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

Optimal. Leaf size=91 \[ \frac{b^2 (3 a-2 b) \sinh (c+d x)}{d}+\frac{(a+5 b) (a-b)^2 \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a-b)^3 \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b^3 \sinh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0957924, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3190, 390, 385, 203} \[ \frac{b^2 (3 a-2 b) \sinh (c+d x)}{d}+\frac{(a+5 b) (a-b)^2 \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{(a-b)^3 \tanh (c+d x) \text{sech}(c+d x)}{2 d}+\frac{b^3 \sinh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3190

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

Rule 390

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

Rule 385

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

Rule 203

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

Rubi steps

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

Mathematica [C]  time = 7.32553, size = 347, normalized size = 3.81 \[ \frac{\text{csch}^5(c+d x) \left (-256 \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2\right \},\left \{1,1,1,\frac{11}{2}\right \},-\sinh ^2(c+d x)\right )+21 \left (3 b \left (753 a^2+18280 a b+10805 b^2\right ) \sinh ^6(c+d x)+3 a \left (491 a^2+16120 a b+36015 b^2\right ) \sinh ^4(c+d x)+5 a^2 (3224 a+21609 b) \sinh ^2(c+d x)+36015 a^3+b^2 (2259 a+17320 b) \sinh ^8(c+d x)+753 b^3 \sinh ^{10}(c+d x)\right )-\frac{315 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right ) \left (3 b \left (a^2+243 a b+625 b^2\right ) \sinh ^8(c+d x)+\left (585 a^2 b-47 a^3+6057 a b^2+2161 b^3\right ) \sinh ^6(c+d x)+3 a \left (81 a^2+1875 a b+2401 b^2\right ) \sinh ^4(c+d x)+3 a^2 (625 a+2401 b) \sinh ^2(c+d x)+2401 a^3+3 b^2 (a+81 b) \sinh ^{10}(c+d x)+b^3 \sinh ^{12}(c+d x)\right )}{\sqrt{-\sinh ^2(c+d x)}}\right )}{30240 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Csch[c + d*x]^5*(-256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8
*(a + b*Sinh[c + d*x]^2)^3 + 21*(36015*a^3 + 5*a^2*(3224*a + 21609*b)*Sinh[c + d*x]^2 + 3*a*(491*a^2 + 16120*a
*b + 36015*b^2)*Sinh[c + d*x]^4 + 3*b*(753*a^2 + 18280*a*b + 10805*b^2)*Sinh[c + d*x]^6 + b^2*(2259*a + 17320*
b)*Sinh[c + d*x]^8 + 753*b^3*Sinh[c + d*x]^10) - (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(2401*a^3 + 3*a^2*(625*a
+ 2401*b)*Sinh[c + d*x]^2 + 3*a*(81*a^2 + 1875*a*b + 2401*b^2)*Sinh[c + d*x]^4 + (-47*a^3 + 585*a^2*b + 6057*
a*b^2 + 2161*b^3)*Sinh[c + d*x]^6 + 3*b*(a^2 + 243*a*b + 625*b^2)*Sinh[c + d*x]^8 + 3*b^2*(a + 81*b)*Sinh[c +
d*x]^10 + b^3*Sinh[c + d*x]^12))/Sqrt[-Sinh[c + d*x]^2]))/(30240*d)

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Maple [B]  time = 0.055, size = 286, normalized size = 3.1 \begin{align*}{\frac{{a}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}-3\,{\frac{{a}^{2}b\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,{a}^{2}b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+3\,{\frac{{a}^{2}b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+3\,{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+9\,{\frac{a{b}^{2}\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9\,a{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}-9\,{\frac{a{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5\,{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-5\,{\frac{{b}^{3}\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{5\,{b}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+5\,{\frac{{b}^{3}\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*a^3*sech(d*x+c)*tanh(d*x+c)+1/d*a^3*arctan(exp(d*x+c))-3/d*a^2*b*sinh(d*x+c)/cosh(d*x+c)^2+3/2/d*a^2*b*s
ech(d*x+c)*tanh(d*x+c)+3/d*a^2*b*arctan(exp(d*x+c))+3/d*a*b^2*sinh(d*x+c)^3/cosh(d*x+c)^2+9/d*a*b^2*sinh(d*x+c
)/cosh(d*x+c)^2-9/2/d*a*b^2*sech(d*x+c)*tanh(d*x+c)-9/d*a*b^2*arctan(exp(d*x+c))+1/3/d*b^3*sinh(d*x+c)^5/cosh(
d*x+c)^2-5/3/d*b^3*sinh(d*x+c)^3/cosh(d*x+c)^2-5/d*b^3*sinh(d*x+c)/cosh(d*x+c)^2+5/2/d*b^3*sech(d*x+c)*tanh(d*
x+c)+5/d*b^3*arctan(exp(d*x+c))

