3.4.10 \(\int \text {sech}^3(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [310]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 = {3269, 398, 393, 209} \begin {gather*} \frac {(a+5 b) (a-b)^2 \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {b^2 (3 a-2 b) \sinh (c+d x)}{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} \end {gather*}

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 209

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

Rule 393

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 398

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 3269

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]

Rubi steps

\begin {align*} \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\text {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 {\text {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 {\text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 5.73, size = 347, normalized size = 3.81 \begin {gather*} \frac {\text {csch}^5(c+d x) \left (-256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3+21 \left (36015 a^3+5 a^2 (3224 a+21609 b) \sinh ^2(c+d x)+3 a \left (491 a^2+16120 a b+36015 b^2\right ) \sinh ^4(c+d x)+3 b \left (753 a^2+18280 a b+10805 b^2\right ) \sinh ^6(c+d x)+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 (2401 a^3+3 a^2 (625 a+2401 b) \sinh ^2(c+d x)+3 a \left (81 a^2+1875 a b+2401 b^2\right ) \sinh ^4(c+d x)+\left (-47 a^3+585 a^2 b+6057 a b^2+2161 b^3\right ) \sinh ^6(c+d x)+3 b \left (a^2+243 a b+625 b^2\right ) \sinh ^8(c+d x)+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} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.50, size = 311, normalized size = 3.42

method	result	size
risch	\(\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}+\frac {3 a \,{\mathrm e}^{d x +c} b^{2}}{2 d}-\frac {9 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 a \,{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {9 b^{3} {\mathrm e}^{-d x -c}}{8 d}-\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {{\mathrm e}^{d x +c} \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 d}-\frac {9 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{2 d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 d}+\frac {9 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{2 d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 d}\)	\(311\)

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,method=_RETURNVERBOSE)

[Out]

1/24/d*exp(3*d*x+3*c)*b^3+3/2*a/d*exp(d*x+c)*b^2-9/8*b^3/d*exp(d*x+c)-3/2*a/d*exp(-d*x-c)*b^2+9/8*b^3/d*exp(-d
*x-c)-1/24*b^3/d*exp(-3*d*x-3*c)+exp(d*x+c)*(a^3-3*a^2*b+3*a*b^2-b^3)*(exp(2*d*x+2*c)-1)/d/(1+exp(2*d*x+2*c))^
2+1/2*I/d*ln(exp(d*x+c)+I)*a^3+3/2*I/d*ln(exp(d*x+c)+I)*a^2*b-9/2*I/d*ln(exp(d*x+c)+I)*a*b^2+5/2*I/d*ln(exp(d*
x+c)+I)*b^3-1/2*I/d*ln(exp(d*x+c)-I)*a^3-3/2*I/d*ln(exp(d*x+c)-I)*a^2*b+9/2*I/d*ln(exp(d*x+c)-I)*a*b^2-5/2*I/d
*ln(exp(d*x+c)-I)*b^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (85) = 170\).
time = 0.51, size = 357, normalized size = 3.92 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1679 vs. \(2 (85) = 170\).
time = 0.42, size = 1679, normalized size = 18.45 \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)^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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (85) = 170\).
time = 0.44, size = 247, normalized size = 2.71 \begin {gather*} \frac {b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 36 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 6 \, {\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 )} + \frac {24 \, {\left (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 )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{24 \, d} \end {gather*}

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]

1/24*(b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 36*a*b^2*(e^(d*x + c) - e^(-d*x - c)) - 24*b^3*(e^(d*x + c) - e^(-d
*x - c)) + 6*(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) + 24*(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

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Mupad [B]
time = 2.28, size = 308, normalized size = 3.38 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}\,\left (4\,a-3\,b\right )}{8\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (4\,a-3\,b\right )}{8\,d}+\frac {\ln \left (-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3+3\,a^2\,b-9\,a\,b^2+5\,b^3\right )-{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}\right )\,{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3+3\,a^2\,b-9\,a\,b^2+5\,b^3\right )+{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}\right )\,{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}}{2\,d} \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)^3,x)

[Out]

(b^3*exp(3*c + 3*d*x))/(24*d) - (b^3*exp(- 3*c - 3*d*x))/(24*d) + (log(- (a - b)^2*(a + 5*b)*1i - exp(d*x)*exp
(c)*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3))*(a - b)^2*(a + 5*b)*1i)/(2*d) - (log((a - b)^2*(a + 5*b)*1i - exp(d*x)*
exp(c)*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3))*(a - b)^2*(a + 5*b)*1i)/(2*d) + (3*b^2*exp(c + d*x)*(4*a - 3*b))/(8*
d) + (exp(c + d*x)*(3*a*b^2 - 3*a^2*b + a^3 - b^3))/(d*(exp(2*c + 2*d*x) + 1)) - (2*exp(c + d*x)*(3*a*b^2 - 3*
a^2*b + a^3 - b^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (3*b^2*exp(- c - d*x)*(4*a - 3*b))/(8*d)

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