3.2 \(\int \sinh ^3(c+d x) (a+b \tanh ^2(c+d x)) \, dx\)

Optimal. Leaf size=47 \[ \frac {(a+b) \cosh ^3(c+d x)}{3 d}-\frac {(a+2 b) \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]

[Out]

-(a+2*b)*cosh(d*x+c)/d+1/3*(a+b)*cosh(d*x+c)^3/d-b*sech(d*x+c)/d

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {3664, 448} \[ \frac {(a+b) \cosh ^3(c+d x)}{3 d}-\frac {(a+2 b) \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + 2*b)*Cosh[c + d*x])/d) + ((a + b)*Cosh[c + d*x]^3)/(3*d) - (b*Sech[c + d*x])/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 3664

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 73, normalized size = 1.55 \[ -\frac {3 a \cosh (c+d x)}{4 d}+\frac {a \cosh (3 (c+d x))}{12 d}-\frac {7 b \cosh (c+d x)}{4 d}+\frac {b \cosh (3 (c+d x))}{12 d}-\frac {b \text {sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cosh[c + d*x])/(4*d) - (7*b*Cosh[c + d*x])/(4*d) + (a*Cosh[3*(c + d*x)])/(12*d) + (b*Cosh[3*(c + d*x)])/
(12*d) - (b*Sech[c + d*x])/d

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fricas [B]  time = 0.78, size = 91, normalized size = 1.94 \[ \frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (2 \, a + 5 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 4 \, a - 10 \, b\right )} \sinh \left (d x + c\right )^{2} - 9 \, a - 45 \, b}{24 \, d \cosh \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/24*((a + b)*cosh(d*x + c)^4 + (a + b)*sinh(d*x + c)^4 - 4*(2*a + 5*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*
x + c)^2 - 4*a - 10*b)*sinh(d*x + c)^2 - 9*a - 45*b)/(d*cosh(d*x + c))

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giac [B]  time = 0.17, size = 120, normalized size = 2.55 \[ -\frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 21 \, b e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - {\left (a e^{\left (3 \, d x + 24 \, c\right )} + b e^{\left (3 \, d x + 24 \, c\right )} - 9 \, a e^{\left (d x + 22 \, c\right )} - 21 \, b e^{\left (d x + 22 \, c\right )}\right )} e^{\left (-21 \, c\right )} + \frac {48 \, b e^{\left (d x + c\right )}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

-1/24*((9*a*e^(2*d*x + 2*c) + 21*b*e^(2*d*x + 2*c) - a - b)*e^(-3*d*x - 3*c) - (a*e^(3*d*x + 24*c) + b*e^(3*d*
x + 24*c) - 9*a*e^(d*x + 22*c) - 21*b*e^(d*x + 22*c))*e^(-21*c) + 48*b*e^(d*x + c)/(e^(2*d*x + 2*c) + 1))/d

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maple [A]  time = 0.26, size = 75, normalized size = 1.60 \[ \frac {a \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh ^{4}\left (d x +c \right )}{3 \cosh \left (d x +c \right )}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2),x)

[Out]

1/d*(a*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+b*(1/3*sinh(d*x+c)^4/cosh(d*x+c)-4/3*sinh(d*x+c)^2/cosh(d*x+c)-8/3
/cosh(d*x+c)))

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maxima [B]  time = 0.39, size = 136, normalized size = 2.89 \[ -\frac {1}{24} \, b {\left (\frac {21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*b*((21*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (20*e^(-2*d*x - 2*c) + 69*e^(-4*d*x - 4*c) - 1)/(d*(e^(-3*d*
x - 3*c) + e^(-5*d*x - 5*c)))) + 1/24*a*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x -
3*c)/d)

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mupad [B]  time = 1.17, size = 99, normalized size = 2.11 \[ \frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (a+b\right )}{24\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+b\right )}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+7\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a+7\,b\right )}{8\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2),x)

[Out]

(exp(- 3*c - 3*d*x)*(a + b))/(24*d) + (exp(3*c + 3*d*x)*(a + b))/(24*d) - (exp(c + d*x)*(3*a + 7*b))/(8*d) - (
exp(- c - d*x)*(3*a + 7*b))/(8*d) - (2*b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**3*(a+b*tanh(d*x+c)**2),x)

[Out]

Integral((a + b*tanh(c + d*x)**2)*sinh(c + d*x)**3, x)

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