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[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)

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Rubi [A]  time = 0.105049, antiderivative size = 104, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [B]  time = 2.30334, 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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Maple [B]  time = 0.051, size = 213, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( -1/3\,{\frac{1}{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}\cosh \left ( dx+c \right ) }}+4/3\,{\frac{1}{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}+8/3\,\tanh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( -1/3\,{\frac{1}{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+2\,{\frac{1}{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+8\, \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{1}{3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8}{3\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+16\, \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) \right ) } \end{align*}

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 = 1.09577, size = 896, normalized size = 8.62 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 2.55822, size = 2480, normalized size = 23.85 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname{csch}^{4}{\left (c + d x \right )}\, dx \end{align*}

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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Giac [B]  time = 1.23254, size = 481, normalized size = 4.62 \begin{align*} \frac{2 \,{\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 )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac{2 \,{\left (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}\right )}}{15 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}} \end{align*}

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]

2/3*(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 + 11*b^
3)/(d*(e^(2*d*x + 2*c) - 1)^3) - 2/15*(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)/(d*(e^(2*d*x + 2*c) + 1)^5)