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Maxima [B]  time = 1.59216, size = 482, normalized size = 5.3 \begin{align*} \frac{1}{24} \, b^{3}{\left (\frac{27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac{120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d{\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac{3}{2} \, a b^{2}{\left (\frac{6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )}}{d} + \frac{4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d{\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} - 3 \, a^{2} b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - a^{3}{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b^3*((27*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 120*arctan(e^(-d*x - c))/d - (25*e^(-2*d*x - 2*c) + 77*e^(-
4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) - 1)/(d*(e^(-3*d*x - 3*c) + 2*e^(-5*d*x - 5*c) + e^(-7*d*x - 7*c)))) + 3/2*a
*b^2*(6*arctan(e^(-d*x - c))/d - e^(-d*x - c)/d + (4*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) + 1)/(d*(e^(-d*x - c)
+ 2*e^(-3*d*x - 3*c) + e^(-5*d*x - 5*c)))) - 3*a^2*b*(arctan(e^(-d*x - c))/d + (e^(-d*x - c) - e^(-3*d*x - 3*
c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) - a^3*(arctan(e^(-d*x - c))/d - (e^(-d*x - c) - e^(-3*d*x
- 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1)))

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Fricas [B]  time = 1.81228, size = 4188, normalized size = 46.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + b^3*sinh(d*x + c)^10 + (36*a*b^2 - 25*b^3)
*cosh(d*x + c)^8 + (45*b^3*cosh(d*x + c)^2 + 36*a*b^2 - 25*b^3)*sinh(d*x + c)^8 + 8*(15*b^3*cosh(d*x + c)^3 +
(36*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^6
+ 2*(105*b^3*cosh(d*x + c)^4 + 12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3 + 14*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^2
)*sinh(d*x + c)^6 + 4*(63*b^3*cosh(d*x + c)^5 + 14*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^3 + 3*(12*a^3 - 36*a^2*b
+ 54*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^
4 + 2*(105*b^3*cosh(d*x + c)^6 + 35*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^4 - 12*a^3 + 36*a^2*b - 54*a*b^2 + 25*b^
3 + 15*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*b^3*cosh(d*x + c)^7 +
7*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^5 + 5*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^3 - (12*a^3 -
36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - b^3 - (36*a*b^2 - 25*b^3)*cosh(d*x + c)^2 + (45
*b^3*cosh(d*x + c)^8 + 28*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^6 + 30*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cos
h(d*x + c)^4 - 36*a*b^2 + 25*b^3 - 12*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 24*((a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 7*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)*s
inh(d*x + c)^6 + (a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*sinh(d*x + c)^7 + 2*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(
d*x + c)^5 + (2*a^3 + 6*a^2*b - 18*a*b^2 + 10*b^3 + 21*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh
(d*x + c)^5 + 5*(7*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 2*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cos
h(d*x + c))*sinh(d*x + c)^4 + (a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (35*(a^3 + 3*a^2*b - 9*a*b^2
+ 5*b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3 + 20*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^3 + (21*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 20*(a^3 + 3*a^2*b - 9*a*b^2 +
5*b^3)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(a^3 + 3*a^2*
b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 10*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 3*(a^3 + 3*a^2*b
- 9*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(5*b^3*cosh(d*x
+ c)^9 + 4*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^7 + 6*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^5 - 4
*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^3 - (36*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x + c))/(
d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7 + 2*d*cosh(d*x + c)^5 + (21*d*cosh(d
*x + c)^2 + 2*d)*sinh(d*x + c)^5 + 5*(7*d*cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sinh(d*x + c)^4 + d*cosh(d*x +
c)^3 + (35*d*cosh(d*x + c)^4 + 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + (21*d*cosh(d*x + c)^5 + 20*d*cosh(d
*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*d*cosh(d*x + c)^6 + 10*d*cosh(d*x + c)^4 + 3*d*cosh(d*x +
c)^2)*sinh(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(sech(d*x+c)**3*(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.43009, size = 354, normalized size = 3.89 \begin{align*} \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )}{\left (a^{3} + 3 \, a^{2} b - 9 \, a b^{2} + 5 \, b^{3}\right )}}{4 \, d} + \frac{a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 3 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 3 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - b^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )} d} + \frac{b^{3} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 36 \, a b^{2} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, b^{3} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{24 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)/d + (a^3*(e^(d*x
+ c) - e^(-d*x - c)) - 3*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 3*a*b^2*(e^(d*x + c) - e^(-d*x - c)) - b^3*(e^(
d*x + c) - e^(-d*x - c)))/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)*d) + 1/24*(b^3*d^2*(e^(d*x + c) - e^(-d*x - c)
)^3 + 36*a*b^2*d^2*(e^(d*x + c) - e^(-d*x - c)) - 24*b^3*d^2*(e^(d*x + c) - e^(-d*x - c)))/d^3