7 d x) a^4+16 \sinh (8 c+7 d x) a^4+576 b \sinh (2 c-d x) a^3-657 b \sinh (2 c+d x) a^3+569 b \sinh (4 c+d x) a^3+111 b \sinh (2 c+3 d x) a^3-152 b \sinh (4 c+3 d x) a^3+192 b \sinh (6 c+3 d x) a^3+72 b \sinh (4 c+5 d x) a^3+27 b \sinh (6 c+5 d x) a^3+45 b \sinh (8 c+5 d x) a^3-83 b \sinh (4 c+7 d x) a^3+27 b \sinh (6 c+7 d x) a^3-56 b \sinh (8 c+7 d x) a^3+124 b^2 \sinh (2 c-d x) a^2-538 b^2 \sinh (2 c+d x) a^2+666 b^2 \sinh (4 c+d x) a^2+360 b^2 \sinh (2 c+3 d x) a^2+146 b^2 \sinh (4 c+3 d x) a^2+558 b^2 \sinh (6 c+3 d x) a^2-598 b^2 \sinh (2 c+5 d x) a^2+150 b^2 \sinh (4 c+5 d x) a^2-388 b^2 \sinh (6 c+5 d x) a^2-60 b^2 \sinh (8 c+5 d x) a^2+6 b^2 \sinh (4 c+7 d x) a^2-6 b^2 \sinh (6 c+7 d x) a^2-2184 b^3 \sinh (2 c-d x) a+984 b^3 \sinh (2 c+d x) a+1704 b^3 \sinh (4 c+d x) a+312 b^3 \sinh (2 c+3 d x) a-728 b^3 \sinh (4 c+3 d x) a-168 b^3 \sinh (6 c+3 d x) a+48 b^3 \sinh (2 c+5 d x) a-48 b^3 \sinh (4 c+5 d x) a+4 \left (44 a^4+122 b a^3+63 b^2 a^2+126 b^3 a+36 b^4\right ) \sinh (d x)+\left (-96 a^4-71 b a^3+344 b^2 a^2-1208 b^3 a+48 b^4\right ) \sinh (3 d x)+144 b^4 \sinh (2 c-d x)+144 b^4 \sinh (2 c+d x)-144 b^4 \sinh (4 c+d x)-48 b^4 \sinh (2 c+3 d x)-48 b^4 \sinh (4 c+3 d x)+48 b^4 \sinh (6 c+3 d x)\right ) \text {csch}^3(c+d x)}{a}+\frac {480 (3 a-4 b) b \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 (c+d x)) a+a+2 b)^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )}{6144 (a+b)^4 d \left (b \text {sech}^2(c+d x)+a\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

-1/6144*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*((480*(3*a - 4*b)*b*ArcTanh[(Sech[d*x]*(Cosh[2*c] - S
inh[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])]*(a + 2*b +
a*Cosh[2*(c + d*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (Csch[c]*Csch[c
+ d*x]^3*Sech[2*c]*(4*(44*a^4 + 122*a^3*b + 63*a^2*b^2 + 126*a*b^3 + 36*b^4)*Sinh[d*x] + (-96*a^4 - 71*a^3*b
+ 344*a^2*b^2 - 1208*a*b^3 + 48*b^4)*Sinh[3*d*x] + 224*a^4*Sinh[2*c - d*x] + 576*a^3*b*Sinh[2*c - d*x] + 124*a
^2*b^2*Sinh[2*c - d*x] - 2184*a*b^3*Sinh[2*c - d*x] + 144*b^4*Sinh[2*c - d*x] - 224*a^4*Sinh[2*c + d*x] - 657*
a^3*b*Sinh[2*c + d*x] - 538*a^2*b^2*Sinh[2*c + d*x] + 984*a*b^3*Sinh[2*c + d*x] + 144*b^4*Sinh[2*c + d*x] + 17
6*a^4*Sinh[4*c + d*x] + 569*a^3*b*Sinh[4*c + d*x] + 666*a^2*b^2*Sinh[4*c + d*x] + 1704*a*b^3*Sinh[4*c + d*x] -
144*b^4*Sinh[4*c + d*x] + 48*a^4*Sinh[2*c + 3*d*x] + 111*a^3*b*Sinh[2*c + 3*d*x] + 360*a^2*b^2*Sinh[2*c + 3*d
*x] + 312*a*b^3*Sinh[2*c + 3*d*x] - 48*b^4*Sinh[2*c + 3*d*x] - 96*a^4*Sinh[4*c + 3*d*x] - 152*a^3*b*Sinh[4*c +
3*d*x] + 146*a^2*b^2*Sinh[4*c + 3*d*x] - 728*a*b^3*Sinh[4*c + 3*d*x] - 48*b^4*Sinh[4*c + 3*d*x] + 48*a^4*Sinh
[6*c + 3*d*x] + 192*a^3*b*Sinh[6*c + 3*d*x] + 558*a^2*b^2*Sinh[6*c + 3*d*x] - 168*a*b^3*Sinh[6*c + 3*d*x] + 48
*b^4*Sinh[6*c + 3*d*x] + 16*a^4*Sinh[2*c + 5*d*x] - 598*a^2*b^2*Sinh[2*c + 5*d*x] + 48*a*b^3*Sinh[2*c + 5*d*x]
+ 72*a^3*b*Sinh[4*c + 5*d*x] + 150*a^2*b^2*Sinh[4*c + 5*d*x] - 48*a*b^3*Sinh[4*c + 5*d*x] + 16*a^4*Sinh[6*c +
5*d*x] + 27*a^3*b*Sinh[6*c + 5*d*x] - 388*a^2*b^2*Sinh[6*c + 5*d*x] + 45*a^3*b*Sinh[8*c + 5*d*x] - 60*a^2*b^2
*Sinh[8*c + 5*d*x] + 16*a^4*Sinh[4*c + 7*d*x] - 83*a^3*b*Sinh[4*c + 7*d*x] + 6*a^2*b^2*Sinh[4*c + 7*d*x] + 27*
a^3*b*Sinh[6*c + 7*d*x] - 6*a^2*b^2*Sinh[6*c + 7*d*x] + 16*a^4*Sinh[8*c + 7*d*x] - 56*a^3*b*Sinh[8*c + 7*d*x])
)/a))/((a + b)^4*d*(a + b*Sech[c + d*x]^2)^3)

