3.4.15 \(\int \text {sech}^8(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [315]

Optimal. Leaf size=80 \[ \frac {a^3 \tanh (c+d x)}{d}-\frac {a^2 (a-b) \tanh ^3(c+d x)}{d}+\frac {3 a (a-b)^2 \tanh ^5(c+d x)}{5 d}-\frac {(a-b)^3 \tanh ^7(c+d x)}{7 d} \]

[Out]

a^3*tanh(d*x+c)/d-a^2*(a-b)*tanh(d*x+c)^3/d+3/5*a*(a-b)^2*tanh(d*x+c)^5/d-1/7*(a-b)^3*tanh(d*x+c)^7/d

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Rubi [A]
time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 = {3270, 200} \begin {gather*} \frac {a^3 \tanh (c+d x)}{d}-\frac {a^2 (a-b) \tanh ^3(c+d x)}{d}-\frac {(a-b)^3 \tanh ^7(c+d x)}{7 d}+\frac {3 a (a-b)^2 \tanh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(a^3*Tanh[c + d*x])/d - (a^2*(a - b)*Tanh[c + d*x]^3)/d + (3*a*(a - b)^2*Tanh[c + d*x]^5)/(5*d) - ((a - b)^3*T
anh[c + d*x]^7)/(7*d)

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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}^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a-(a-b) x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^3-3 a^2 (a-b) x^2+3 a (a-b)^2 x^4-(a-b)^3 x^6\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a^3 \tanh (c+d x)}{d}-\frac {a^2 (a-b) \tanh ^3(c+d x)}{d}+\frac {3 a (a-b)^2 \tanh ^5(c+d x)}{5 d}-\frac {(a-b)^3 \tanh ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(80)=160\).
time = 0.60, size = 163, normalized size = 2.04 \begin {gather*} \frac {\left (512 a^3-304 a^2 b+192 a b^2-50 b^3+\left (464 a^3+232 a^2 b-246 a b^2+75 b^3\right ) \cosh (2 (c+d x))+2 \left (64 a^3+32 a^2 b+24 a b^2-15 b^3\right ) \cosh (4 (c+d x))+16 a^3 \cosh (6 (c+d x))+8 a^2 b \cosh (6 (c+d x))+6 a b^2 \cosh (6 (c+d x))+5 b^3 \cosh (6 (c+d x))\right ) \text {sech}^6(c+d x) \tanh (c+d x)}{1120 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((512*a^3 - 304*a^2*b + 192*a*b^2 - 50*b^3 + (464*a^3 + 232*a^2*b - 246*a*b^2 + 75*b^3)*Cosh[2*(c + d*x)] + 2*
(64*a^3 + 32*a^2*b + 24*a*b^2 - 15*b^3)*Cosh[4*(c + d*x)] + 16*a^3*Cosh[6*(c + d*x)] + 8*a^2*b*Cosh[6*(c + d*x
)] + 6*a*b^2*Cosh[6*(c + d*x)] + 5*b^3*Cosh[6*(c + d*x)])*Sech[c + d*x]^6*Tanh[c + d*x])/(1120*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(260\) vs. \(2(76)=152\).
time = 1.76, size = 261, normalized size = 3.26

