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

Optimal. Leaf size=104 \[ -\frac {b^2 (3 a+4 b) \tanh ^3(c+d x)}{3 d}+\frac {3 b (a+b) (a+2 b) \tanh (c+d x)}{d}-\frac {(a+b)^3 \coth ^3(c+d x)}{3 d}+\frac {(a+b)^2 (a+4 b) \coth (c+d x)}{d}+\frac {b^3 \tanh ^5(c+d x)}{5 d} \]

[Out]

(a+b)^2*(a+4*b)*coth(d*x+c)/d-1/3*(a+b)^3*coth(d*x+c)^3/d+3*b*(a+b)*(a+2*b)*tanh(d*x+c)/d-1/3*b^2*(3*a+4*b)*ta
nh(d*x+c)^3/d+1/5*b^3*tanh(d*x+c)^5/d

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Rubi [A]  time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4132, 448} \[ -\frac {b^2 (3 a+4 b) \tanh ^3(c+d x)}{3 d}+\frac {3 b (a+b) (a+2 b) \tanh (c+d x)}{d}-\frac {(a+b)^3 \coth ^3(c+d x)}{3 d}+\frac {(a+b)^2 (a+4 b) \coth (c+d x)}{d}+\frac {b^3 \tanh ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 448

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

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

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Mathematica [B]  time = 2.47, size = 213, normalized size = 2.05 \[ -\frac {8 \tanh (c) \text {sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (\text {csch}(c) \sinh (d x) \cosh ^4(c+d x) \left (5 (a+b)^2 (2 a+11 b) \coth (c) \coth (c+d x)-b \left (45 a^2+120 a b+73 b^2\right )\right )+\cosh ^3(c+d x) \left (5 (a+b)^3 \coth ^2(c) \coth ^2(c+d x)-b^2 (15 a+14 b)\right )-\text {csch}(c) \sinh (d x) \cosh ^2(c+d x) \left (b^2 (15 a+14 b)+5 (a+b)^3 \coth (c) \coth ^3(c+d x)\right )-3 b^3 \cosh (c+d x)-3 b^3 \text {csch}(c) \sinh (d x)\right )}{15 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^5*(-3*b^3*Cosh[c + d*x] + Cosh[c + d*x]^3*(-(b^2*(15*a + 14*b)) +
5*(a + b)^3*Coth[c]^2*Coth[c + d*x]^2) - 3*b^3*Csch[c]*Sinh[d*x] + Cosh[c + d*x]^4*(-(b*(45*a^2 + 120*a*b + 73
*b^2)) + 5*(a + b)^2*(2*a + 11*b)*Coth[c]*Coth[c + d*x])*Csch[c]*Sinh[d*x] - Cosh[c + d*x]^2*(b^2*(15*a + 14*b
) + 5*(a + b)^3*Coth[c]*Coth[c + d*x]^3)*Csch[c]*Sinh[d*x])*Tanh[c])/(15*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

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fricas [B]  time = 0.59, size = 955, normalized size = 9.18 \[ \text {result too large to display} \]

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]

-8/15*((5*a^3 - 30*a^2*b - 60*a*b^2 - 32*b^3)*cosh(d*x + c)^6 + 12*(5*a^3 + 15*a^2*b + 30*a*b^2 + 16*b^3)*cosh
(d*x + c)*sinh(d*x + c)^5 + (5*a^3 - 30*a^2*b - 60*a*b^2 - 32*b^3)*sinh(d*x + c)^6 + 2*(15*a^3 - 60*a*b^2 - 32
*b^3)*cosh(d*x + c)^4 + (30*a^3 - 120*a*b^2 - 64*b^3 + 15*(5*a^3 - 30*a^2*b - 60*a*b^2 - 32*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^4 + 8*(5*(5*a^3 + 15*a^2*b + 30*a*b^2 + 16*b^3)*cosh(d*x + c)^3 + 4*(5*a^3 + 15*a^2*b + 15*a
*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 50*a^3 + 240*a^2*b + 360*a*b^2 + 192*b^3 + (75*a^3 + 270*a^2*b
+ 300*a*b^2 + 64*b^3)*cosh(d*x + c)^2 + (15*(5*a^3 - 30*a^2*b - 60*a*b^2 - 32*b^3)*cosh(d*x + c)^4 + 75*a^3 +
270*a^2*b + 300*a*b^2 + 64*b^3 + 12*(15*a^3 - 60*a*b^2 - 32*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*(5*a^
3 + 15*a^2*b + 30*a*b^2 + 16*b^3)*cosh(d*x + c)^5 + 8*(5*a^3 + 15*a^2*b + 15*a*b^2 + 8*b^3)*cosh(d*x + c)^3 +
(25*a^3 + 75*a^2*b + 30*a*b^2 - 32*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)
*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 + 2*d*cosh(d*x + c)^8 + (45*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8
*(15*d*cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sinh(d*x + c)^7 - 3*d*cosh(d*x + c)^6 + (210*d*cosh(d*x + c)^4 + 5
6*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^6 + 2*(126*d*cosh(d*x + c)^5 + 56*d*cosh(d*x + c)^3 - 3*d*cosh(d*x +
c))*sinh(d*x + c)^5 - 8*d*cosh(d*x + c)^4 + (210*d*cosh(d*x + c)^6 + 140*d*cosh(d*x + c)^4 - 45*d*cosh(d*x + c
)^2 - 8*d)*sinh(d*x + c)^4 + 4*(30*d*cosh(d*x + c)^7 + 28*d*cosh(d*x + c)^5 - 5*d*cosh(d*x + c)^3 - 4*d*cosh(d
*x + c))*sinh(d*x + c)^3 + 2*d*cosh(d*x + c)^2 + (45*d*cosh(d*x + c)^8 + 56*d*cosh(d*x + c)^6 - 45*d*cosh(d*x
+ c)^4 - 48*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x + c)^9 + 8*d*cosh(d*x + c)^7 - 3*d*cosh
(d*x + c)^5 - 8*d*cosh(d*x + c)^3 - 2*d*cosh(d*x + c))*sinh(d*x + c) + 6*d)

