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

Optimal. Leaf size=112 \[ -\frac{3 a^2 b \tanh (c+d x)}{d}-\frac{1}{2} a^2 x (a-6 b)+\frac{a^3}{4 d (1-\tanh (c+d x))}-\frac{a^3}{4 d (\tanh (c+d x)+1)}+\frac{b^2 (3 a+b) \tanh ^3(c+d x)}{3 d}-\frac{b^3 \tanh ^5(c+d x)}{5 d} \]

[Out]

-(a^2*(a - 6*b)*x)/2 + a^3/(4*d*(1 - Tanh[c + d*x])) - (3*a^2*b*Tanh[c + d*x])/d + (b^2*(3*a + b)*Tanh[c + d*x
]^3)/(3*d) - (b^3*Tanh[c + d*x]^5)/(5*d) - a^3/(4*d*(1 + Tanh[c + d*x]))

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Rubi [A]  time = 0.192742, antiderivative size = 143, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4132, 467, 528, 388, 206} \[ -\frac{b \left (81 a^2-28 a b-4 b^2\right ) \tanh (c+d x)}{30 d}-\frac{1}{2} a^2 x (a-6 b)-\frac{7 b \tanh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )^2}{10 d}-\frac{b (33 a-2 b) \tanh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )}{30 d}+\frac{\sinh (c+d x) \cosh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )^3}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*(a - 6*b)*x)/2 - (b*(81*a^2 - 28*a*b - 4*b^2)*Tanh[c + d*x])/(30*d) - ((33*a - 2*b)*b*Tanh[c + d*x]*(a +
b - b*Tanh[c + d*x]^2))/(30*d) - (7*b*Tanh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^2)/(10*d) + (Cosh[c + d*x]*Si
nh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^3)/(2*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 467

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^3 \sinh ^2(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a+b-b x^2\right )^3}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^3}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-7 b x^2\right ) \left (a+b-b x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac{7 b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^2}{10 d}+\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^3}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right ) \left (-(5 a-2 b) (a+b)+(33 a-2 b) b x^2\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{10 d}\\ &=-\frac{(33 a-2 b) b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{30 d}-\frac{7 b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^2}{10 d}+\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^3}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+b) \left (15 a^2-24 a b-4 b^2\right )-b \left (81 a^2-28 a b-4 b^2\right ) x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{30 d}\\ &=-\frac{b \left (81 a^2-28 a b-4 b^2\right ) \tanh (c+d x)}{30 d}-\frac{(33 a-2 b) b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{30 d}-\frac{7 b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^2}{10 d}+\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^3}{2 d}-\frac{\left (a^2 (a-6 b)\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac{1}{2} a^2 (a-6 b) x-\frac{b \left (81 a^2-28 a b-4 b^2\right ) \tanh (c+d x)}{30 d}-\frac{(33 a-2 b) b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )}{30 d}-\frac{7 b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^2}{10 d}+\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^3}{2 d}\\ \end{align*}