method	result	size
risch	\(-\frac {2 \left (35 b^{3} {\mathrm e}^{12 d x +12 c}+210 a \,b^{2} {\mathrm e}^{10 d x +10 c}+560 a^{2} b \,{\mathrm e}^{8 d x +8 c}-210 a \,b^{2} {\mathrm e}^{8 d x +8 c}+175 b^{3} {\mathrm e}^{8 d x +8 c}+560 a^{3} {\mathrm e}^{6 d x +6 c}-280 a^{2} b \,{\mathrm e}^{6 d x +6 c}+420 a \,b^{2} {\mathrm e}^{6 d x +6 c}+336 a^{3} {\mathrm e}^{4 d x +4 c}+168 a^{2} b \,{\mathrm e}^{4 d x +4 c}-84 a \,b^{2} {\mathrm e}^{4 d x +4 c}+105 b^{3} {\mathrm e}^{4 d x +4 c}+112 a^{3} {\mathrm e}^{2 d x +2 c}+56 a^{2} b \,{\mathrm e}^{2 d x +2 c}+42 a \,b^{2} {\mathrm e}^{2 d x +2 c}+16 a^{3}+8 a^{2} b +6 a \,b^{2}+5 b^{3}\right )}{35 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{7}}\)	\(261\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^8*(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-2/35*(35*b^3*exp(12*d*x+12*c)+210*a*b^2*exp(10*d*x+10*c)+560*a^2*b*exp(8*d*x+8*c)-210*a*b^2*exp(8*d*x+8*c)+17
5*b^3*exp(8*d*x+8*c)+560*a^3*exp(6*d*x+6*c)-280*a^2*b*exp(6*d*x+6*c)+420*a*b^2*exp(6*d*x+6*c)+336*a^3*exp(4*d*
x+4*c)+168*a^2*b*exp(4*d*x+4*c)-84*a*b^2*exp(4*d*x+4*c)+105*b^3*exp(4*d*x+4*c)+112*a^3*exp(2*d*x+2*c)+56*a^2*b
*exp(2*d*x+2*c)+42*a*b^2*exp(2*d*x+2*c)+16*a^3+8*a^2*b+6*a*b^2+5*b^3)/d/(1+exp(2*d*x+2*c))^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1754 vs. \(2 (76) = 152\).
time = 0.30, size = 1754, normalized size = 21.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^8*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

32/35*a^3*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*
x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 21*e^(-4*d*x - 4*c)/(d*(7
*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) +
7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x -
4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x -
14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21
*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1))) + 16/35*a^2*b*(7*e^(-2*d*x - 2*c)/(d*(7
*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) +
7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 21*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x -
4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x -
14*c) + 1)) - 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(
-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 70*e^(-8*d*x - 8*c)/
(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*
c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(
-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)))
+ 12/35*a*b^2*(7*e^(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(
-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) - 14*e^(-4*d*x - 4*c)/
(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*
c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 70*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d
*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*
d*x - 14*c) + 1)) - 35*e^(-8*d*x - 8*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 3
5*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 35*e^(-10*d*x -
10*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*
x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) +
35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c)
+ 1))) + 2/7*b^3*(21*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35
*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 35*e^(-8*d*x - 8
*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x -
10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 7*e^(-12*d*x - 12*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^
(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^
(-14*d*x - 14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x -
8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (76) = 152\).
time = 0.54, size = 814, normalized size = 10.18 \begin {gather*} -\frac {4 \, {\left ({\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} - 6 \, {\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} + 20 \, b^{3}\right )} \sinh \left (d x + c\right )^{6} + 14 \, {\left (4 \, a^{3} + 2 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (56 \, a^{3} + 28 \, a^{2} b + 126 \, a b^{2} + 15 \, {\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (5 \, {\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (2 \, a^{3} + a^{2} b - 3 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 280 \, a^{3} - 140 \, a^{2} b + 210 \, a b^{2} + 7 \, {\left (24 \, a^{3} + 52 \, a^{2} b - 21 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 168 \, a^{3} + 364 \, a^{2} b - 147 \, a b^{2} + 140 \, b^{3} + 84 \, {\left (4 \, a^{3} + 2 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (3 \, {\left (8 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 56 \, {\left (2 \, a^{3} + a^{2} b - 3 \, a b^{2}\right )} \cosh \left (d x + c\right )^{3} + 7 \, {\left (24 \, a^{3} - 28 \, a^{2} b + 9 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (14 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} + 60 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 15 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 56 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} + 30 \, d \cosh \left (d x + c\right )^{4} + 42 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (2 \, d \cosh \left (d x + c\right )^{7} + 9 \, d \cosh \left (d x + c\right )^{5} + 14 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 35 \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^8*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-4/35*((8*a^3 + 4*a^2*b + 3*a*b^2 + 20*b^3)*cosh(d*x + c)^6 - 6*(8*a^3 + 4*a^2*b + 3*a*b^2 - 15*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^5 + (8*a^3 + 4*a^2*b + 3*a*b^2 + 20*b^3)*sinh(d*x + c)^6 + 14*(4*a^3 + 2*a^2*b + 9*a*b^2)*c
osh(d*x + c)^4 + (56*a^3 + 28*a^2*b + 126*a*b^2 + 15*(8*a^3 + 4*a^2*b + 3*a*b^2 + 20*b^3)*cosh(d*x + c)^2)*sin
h(d*x + c)^4 - 4*(5*(8*a^3 + 4*a^2*b + 3*a*b^2 - 15*b^3)*cosh(d*x + c)^3 + 28*(2*a^3 + a^2*b - 3*a*b^2)*cosh(d
*x + c))*sinh(d*x + c)^3 + 280*a^3 - 140*a^2*b + 210*a*b^2 + 7*(24*a^3 + 52*a^2*b - 21*a*b^2 + 20*b^3)*cosh(d*
x + c)^2 + (15*(8*a^3 + 4*a^2*b + 3*a*b^2 + 20*b^3)*cosh(d*x + c)^4 + 168*a^3 + 364*a^2*b - 147*a*b^2 + 140*b^
3 + 84*(4*a^3 + 2*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*(3*(8*a^3 + 4*a^2*b + 3*a*b^2 - 15*b^3
)*cosh(d*x + c)^5 + 56*(2*a^3 + a^2*b - 3*a*b^2)*cosh(d*x + c)^3 + 7*(24*a^3 - 28*a^2*b + 9*a*b^2 - 5*b^3)*cos
h(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 8*d*co
sh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^6 + 4*(14*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*s
inh(d*x + c)^5 + 28*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 + 60*d*cosh(d*x + c)^2 + 14*d)*sinh(d*x + c)^4
+ 8*(7*d*cosh(d*x + c)^5 + 15*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x + c)^3 + 56*d*cosh(d*x + c)^2 +
4*(7*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 + 42*d*cosh(d*x + c)^2 + 14*d)*sinh(d*x + c)^2 + 4*(2*d*cosh(d*
x + c)^7 + 9*d*cosh(d*x + c)^5 + 14*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x + c) + 35*d)