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fricas [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(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 1.66, size = 406, normalized size = 2.46 \[ -\frac {\frac {15 \, {\left (3 \, a b - 4 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 20 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 66 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b - 2 \, a^{2} b^{2}\right )}}{{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} + \frac {16 \, {\left (9 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a + 7 \, b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/24*(15*(3*a*b - 4*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^4 + 4*a^3*b + 6*a^2*b
^2 + 4*a*b^3 + b^4)*sqrt(-a*b - b^2)) - 6*(9*a^3*b*e^(6*d*x + 6*c) + 20*a^2*b^2*e^(6*d*x + 6*c) + 27*a^3*b*e^(
4*d*x + 4*c) + 66*a^2*b^2*e^(4*d*x + 4*c) + 56*a*b^3*e^(4*d*x + 4*c) - 16*b^4*e^(4*d*x + 4*c) + 27*a^3*b*e^(2*
d*x + 2*c) + 44*a^2*b^2*e^(2*d*x + 2*c) - 16*a*b^3*e^(2*d*x + 2*c) + 9*a^3*b - 2*a^2*b^2)/((a^5 + 4*a^4*b + 6*
a^3*b^2 + 4*a^2*b^3 + a*b^4)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2) + 16*(9*b*
e^(4*d*x + 4*c) + 6*a*e^(2*d*x + 2*c) - 12*b*e^(2*d*x + 2*c) - 2*a + 7*b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^
3 + b^4)*(e^(2*d*x + 2*c) - 1)^3))/d

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maple [B]  time = 0.48, size = 1443, normalized size = 8.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)

[Out]

-1/24/d/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+b)*a*tanh(1/2*d*x+1/2*c)^3-1/24/d/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+b)*tanh(1/
2*d*x+1/2*c)^3*b+3/8/d/(a^3+3*a^2*b+3*a*b^2+b^3)/(a+b)*a*tanh(1/2*d*x+1/2*c)-9/8/d/(a^3+3*a^2*b+3*a*b^2+b^3)/(
a+b)*tanh(1/2*d*x+1/2*c)*b-9/4/d*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2
*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7*a^2-5/4/d*b^2/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+
b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7*a+1/d
*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^
2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^7-27/4/d*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*
d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^5*a^2+13/4/d*b^2/(a+b)^4/(tanh(1/2*d*x+1/2
*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)
^5*a-1/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x
+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^5-27/4/d*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*t
anh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^3*a^2+13/4/d*b^2/(a+b)^4/(tanh(1/2
*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*
x+1/2*c)^3*a-1/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh
(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^3-9/4/d*b/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c
)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)*a^2-5/4/d*b^2/(a+b)^4/(tanh
(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/
2*d*x+1/2*c)*a+1/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*ta
nh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)+15/16/d*b^(1/2)/(a+b)^(9/2)*a*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/
2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)-(a+b)^(1/2))-15/16/d*b^(1/2)/(a+b)^(9/2)*a*ln((a+b)^(1/2)*tanh(1/2*d*x+1/
2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)+(a+b)^(1/2))-5/4/d*b^(3/2)/(a+b)^(9/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c
)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)-(a+b)^(1/2))+5/4/d*b^(3/2)/(a+b)^(9/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+
2*b^(1/2)*tanh(1/2*d*x+1/2*c)+(a+b)^(1/2))-1/24/d/(a+b)^3/tanh(1/2*d*x+1/2*c)^3+3/8/d/(a+b)^4/tanh(1/2*d*x+1/2
*c)*a-9/8/d/(a+b)^4/tanh(1/2*d*x+1/2*c)*b