Mathematica [B]  time = 1.81971, size = 480, normalized size = 4.29 \[ \frac{\text{sech}(c) \text{sech}^5(c+d x) \left (2880 a^2 b \sinh (2 c+d x)-2880 a^2 b \sinh (2 c+3 d x)+720 a^2 b \sinh (4 c+3 d x)-720 a^2 b \sinh (4 c+5 d x)-600 a^2 d x (a-6 b) \cosh (2 c+d x)+1800 a^2 b d x \cosh (2 c+3 d x)+1800 a^2 b d x \cosh (4 c+3 d x)+360 a^2 b d x \cosh (4 c+5 d x)+360 a^2 b d x \cosh (6 c+5 d x)-4320 a^2 b \sinh (d x)-600 a^2 d x (a-6 b) \cosh (d x)+75 a^3 \sinh (2 c+d x)+135 a^3 \sinh (2 c+3 d x)+135 a^3 \sinh (4 c+3 d x)+75 a^3 \sinh (4 c+5 d x)+75 a^3 \sinh (6 c+5 d x)+15 a^3 \sinh (6 c+7 d x)+15 a^3 \sinh (8 c+7 d x)-300 a^3 d x \cosh (2 c+3 d x)-300 a^3 d x \cosh (4 c+3 d x)-60 a^3 d x \cosh (4 c+5 d x)-60 a^3 d x \cosh (6 c+5 d x)+75 a^3 \sinh (d x)-1440 a b^2 \sinh (2 c+d x)+480 a b^2 \sinh (2 c+3 d x)-720 a b^2 \sinh (4 c+3 d x)+240 a b^2 \sinh (4 c+5 d x)+960 a b^2 \sinh (d x)-480 b^3 \sinh (2 c+d x)+160 b^3 \sinh (2 c+3 d x)+32 b^3 \sinh (4 c+5 d x)-160 b^3 \sinh (d x)\right )}{3840 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]^5*(-600*a^2*(a - 6*b)*d*x*Cosh[d*x] - 600*a^2*(a - 6*b)*d*x*Cosh[2*c + d*x] - 300*a^3*d
*x*Cosh[2*c + 3*d*x] + 1800*a^2*b*d*x*Cosh[2*c + 3*d*x] - 300*a^3*d*x*Cosh[4*c + 3*d*x] + 1800*a^2*b*d*x*Cosh[
4*c + 3*d*x] - 60*a^3*d*x*Cosh[4*c + 5*d*x] + 360*a^2*b*d*x*Cosh[4*c + 5*d*x] - 60*a^3*d*x*Cosh[6*c + 5*d*x] +
360*a^2*b*d*x*Cosh[6*c + 5*d*x] + 75*a^3*Sinh[d*x] - 4320*a^2*b*Sinh[d*x] + 960*a*b^2*Sinh[d*x] - 160*b^3*Sin
h[d*x] + 75*a^3*Sinh[2*c + d*x] + 2880*a^2*b*Sinh[2*c + d*x] - 1440*a*b^2*Sinh[2*c + d*x] - 480*b^3*Sinh[2*c +
d*x] + 135*a^3*Sinh[2*c + 3*d*x] - 2880*a^2*b*Sinh[2*c + 3*d*x] + 480*a*b^2*Sinh[2*c + 3*d*x] + 160*b^3*Sinh[
2*c + 3*d*x] + 135*a^3*Sinh[4*c + 3*d*x] + 720*a^2*b*Sinh[4*c + 3*d*x] - 720*a*b^2*Sinh[4*c + 3*d*x] + 75*a^3*
Sinh[4*c + 5*d*x] - 720*a^2*b*Sinh[4*c + 5*d*x] + 240*a*b^2*Sinh[4*c + 5*d*x] + 32*b^3*Sinh[4*c + 5*d*x] + 75*
a^3*Sinh[6*c + 5*d*x] + 15*a^3*Sinh[6*c + 7*d*x] + 15*a^3*Sinh[8*c + 7*d*x]))/(3840*d)

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Maple [A]  time = 0.039, size = 145, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) +3\,{a}^{2}b \left ( dx+c-\tanh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( -1/2\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+1/2\, \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{\sinh \left ( dx+c \right ) }{4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{\tanh \left ( dx+c \right ) }{4} \left ({\frac{8}{15}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+3*a^2*b*(d*x+c-tanh(d*x+c))+3*a*b^2*(-1/2*sinh(d*x+c)/cos
h(d*x+c)^3+1/2*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/4*sinh(d*x+c)/cosh(d*x+c)^5+1/4*(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 = 1.05875, size = 598, normalized size = 5.34 \begin{align*} -\frac{1}{8} \, a^{3}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 3 \, a^{2} b{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac{4}{15} \, b^{3}{\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 )} + 2 \, a b^{2}{\left (\frac{3 \, e^{\left (-4 \, d x - 4 \, 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((a+b*sech(d*x+c)^2)^3*sinh(d*x+c)^2,x, algorithm="maxima")