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giac [B]  time = 0.19, size = 355, normalized size = 3.41 \[ \frac {2 \, {\left (\frac {5 \, {\left (9 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 36 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 54 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} + 15 \, a^{2} b + 24 \, a b^{2} + 11 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac {45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 90 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 45 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 450 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 240 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 750 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 490 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 510 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 320 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b + 120 \, a b^{2} + 73 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}\right )}}{15 \, d} \]

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]

2/15*(5*(9*a^2*b*e^(4*d*x + 4*c) + 18*a*b^2*e^(4*d*x + 4*c) + 9*b^3*e^(4*d*x + 4*c) - 6*a^3*e^(2*d*x + 2*c) -
36*a^2*b*e^(2*d*x + 2*c) - 54*a*b^2*e^(2*d*x + 2*c) - 24*b^3*e^(2*d*x + 2*c) + 2*a^3 + 15*a^2*b + 24*a*b^2 + 1
1*b^3)/(e^(2*d*x + 2*c) - 1)^3 - (45*a^2*b*e^(8*d*x + 8*c) + 90*a*b^2*e^(8*d*x + 8*c) + 45*b^3*e^(8*d*x + 8*c)
+ 180*a^2*b*e^(6*d*x + 6*c) + 450*a*b^2*e^(6*d*x + 6*c) + 240*b^3*e^(6*d*x + 6*c) + 270*a^2*b*e^(4*d*x + 4*c)
+ 750*a*b^2*e^(4*d*x + 4*c) + 490*b^3*e^(4*d*x + 4*c) + 180*a^2*b*e^(2*d*x + 2*c) + 510*a*b^2*e^(2*d*x + 2*c)
+ 320*b^3*e^(2*d*x + 2*c) + 45*a^2*b + 120*a*b^2 + 73*b^3)/(e^(2*d*x + 2*c) + 1)^5)/d

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maple [B]  time = 0.66, size = 213, normalized size = 2.05 \[ \frac {a^{3} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )}+\frac {4}{3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}+\frac {8 \tanh \left (d x +c \right )}{3}\right )+3 a \,b^{2} \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )^{3}}+\frac {2}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}+8 \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )+b^{3} \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )^{5}}+\frac {8}{3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{5}}+16 \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )\right )}{d} \]

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/d*(a^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(-1/3/sinh(d*x+c)^3/cosh(d*x+c)+4/3/sinh(d*x+c)/cosh(d*x+
c)+8/3*tanh(d*x+c))+3*a*b^2*(-1/3/sinh(d*x+c)^3/cosh(d*x+c)^3+2/sinh(d*x+c)/cosh(d*x+c)^3+8*(2/3+1/3*sech(d*x+
c)^2)*tanh(d*x+c))+b^3*(-1/3/sinh(d*x+c)^3/cosh(d*x+c)^5+8/3/sinh(d*x+c)/cosh(d*x+c)^5+16*(8/15+1/5*sech(d*x+c
)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c)))