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maxima [B]  time = 0.60, size = 782, normalized size = 4.74 \[ \frac {5 \, {\left (3 \, a b - 4 \, 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 )}{16 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {16 \, a^{4} - 83 \, a^{3} b + 6 \, a^{2} b^{2} + 2 \, {\left (8 \, a^{4} - 299 \, a^{2} b^{2} + 24 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (96 \, a^{4} + 71 \, a^{3} b - 344 \, a^{2} b^{2} + 1208 \, a b^{3} - 48 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (56 \, a^{4} + 144 \, a^{3} b + 31 \, a^{2} b^{2} - 546 \, a b^{3} + 36 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (176 \, a^{4} + 569 \, a^{3} b + 666 \, a^{2} b^{2} + 1704 \, a b^{3} - 144 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - 6 \, {\left (8 \, a^{4} + 32 \, a^{3} b + 93 \, a^{2} b^{2} - 28 \, a b^{3} + 8 \, b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} - 15 \, {\left (3 \, a^{3} b - 4 \, a^{2} b^{2}\right )} e^{\left (-12 \, d x - 12 \, c\right )}}{12 \, {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4} + {\left (a^{7} + 12 \, a^{6} b + 38 \, a^{5} b^{2} + 52 \, a^{4} b^{3} + 33 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (3 \, a^{7} + 20 \, a^{6} b + 34 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - 61 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 16 \, a b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{7} + 28 \, a^{6} b + 130 \, a^{5} b^{2} + 300 \, a^{4} b^{3} + 355 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{7} + 28 \, a^{6} b + 130 \, a^{5} b^{2} + 300 \, a^{4} b^{3} + 355 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (3 \, a^{7} + 20 \, a^{6} b + 34 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - 61 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 16 \, a b^{6}\right )} e^{\left (-10 \, d x - 10 \, c\right )} - {\left (a^{7} + 12 \, a^{6} b + 38 \, a^{5} b^{2} + 52 \, a^{4} b^{3} + 33 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-12 \, d x - 12 \, c\right )} - {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} e^{\left (-14 \, d x - 14 \, c\right )}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

5/16*(3*a*b - 4*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^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 1/12*(16*a^4 - 83*a^3*b +
6*a^2*b^2 + 2*(8*a^4 - 299*a^2*b^2 + 24*a*b^3)*e^(-2*d*x - 2*c) - (96*a^4 + 71*a^3*b - 344*a^2*b^2 + 1208*a*b^
3 - 48*b^4)*e^(-4*d*x - 4*c) - 4*(56*a^4 + 144*a^3*b + 31*a^2*b^2 - 546*a*b^3 + 36*b^4)*e^(-6*d*x - 6*c) - (17
6*a^4 + 569*a^3*b + 666*a^2*b^2 + 1704*a*b^3 - 144*b^4)*e^(-8*d*x - 8*c) - 6*(8*a^4 + 32*a^3*b + 93*a^2*b^2 -
28*a*b^3 + 8*b^4)*e^(-10*d*x - 10*c) - 15*(3*a^3*b - 4*a^2*b^2)*e^(-12*d*x - 12*c))/((a^7 + 4*a^6*b + 6*a^5*b^
2 + 4*a^4*b^3 + a^3*b^4 + (a^7 + 12*a^6*b + 38*a^5*b^2 + 52*a^4*b^3 + 33*a^3*b^4 + 8*a^2*b^5)*e^(-2*d*x - 2*c)
- (3*a^7 + 20*a^6*b + 34*a^5*b^2 - 4*a^4*b^3 - 61*a^3*b^4 - 56*a^2*b^5 - 16*a*b^6)*e^(-4*d*x - 4*c) - (3*a^7
+ 28*a^6*b + 130*a^5*b^2 + 300*a^4*b^3 + 355*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*e^(-6*d*x - 6*c) + (3*a^7 + 28*
a^6*b + 130*a^5*b^2 + 300*a^4*b^3 + 355*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*e^(-8*d*x - 8*c) + (3*a^7 + 20*a^6*b
+ 34*a^5*b^2 - 4*a^4*b^3 - 61*a^3*b^4 - 56*a^2*b^5 - 16*a*b^6)*e^(-10*d*x - 10*c) - (a^7 + 12*a^6*b + 38*a^5*
b^2 + 52*a^4*b^3 + 33*a^3*b^4 + 8*a^2*b^5)*e^(-12*d*x - 12*c) - (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b
^4)*e^(-14*d*x - 14*c))*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

int(cosh(c + d*x)^6/(sinh(c + d*x)^4*(b + a*cosh(c + d*x)^2)^3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral(csch(c + d*x)**4/(a + b*sech(c + d*x)**2)**3, x)

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