3.56 $$\int \text{csch}^4(c+d x) (a+b \tanh ^3(c+d x)) \, dx$$

Optimal. Leaf size=56 $-\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}-\frac{b \tanh ^2(c+d x)}{2 d}+\frac{b \log (\tanh (c+d x))}{d}$

[Out]

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

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Rubi [A]  time = 0.0582295, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.095, Rules used = {3663, 1802} $-\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}-\frac{b \tanh ^2(c+d x)}{2 d}+\frac{b \log (\tanh (c+d x))}{d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3663

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

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.196557, size = 74, normalized size = 1.32 $\frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}-\frac{b \left (-\text{sech}^2(c+d x)-2 \log (\sinh (c+d x))+2 \log (\cosh (c+d x))\right )}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*a*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*(2*Log[Cosh[c + d*x]] - 2*Log[Sinh[c
+ d*x]] - Sech[c + d*x]^2))/(2*d)

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Maple [A]  time = 0.061, size = 60, normalized size = 1.1 \begin{align*}{\frac{2\,{\rm coth} \left (dx+c\right )a}{3\,d}}-{\frac{{\rm coth} \left (dx+c\right )a \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3\,d}}+{\frac{b}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3),x)

[Out]

2/3*a*coth(d*x+c)/d-1/3/d*a*coth(d*x+c)*csch(d*x+c)^2+1/2/d*b/cosh(d*x+c)^2+b*ln(tanh(d*x+c))/d

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Maxima [B]  time = 1.5771, size = 248, normalized size = 4.43 \begin{align*} b{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac{4}{3} \, a{\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 )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*
e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 4/3*a*(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)))

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Fricas [B]  time = 2.45432, size = 4702, normalized size = 83.96 \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*tanh(d*x+c)^3),x, algorithm="fricas")

[Out]