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (76) = 152\).
time = 0.48, size = 260, normalized size = 3.25 \begin {gather*} -\frac {2 \, {\left (35 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 210 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 560 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 210 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 175 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 280 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 420 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 336 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 168 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 84 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 105 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 112 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 56 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 42 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a^{3} + 8 \, a^{2} b + 6 \, a b^{2} + 5 \, b^{3}\right )}}{35 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^8*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-2/35*(35*b^3*e^(12*d*x + 12*c) + 210*a*b^2*e^(10*d*x + 10*c) + 560*a^2*b*e^(8*d*x + 8*c) - 210*a*b^2*e^(8*d*x
+ 8*c) + 175*b^3*e^(8*d*x + 8*c) + 560*a^3*e^(6*d*x + 6*c) - 280*a^2*b*e^(6*d*x + 6*c) + 420*a*b^2*e^(6*d*x +
6*c) + 336*a^3*e^(4*d*x + 4*c) + 168*a^2*b*e^(4*d*x + 4*c) - 84*a*b^2*e^(4*d*x + 4*c) + 105*b^3*e^(4*d*x + 4*
c) + 112*a^3*e^(2*d*x + 2*c) + 56*a^2*b*e^(2*d*x + 2*c) + 42*a*b^2*e^(2*d*x + 2*c) + 16*a^3 + 8*a^2*b + 6*a*b^
2 + 5*b^3)/(d*(e^(2*d*x + 2*c) + 1)^7)

