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

Optimal. Leaf size=70 \[ \frac{b^2 (a+b) \tanh ^3(c+d x)}{d}-\frac{3 b (a+b)^2 \tanh (c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d}-\frac{b^3 \tanh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]  time = 0.0734432, antiderivative size = 70, 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, 270} \[ \frac{b^2 (a+b) \tanh ^3(c+d x)}{d}-\frac{3 b (a+b)^2 \tanh (c+d x)}{d}-\frac{(a+b)^3 \coth (c+d x)}{d}-\frac{b^3 \tanh ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b)^3*Coth[c + d*x])/d) - (3*b*(a + b)^2*Tanh[c + d*x])/d + (b^2*(a + b)*Tanh[c + d*x]^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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [B]  time = 2.73583, size = 380, normalized size = 5.43 \[ -\frac{\text{csch}(c) \text{sech}(c) \coth (c+d x) \left (10 a \left (5 a^2+12 a b+8 b^2\right ) \sinh (2 c)-10 \left (18 a^2 b+5 a^3+20 a b^2+8 b^3\right ) \sinh (2 d x)-120 a^2 b \sinh (2 (c+2 d x))+30 a^2 b \sinh (4 c+2 d x)-30 a^2 b \sinh (4 c+6 d x)-25 a^3 \sinh (2 (c+d x))-20 a^3 \sinh (4 (c+d x))-5 a^3 \sinh (6 (c+d x))-25 a^3 \sinh (2 (c+2 d x))+25 a^3 \sinh (4 c+2 d x)+5 a^3 \sinh (6 c+4 d x)-5 a^3 \sinh (4 c+6 d x)+50 a b^2 \sinh (2 (c+d x))+40 a b^2 \sinh (4 (c+d x))+10 a b^2 \sinh (6 (c+d x))-160 a b^2 \sinh (2 (c+2 d x))-40 a b^2 \sinh (4 c+6 d x)+30 b^3 \sinh (2 (c+d x))+24 b^3 \sinh (4 (c+d x))+6 b^3 \sinh (6 (c+d x))-64 b^3 \sinh (2 (c+2 d x))-16 b^3 \sinh (4 c+6 d x)\right ) \left (a+b \text{sech}^2(c+d x)\right )^3}{40 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Coth[c + d*x]*Csch[c]*Sech[c]*(a + b*Sech[c + d*x]^2)^3*(10*a*(5*a^2 + 12*a*b + 8*b^2)*Sinh[2*c] - 10*(5*a^3
+ 18*a^2*b + 20*a*b^2 + 8*b^3)*Sinh[2*d*x] - 25*a^3*Sinh[2*(c + d*x)] + 50*a*b^2*Sinh[2*(c + d*x)] + 30*b^3*S
inh[2*(c + d*x)] - 20*a^3*Sinh[4*(c + d*x)] + 40*a*b^2*Sinh[4*(c + d*x)] + 24*b^3*Sinh[4*(c + d*x)] - 5*a^3*Si
nh[6*(c + d*x)] + 10*a*b^2*Sinh[6*(c + d*x)] + 6*b^3*Sinh[6*(c + d*x)] - 25*a^3*Sinh[2*(c + 2*d*x)] - 120*a^2*
b*Sinh[2*(c + 2*d*x)] - 160*a*b^2*Sinh[2*(c + 2*d*x)] - 64*b^3*Sinh[2*(c + 2*d*x)] + 25*a^3*Sinh[4*c + 2*d*x]
+ 30*a^2*b*Sinh[4*c + 2*d*x] + 5*a^3*Sinh[6*c + 4*d*x] - 5*a^3*Sinh[4*c + 6*d*x] - 30*a^2*b*Sinh[4*c + 6*d*x]
- 40*a*b^2*Sinh[4*c + 6*d*x] - 16*b^3*Sinh[4*c + 6*d*x]))/(40*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

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Maple [B]  time = 0.043, size = 148, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ( -{a}^{3}{\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( -{\frac{1}{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }}-2\,\tanh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( -{\frac{1}{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-4\, \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}{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-6\, \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)^2*(a+b*sech(d*x+c)^2)^3,x)

[Out]