1/3*(6*b*cosh(d*x + c)^8 + 48*b*cosh(d*x + c)*sinh(d*x + c)^7 + 6*b*sinh(d*x + c)^8 - 6*(2*a + 3*b)*cosh(d*x +
c)^6 + 6*(28*b*cosh(d*x + c)^2 - 2*a - 3*b)*sinh(d*x + c)^6 + 12*(28*b*cosh(d*x + c)^3 - 3*(2*a + 3*b)*cosh(d
*x + c))*sinh(d*x + c)^5 - 2*(10*a - 9*b)*cosh(d*x + c)^4 + 2*(210*b*cosh(d*x + c)^4 - 45*(2*a + 3*b)*cosh(d*x
+ c)^2 - 10*a + 9*b)*sinh(d*x + c)^4 + 8*(42*b*cosh(d*x + c)^5 - 15*(2*a + 3*b)*cosh(d*x + c)^3 - (10*a - 9*b
)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a + 3*b)*cosh(d*x + c)^2 + 2*(84*b*cosh(d*x + c)^6 - 45*(2*a + 3*b)*co
sh(d*x + c)^4 - 6*(10*a - 9*b)*cosh(d*x + c)^2 - 2*a - 3*b)*sinh(d*x + c)^2 - 3*(b*cosh(d*x + c)^10 + 10*b*cos
h(d*x + c)*sinh(d*x + c)^9 + b*sinh(d*x + c)^10 - b*cosh(d*x + c)^8 + (45*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)
^8 + 8*(15*b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c)^7 - 2*b*cosh(d*x + c)^6 + 2*(105*b*cosh(d*x + c)
^4 - 14*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^6 + 4*(63*b*cosh(d*x + c)^5 - 14*b*cosh(d*x + c)^3 - 3*b*cosh(d*x
+ c))*sinh(d*x + c)^5 + 2*b*cosh(d*x + c)^4 + 2*(105*b*cosh(d*x + c)^6 - 35*b*cosh(d*x + c)^4 - 15*b*cosh(d*x
+ c)^2 + b)*sinh(d*x + c)^4 + 8*(15*b*cosh(d*x + c)^7 - 7*b*cosh(d*x + c)^5 - 5*b*cosh(d*x + c)^3 + b*cosh(d*
x + c))*sinh(d*x + c)^3 + b*cosh(d*x + c)^2 + (45*b*cosh(d*x + c)^8 - 28*b*cosh(d*x + c)^6 - 30*b*cosh(d*x + c
)^4 + 12*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 2*(5*b*cosh(d*x + c)^9 - 4*b*cosh(d*x + c)^7 - 6*b*cosh(d*x
+ c)^5 + 4*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d
*x + c))) + 3*(b*cosh(d*x + c)^10 + 10*b*cosh(d*x + c)*sinh(d*x + c)^9 + b*sinh(d*x + c)^10 - b*cosh(d*x + c)^
8 + (45*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^8 + 8*(15*b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c)^7 -
2*b*cosh(d*x + c)^6 + 2*(105*b*cosh(d*x + c)^4 - 14*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^6 + 4*(63*b*cosh(d*x
+ c)^5 - 14*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^5 + 2*b*cosh(d*x + c)^4 + 2*(105*b*cosh(d*x +
c)^6 - 35*b*cosh(d*x + c)^4 - 15*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^4 + 8*(15*b*cosh(d*x + c)^7 - 7*b*cosh(
d*x + c)^5 - 5*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c)^3 + b*cosh(d*x + c)^2 + (45*b*cosh(d*x + c)^
8 - 28*b*cosh(d*x + c)^6 - 30*b*cosh(d*x + c)^4 + 12*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 2*(5*b*cosh(d*x
+ c)^9 - 4*b*cosh(d*x + c)^7 - 6*b*cosh(d*x + c)^5 + 4*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)
*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(12*b*cosh(d*x + c)^7 - 9*(2*a + 3*b)*cosh(d*x + c)^
5 - 2*(10*a - 9*b)*cosh(d*x + c)^3 - (2*a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4*a)/(d*cosh(d*x + c)^10 + 10*
d*cosh(d*x + c)*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 - d*cosh(d*x + c)^8 + (45*d*cosh(d*x + c)^2 - d)*sinh(d*x
+ c)^8 + 8*(15*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^7 - 2*d*cosh(d*x + c)^6 + 2*(105*d*cosh(d*x
+ c)^4 - 14*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^6 + 4*(63*d*cosh(d*x + c)^5 - 14*d*cosh(d*x + c)^3 - 3*d*cos
h(d*x + c))*sinh(d*x + c)^5 + 2*d*cosh(d*x + c)^4 + 2*(105*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 - 15*d*cos
h(d*x + c)^2 + d)*sinh(d*x + c)^4 + 8*(15*d*cosh(d*x + c)^7 - 7*d*cosh(d*x + c)^5 - 5*d*cosh(d*x + c)^3 + d*co
sh(d*x + c))*sinh(d*x + c)^3 + d*cosh(d*x + c)^2 + (45*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 - 30*d*cosh(d*
x + c)^4 + 12*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x + c)^9 - 4*d*cosh(d*x + c)^7 - 6*d*cosh
(d*x + c)^5 + 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) - d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \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*tanh(d*x+c)**3),x)

[Out]

Integral((a + b*tanh(c + d*x)**3)*csch(c + d*x)**4, x)

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Giac [B]  time = 1.2359, size = 201, normalized size = 3.59 \begin{align*} -\frac{6 \, b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 6 \, b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) - \frac{3 \,{\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}} + \frac{11 \, b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a e^{\left (2 \, d x + 2 \, c\right )} + 33 \, b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a - 11 \, b}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*b*log(e^(2*d*x + 2*c) + 1) - 6*b*log(abs(e^(2*d*x + 2*c) - 1)) - 3*(3*b*e^(4*d*x + 4*c) + 10*b*e^(2*d*
x + 2*c) + 3*b)/(e^(2*d*x + 2*c) + 1)^2 + (11*b*e^(6*d*x + 6*c) - 33*b*e^(4*d*x + 4*c) + 24*a*e^(2*d*x + 2*c)
+ 33*b*e^(2*d*x + 2*c) - 8*a - 11*b)/(e^(2*d*x + 2*c) - 1)^3)/d