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Mupad [B]
time = 0.87, size = 994, normalized size = 12.42 \begin {gather*} -\frac {\frac {2\,b^3}{7\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (16\,a^3-24\,a^2\,b+18\,a\,b^2-5\,b^3\right )}{7\,d}+\frac {2\,b^3\,{\mathrm {e}}^{12\,c+12\,d\,x}}{7\,d}+\frac {6\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{7\,d}+\frac {6\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{7\,d}+\frac {12\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a-b\right )}{7\,d}+\frac {12\,b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (2\,a-b\right )}{7\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1}-\frac {\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (16\,a^3-24\,a^2\,b+18\,a\,b^2-5\,b^3\right )}{7\,d}+\frac {2\,b^3\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}+\frac {2\,b^2\,\left (2\,a-b\right )}{7\,d}+\frac {2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{7\,d}+\frac {4\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{7\,d}+\frac {10\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (2\,a-b\right )}{7\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {\frac {2\,\left (16\,a^3-24\,a^2\,b+18\,a\,b^2-5\,b^3\right )}{35\,d}+\frac {2\,b^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}+\frac {6\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{35\,d}+\frac {6\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a-b\right )}{7\,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 {2\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}+\frac {2\,b^2\,\left (2\,a-b\right )}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,b\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{35\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (16\,a^3-24\,a^2\,b+18\,a\,b^2-5\,b^3\right )}{35\,d}+\frac {2\,b^3\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}+\frac {12\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{35\,d}+\frac {8\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (2\,a-b\right )}{7\,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\,\left (16\,a^2-16\,a\,b+5\,b^2\right )}{35\,d}+\frac {2\,b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}+\frac {4\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a-b\right )}{7\,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^3}{7\,d\,\left ({\mathrm {e}}^{2\,c+2\,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)^3/cosh(c + d*x)^8,x)

[Out]

- ((2*b^3)/(7*d) + (8*exp(6*c + 6*d*x)*(18*a*b^2 - 24*a^2*b + 16*a^3 - 5*b^3))/(7*d) + (2*b^3*exp(12*c + 12*d*
x))/(7*d) + (6*b*exp(4*c + 4*d*x)*(16*a^2 - 16*a*b + 5*b^2))/(7*d) + (6*b*exp(8*c + 8*d*x)*(16*a^2 - 16*a*b +
5*b^2))/(7*d) + (12*b^2*exp(2*c + 2*d*x)*(2*a - b))/(7*d) + (12*b^2*exp(10*c + 10*d*x)*(2*a - b))/(7*d))/(7*ex
p(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*e
xp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) - ((4*exp(4*c + 4*d*x)*(18*a*b^2 - 24*a^2*b + 16*a^3 - 5*b^3))/(7*
d) + (2*b^3*exp(10*c + 10*d*x))/(7*d) + (2*b^2*(2*a - b))/(7*d) + (2*b*exp(2*c + 2*d*x)*(16*a^2 - 16*a*b + 5*b
^2))/(7*d) + (4*b*exp(6*c + 6*d*x)*(16*a^2 - 16*a*b + 5*b^2))/(7*d) + (10*b^2*exp(8*c + 8*d*x)*(2*a - b))/(7*d
))/(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x
) + exp(12*c + 12*d*x) + 1) - ((2*(18*a*b^2 - 24*a^2*b + 16*a^3 - 5*b^3))/(35*d) + (2*b^3*exp(6*c + 6*d*x))/(7
*d) + (6*b*exp(2*c + 2*d*x)*(16*a^2 - 16*a*b + 5*b^2))/(35*d) + (6*b^2*exp(4*c + 4*d*x)*(2*a - b))/(7*d))/(4*e
xp(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) - ((2*b^3*exp(2*c + 2*d*x))/
(7*d) + (2*b^2*(2*a - b))/(7*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - ((2*b*(16*a^2 - 16*a*b + 5*b^2)
)/(35*d) + (8*exp(2*c + 2*d*x)*(18*a*b^2 - 24*a^2*b + 16*a^3 - 5*b^3))/(35*d) + (2*b^3*exp(8*c + 8*d*x))/(7*d)
+ (12*b*exp(4*c + 4*d*x)*(16*a^2 - 16*a*b + 5*b^2))/(35*d) + (8*b^2*exp(6*c + 6*d*x)*(2*a - b))/(7*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*(16*a^2 - 16*a*b + 5*b^2))/(35*d) + (2*b^3*exp(4*c + 4*d*x))/(7*d) + (4*b^2*exp(2*c + 2*d*x)*(2*a - b))/(7*
d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - (2*b^3)/(7*d*(exp(2*c + 2*d*x) + 1))

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