[Out]

-1/8*a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + 3*a^2*b*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1))) + 4
/15*b^3*(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^(-10*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))) + 2*a*b^2*(3*e
^(-4*d*x - 4*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 = 2.55223, size = 1508, normalized size = 13.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*a^3*sinh(d*x + c)^7 + 4*(90*a^2*b - 30*a*b^2 - 4*b^3 - 15*(a^3 - 6*a^2*b)*d*x)*cosh(d*x + c)^5 + 20*
(90*a^2*b - 30*a*b^2 - 4*b^3 - 15*(a^3 - 6*a^2*b)*d*x)*cosh(d*x + c)*sinh(d*x + c)^4 + (315*a^3*cosh(d*x + c)^
2 + 75*a^3 - 360*a^2*b + 120*a*b^2 + 16*b^3)*sinh(d*x + c)^5 + 20*(90*a^2*b - 30*a*b^2 - 4*b^3 - 15*(a^3 - 6*a
^2*b)*d*x)*cosh(d*x + c)^3 + 5*(105*a^3*cosh(d*x + c)^4 + 27*a^3 - 216*a^2*b - 24*a*b^2 + 16*b^3 + 2*(75*a^3 -
360*a^2*b + 120*a*b^2 + 16*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 20*(2*(90*a^2*b - 30*a*b^2 - 4*b^3 - 15*(a
^3 - 6*a^2*b)*d*x)*cosh(d*x + c)^3 + 3*(90*a^2*b - 30*a*b^2 - 4*b^3 - 15*(a^3 - 6*a^2*b)*d*x)*cosh(d*x + c))*s
inh(d*x + c)^2 + 40*(90*a^2*b - 30*a*b^2 - 4*b^3 - 15*(a^3 - 6*a^2*b)*d*x)*cosh(d*x + c) + 5*(21*a^3*cosh(d*x
+ c)^6 + (75*a^3 - 360*a^2*b + 120*a*b^2 + 16*b^3)*cosh(d*x + c)^4 + 15*a^3 - 144*a^2*b - 48*a*b^2 - 64*b^3 +
3*(27*a^3 - 216*a^2*b - 24*a*b^2 + 16*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x +
c)*sinh(d*x + c)^4 + 5*d*cosh(d*x + c)^3 + 5*(2*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + 10*d
*cosh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.19006, size = 385, normalized size = 3.44 \begin{align*} \frac{a^{3} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac{{\left (a^{3} - 6 \, a^{2} b\right )}{\left (d x + c\right )}}{2 \, d} + \frac{{\left (2 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 12 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - a^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} + \frac{2 \,{\left (45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 45 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 180 \, 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 )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 60 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 30 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b - 15 \, a b^{2} - 2 \, b^{3}\right )}}{15 \, 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((a+b*sech(d*x+c)^2)^3*sinh(d*x+c)^2,x, algorithm="giac")

[Out]

1/8*a^3*e^(2*d*x + 2*c)/d - 1/2*(a^3 - 6*a^2*b)*(d*x + c)/d + 1/8*(2*a^3*e^(2*d*x + 2*c) - 12*a^2*b*e^(2*d*x +
2*c) - a^3)*e^(-2*d*x - 2*c)/d + 2/15*(45*a^2*b*e^(8*d*x + 8*c) - 45*a*b^2*e^(8*d*x + 8*c) + 180*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) + 270*a^2*b*e^(4*d*x + 4*c) - 60*a*b^2*e^(4*d*x
+ 4*c) + 10*b^3*e^(4*d*x + 4*c) + 180*a^2*b*e^(2*d*x + 2*c) - 30*a*b^2*e^(2*d*x + 2*c) - 10*b^3*e^(2*d*x + 2*c
) + 45*a^2*b - 15*a*b^2 - 2*b^3)/(d*(e^(2*d*x + 2*c) + 1)^5)