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maxima [B]  time = 0.34, size = 664, normalized size = 6.38 \[ \frac {4}{3} \, a^{3} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {256}{15} \, b^{3} {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, e^{\left (-6 \, d x - 6 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + 2 \, e^{\left (-12 \, d x - 12 \, c\right )} - 2 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} - \frac {2 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, e^{\left (-6 \, d x - 6 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + 2 \, e^{\left (-12 \, d x - 12 \, c\right )} - 2 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} - \frac {6 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, e^{\left (-6 \, d x - 6 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + 2 \, e^{\left (-12 \, d x - 12 \, c\right )} - 2 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}} + \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-4 \, d x - 4 \, c\right )} - 6 \, e^{\left (-6 \, d x - 6 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + 2 \, e^{\left (-12 \, d x - 12 \, c\right )} - 2 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} + 16 \, a^{2} b {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} - \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 32 \, a b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} \]

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]

4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(
-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) + 256/15*b^3*(2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x
- 2*c) - 2*e^(-4*d*x - 4*c) - 6*e^(-6*d*x - 6*c) + 6*e^(-10*d*x - 10*c) + 2*e^(-12*d*x - 12*c) - 2*e^(-14*d*x
- 14*c) - e^(-16*d*x - 16*c) + 1)) - 2*e^(-4*d*x - 4*c)/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-4*d*x - 4*c) - 6*e^(-6*
d*x - 6*c) + 6*e^(-10*d*x - 10*c) + 2*e^(-12*d*x - 12*c) - 2*e^(-14*d*x - 14*c) - e^(-16*d*x - 16*c) + 1)) - 6
*e^(-6*d*x - 6*c)/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-4*d*x - 4*c) - 6*e^(-6*d*x - 6*c) + 6*e^(-10*d*x - 10*c) + 2*
e^(-12*d*x - 12*c) - 2*e^(-14*d*x - 14*c) - e^(-16*d*x - 16*c) + 1)) + 1/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-4*d*x
- 4*c) - 6*e^(-6*d*x - 6*c) + 6*e^(-10*d*x - 10*c) + 2*e^(-12*d*x - 12*c) - 2*e^(-14*d*x - 14*c) - e^(-16*d*x
- 16*c) + 1))) + 16*a^2*b*(2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) -
1)) - 1/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) - 1))) + 32*a*b^2*(3*e^(-4*d*x - 4*c)/
(d*(3*e^(-4*d*x - 4*c) - 3*e^(-8*d*x - 8*c) + e^(-12*d*x - 12*c) - 1)) - 1/(d*(3*e^(-4*d*x - 4*c) - 3*e^(-8*d*
x - 8*c) + e^(-12*d*x - 12*c) - 1)))

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mupad [B]  time = 1.60, size = 745, normalized size = 7.16 \[ \frac {6\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {\frac {2\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b+30\,a\,b^2+25\,b^3\right )}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {6\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b+30\,a\,b^2+25\,b^3\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {\frac {2\,\left (9\,a^2\,b+30\,a\,b^2+25\,b^3\right )}{15\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {\frac {2\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {6\,\left (a^2\,b+2\,a\,b^2+b^3\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*(2*a*b^2 + a^2*b + b^3))/(d*(exp(2*c + 2*d*x) - 1)) - ((2*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (6*exp(6*c +
6*d*x)*(2*a*b^2 + a^2*b + b^3))/(5*d) + (6*exp(4*c + 4*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (2*exp(2*c +
2*d*x)*(30*a*b^2 + 9*a^2*b + 25*b^3))/(5*d))/(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) +
exp(8*c + 8*d*x) + 1) - ((6*(2*a*b^2 + a^2*b + b^3))/(5*d) + (6*exp(8*c + 8*d*x)*(2*a*b^2 + a^2*b + b^3))/(5*d
) + (8*exp(2*c + 2*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (8*exp(6*c + 6*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/
(5*d) + (4*exp(4*c + 4*d*x)*(30*a*b^2 + 9*a^2*b + 25*b^3))/(5*d))/(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) +
10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1) - ((2*(30*a*b^2 + 9*a^2*b + 25*b^3))/(15*d)
+ (6*exp(4*c + 4*d*x)*(2*a*b^2 + a^2*b + b^3))/(5*d) + (4*exp(2*c + 2*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d)
)/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - (4*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(d*(e
xp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - ((2*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (6*exp(2*c + 2*d*x)*(2*a
*b^2 + a^2*b + b^3))/(5*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - (6*(2*a*b^2 + a^2*b + b^3))/(5*d*(ex
p(2*c + 2*d*x) + 1)) - (8*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp
(6*c + 6*d*x) - 1))

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

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

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