3.3.92 \(\int \text {sech}^6(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [292]

Optimal. Leaf size=54 \[ \frac {a \tanh (c+d x)}{d}-\frac {(2 a-b) \tanh ^3(c+d x)}{3 d}+\frac {(a-b) \tanh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 54, 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 = {3270, 380} \begin {gather*} \frac {(a-b) \tanh ^5(c+d x)}{5 d}-\frac {(2 a-b) \tanh ^3(c+d x)}{3 d}+\frac {a \tanh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 380

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

Rule 3270

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \text {sech}^6(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (a-(a-b) x^2\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a-(2 a-b) x^2+(a-b) x^4\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a \tanh (c+d x)}{d}-\frac {(2 a-b) \tanh ^3(c+d x)}{3 d}+\frac {(a-b) \tanh ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 102, normalized size = 1.89 \begin {gather*} \frac {a \tanh (c+d x)}{d}+\frac {2 b \tanh (c+d x)}{15 d}+\frac {b \text {sech}^2(c+d x) \tanh (c+d x)}{15 d}-\frac {b \text {sech}^4(c+d x) \tanh (c+d x)}{5 d}-\frac {2 a \tanh ^3(c+d x)}{3 d}+\frac {a \tanh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2),x]

[Out]

(a*Tanh[c + d*x])/d + (2*b*Tanh[c + d*x])/(15*d) + (b*Sech[c + d*x]^2*Tanh[c + d*x])/(15*d) - (b*Sech[c + d*x]
^4*Tanh[c + d*x])/(5*d) - (2*a*Tanh[c + d*x]^3)/(3*d) + (a*Tanh[c + d*x]^5)/(5*d)

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Maple [A]
time = 1.44, size = 84, normalized size = 1.56

method	result	size
risch	\(-\frac {4 \left (15 b \,{\mathrm e}^{6 d x +6 c}+40 a \,{\mathrm e}^{4 d x +4 c}-5 b \,{\mathrm e}^{4 d x +4 c}+20 a \,{\mathrm e}^{2 d x +2 c}+5 b \,{\mathrm e}^{2 d x +2 c}+4 a +b \right )}{15 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{5}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^6*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-4/15*(15*b*exp(6*d*x+6*c)+40*a*exp(4*d*x+4*c)-5*b*exp(4*d*x+4*c)+20*a*exp(2*d*x+2*c)+5*b*exp(2*d*x+2*c)+4*a+b
)/d/(1+exp(2*d*x+2*c))^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (50) = 100\).
time = 0.27, size = 486, normalized size = 9.00 \begin {gather*} \frac {16}{15} \, a {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {4}{15} \, b {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {5 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

16/15*a*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x -
8*c) + e^(-10*d*x - 10*c) + 1)) + 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6
*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) +
10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + 4/15*b*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2
*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) - 5*e^
(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-1
0*d*x - 10*c) + 1)) + 15*e^(-6*d*x - 6*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) +
5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x
- 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (50) = 100\).
time = 0.48, size = 343, normalized size = 6.35 \begin {gather*} -\frac {8 \, {\left (2 \, {\left (a + 4 \, b\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (a + 4 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (2 \, a - 7 \, b\right )} \sinh \left (d x + c\right )^{3} + 30 \, a \cosh \left (d x + c\right ) - {\left (3 \, {\left (2 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{2} - 10 \, a + 5 \, b\right )} \sinh \left (d x + c\right )\right )}}{15 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{7} + 5 \, d \cosh \left (d x + c\right )^{5} + {\left (21 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 11 \, d \cosh \left (d x + c\right )^{3} + {\left (35 \, d \cosh \left (d x + c\right )^{4} + 50 \, d \cosh \left (d x + c\right )^{2} + 9 \, d\right )} \sinh \left (d x + c\right )^{3} + {\left (21 \, d \cosh \left (d x + c\right )^{5} + 50 \, d \cosh \left (d x + c\right )^{3} + 33 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 15 \, d \cosh \left (d x + c\right ) + {\left (7 \, d \cosh \left (d x + c\right )^{6} + 25 \, d \cosh \left (d x + c\right )^{4} + 27 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

-8/15*(2*(a + 4*b)*cosh(d*x + c)^3 + 6*(a + 4*b)*cosh(d*x + c)*sinh(d*x + c)^2 - (2*a - 7*b)*sinh(d*x + c)^3 +
30*a*cosh(d*x + c) - (3*(2*a - 7*b)*cosh(d*x + c)^2 - 10*a + 5*b)*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cos
h(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7 + 5*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 + 5*d*cosh(d*x + c))*sinh(d*x + c)^4 + 11*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 + (21*d*cosh(d*x + c)^5 + 50*d*cosh(d*x + c)^3 + 33*d*cosh(
d*x + c))*sinh(d*x + c)^2 + 15*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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**6*(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [A]
time = 0.41, size = 83, normalized size = 1.54 \begin {gather*} -\frac {4 \, {\left (15 \, b e^{\left (6 \, d x + 6 \, c\right )} + 40 \, a e^{\left (4 \, d x + 4 \, c\right )} - 5 \, b e^{\left (4 \, d x + 4 \, c\right )} + 20 \, a e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + b\right )}}{15 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

-4/15*(15*b*e^(6*d*x + 6*c) + 40*a*e^(4*d*x + 4*c) - 5*b*e^(4*d*x + 4*c) + 20*a*e^(2*d*x + 2*c) + 5*b*e^(2*d*x
+ 2*c) + 4*a + b)/(d*(e^(2*d*x + 2*c) + 1)^5)

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Mupad [B]
time = 0.82, size = 298, normalized size = 5.52 \begin {gather*} -\frac {\frac {8\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}+\frac {8\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a-b\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\,b}{5\,d}+\frac {6\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a-b\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 {8\,\left (2\,a-b\right )}{15\,d}+\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{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 {2\,b}{5\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^2)/cosh(c + d*x)^6,x)

[Out]

- ((8*b*exp(2*c + 2*d*x))/(5*d) + (8*b*exp(6*c + 6*d*x))/(5*d) + (16*exp(4*c + 4*d*x)*(2*a - b))/(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
*b)/(5*d) + (6*b*exp(4*c + 4*d*x))/(5*d) + (8*exp(2*c + 2*d*x)*(2*a - b))/(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) - ((8*(2*a - b))/(15*d) + (4*b*exp(2*c + 2*d*x))/(5*d
))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - (2*b)/(5*d*(2*exp(2*c + 2*d*x) + exp(4*c
+ 4*d*x) + 1))

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