1/d*(-a^3*coth(d*x+c)+3*a^2*b*(-1/sinh(d*x+c)/cosh(d*x+c)-2*tanh(d*x+c))+3*a*b^2*(-1/sinh(d*x+c)/cosh(d*x+c)^3
-4*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/sinh(d*x+c)/cosh(d*x+c)^5-6*(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 = 1.05741, size = 483, normalized size = 6.9 \begin{align*} -\frac{32}{5} \, b^{3}{\left (\frac{4 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} - 4 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}} + \frac{5 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} - 4 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}} + \frac{1}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} - 4 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - 16 \, a b^{2}{\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 )} + \frac{2 \, a^{3}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac{12 \, a^{2} b}{d{\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/5*b^3*(4*e^(-2*d*x - 2*c)/(d*(4*e^(-2*d*x - 2*c) + 5*e^(-4*d*x - 4*c) - 5*e^(-8*d*x - 8*c) - 4*e^(-10*d*x
- 10*c) - e^(-12*d*x - 12*c) + 1)) + 5*e^(-4*d*x - 4*c)/(d*(4*e^(-2*d*x - 2*c) + 5*e^(-4*d*x - 4*c) - 5*e^(-8*
d*x - 8*c) - 4*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) + 1)) + 1/(d*(4*e^(-2*d*x - 2*c) + 5*e^(-4*d*x - 4*c) -
5*e^(-8*d*x - 8*c) - 4*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) + 1))) - 16*a*b^2*(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))) + 2*a^3/(d*(e^(-2*d*x - 2*c) - 1)) + 12*a^2*b/(d*(e^(-4*d*x - 4*c) - 1))

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Fricas [B]  time = 2.54886, size = 1580, normalized size = 22.57 \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)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-4/5*((5*a^3 + 15*a^2*b + 20*a*b^2 + 8*b^3)*cosh(d*x + c)^5 + 5*(5*a^3 + 15*a^2*b + 20*a*b^2 + 8*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^4 - (15*a^2*b + 20*a*b^2 + 8*b^3)*sinh(d*x + c)^5 + (25*a^3 + 75*a^2*b + 80*a*b^2 + 32*b^3
)*cosh(d*x + c)^3 - (45*a^2*b + 80*a*b^2 + 32*b^3 + 10*(15*a^2*b + 20*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^3 + (10*(5*a^3 + 15*a^2*b + 20*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + 3*(25*a^3 + 75*a^2*b + 80*a*b^2 + 32*b^3
)*cosh(d*x + c))*sinh(d*x + c)^2 + 10*(5*a^3 + 15*a^2*b + 14*a*b^2 + 4*b^3)*cosh(d*x + c) - (5*(15*a^2*b + 20*
a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 30*a^2*b + 60*a*b^2 + 40*b^3 + 3*(45*a^2*b + 80*a*b^2 + 32*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7 + 3*d*cosh(d*x +
c)^5 + (21*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^5 + 5*(7*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x +
c)^4 + d*cosh(d*x + c)^3 + (35*d*cosh(d*x + c)^4 + 50*d*cosh(d*x + c)^2 + 9*d)*sinh(d*x + c)^3 + 3*(7*d*cosh(d
*x + c)^5 + 10*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^2 - 5*d*cosh(d*x + c) + (7*d*cosh(d*x + c)^6
+ 25*d*cosh(d*x + c)^4 + 27*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c))

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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}^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.19468, size = 338, normalized size = 4.83 \begin{align*} -\frac{2 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} + \frac{2 \,{\left (15 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 15 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 5 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 30 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 160 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 110 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 50 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} b + 25 \, a b^{2} + 11 \, b^{3}\right )}}{5 \, 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)^2*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(d*(e^(2*d*x + 2*c) - 1)) + 2/5*(15*a^2*b*e^(8*d*x + 8*c) + 15*a*b^2*e^(8*d
*x + 8*c) + 5*b^3*e^(8*d*x + 8*c) + 60*a^2*b*e^(6*d*x + 6*c) + 90*a*b^2*e^(6*d*x + 6*c) + 30*b^3*e^(6*d*x + 6*
c) + 90*a^2*b*e^(4*d*x + 4*c) + 160*a*b^2*e^(4*d*x + 4*c) + 80*b^3*e^(4*d*x + 4*c) + 60*a^2*b*e^(2*d*x + 2*c)
+ 110*a*b^2*e^(2*d*x + 2*c) + 50*b^3*e^(2*d*x + 2*c) + 15*a^2*b + 25*a*b^2 + 11*b^3)/(d*(e^(2*d*x + 2*c) + 1)^
5)