3.4.8 \(\int \text {sech}(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [308]

Optimal. Leaf size=86 \[ \frac {(a-b)^3 \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {(3 a-b) b^2 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \sinh ^5(c+d x)}{5 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3269, 398, 209} \begin {gather*} \frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {(a-b)^3 \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b^2 (3 a-b) \sinh ^3(c+d x)}{3 d}+\frac {b^3 \sinh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)^3*ArcTan[Sinh[c + d*x]])/d + (b*(3*a^2 - 3*a*b + b^2)*Sinh[c + d*x])/d + ((3*a - b)*b^2*Sinh[c + d*x]
^3)/(3*d) + (b^3*Sinh[c + d*x]^5)/(5*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 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}(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}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b \left (3 a^2-3 a b+b^2\right )+(3 a-b) b^2 x^2+b^3 x^4+\frac {(a-b)^3}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {(3 a-b) b^2 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \sinh ^5(c+d x)}{5 d}+\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b)^3 \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {(3 a-b) b^2 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \sinh ^5(c+d x)}{5 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.39, size = 100, normalized size = 1.16 \begin {gather*} \frac {\sinh (c+d x) \left (\frac {15 (a-b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right )}{\sqrt {-\sinh ^2(c+d x)}}+b \left (45 a^2+15 a b \left (-3+\sinh ^2(c+d x)\right )+b^2 \left (15-5 \sinh ^2(c+d x)+3 \sinh ^4(c+d x)\right )\right )\right )}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sinh[c + d*x]*((15*(a - b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/Sqrt[-Sinh[c + d*x]^2] + b*(45*a^2 + 15*a*b*(-3
+ Sinh[c + d*x]^2) + b^2*(15 - 5*Sinh[c + d*x]^2 + 3*Sinh[c + d*x]^4))))/(15*d)

________________________________________________________________________________________

Maple [A]
time = 1.00, size = 114, normalized size = 1.33

method	result	size
derivativedivides	\(\frac {2 a^{3} \arctan \left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{3}-\sinh \left (d x +c \right )+2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{3}+\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(114\)
default	\(\frac {2 a^{3} \arctan \left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{3}-\sinh \left (d x +c \right )+2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{3}+\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(114\)
risch	\(\frac {{\mathrm e}^{5 d x +5 c} b^{3}}{160 d}-\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{3}}{96 d}+\frac {a \,b^{2} {\mathrm e}^{3 d x +3 c}}{8 d}+\frac {3 b \,{\mathrm e}^{d x +c} a^{2}}{2 d}-\frac {15 a \,{\mathrm e}^{d x +c} b^{2}}{8 d}+\frac {11 b^{3} {\mathrm e}^{d x +c}}{16 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}+\frac {15 a \,{\mathrm e}^{-d x -c} b^{2}}{8 d}-\frac {11 b^{3} {\mathrm e}^{-d x -c}}{16 d}+\frac {7 b^{3} {\mathrm e}^{-3 d x -3 c}}{96 d}-\frac {a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{8 d}-\frac {{\mathrm e}^{-5 d x -5 c} b^{3}}{160 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{d}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{d}\)	\(359\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (82) = 164\).
time = 0.47, size = 233, normalized size = 2.71 \begin {gather*} -\frac {1}{480} \, b^{3} {\left (\frac {{\left (35 \, e^{\left (-2 \, d x - 2 \, c\right )} - 330 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {330 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d} - \frac {960 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/480*b^3*((35*e^(-2*d*x - 2*c) - 330*e^(-4*d*x - 4*c) - 3)*e^(5*d*x + 5*c)/d + (330*e^(-d*x - c) - 35*e^(-3*
d*x - 3*c) + 3*e^(-5*d*x - 5*c))/d - 960*arctan(e^(-d*x - c))/d) - 1/8*a*b^2*((15*e^(-2*d*x - 2*c) - 1)*e^(3*d
*x + 3*c)/d - (15*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + 48*arctan(e^(-d*x - c))/d) + 3/2*a^2*b*(4*arctan(e^(-d*
x - c))/d + e^(d*x + c)/d - e^(-d*x - c)/d) + a^3*arctan(sinh(d*x + c))/d

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1114 vs. \(2 (82) = 164\).
time = 0.39, size = 1114, normalized size = 12.95 \begin {gather*} \frac {3 \, b^{3} \cosh \left (d x + c\right )^{10} + 30 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 3 \, b^{3} \sinh \left (d x + c\right )^{10} + 5 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{8} + 5 \, {\left (27 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 7 \, b^{3}\right )} \sinh \left (d x + c\right )^{8} + 40 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 30 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 10 \, {\left (63 \, b^{3} \cosh \left (d x + c\right )^{4} + 72 \, a^{2} b - 90 \, a b^{2} + 33 \, b^{3} + 14 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (189 \, b^{3} \cosh \left (d x + c\right )^{5} + 70 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 45 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} - 30 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (63 \, b^{3} \cosh \left (d x + c\right )^{6} + 35 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 72 \, a^{2} b + 90 \, a b^{2} - 33 \, b^{3} + 45 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 40 \, {\left (9 \, b^{3} \cosh \left (d x + c\right )^{7} + 7 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 15 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, b^{3} - 5 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 5 \, {\left (27 \, b^{3} \cosh \left (d x + c\right )^{8} + 28 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 90 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 12 \, a b^{2} + 7 \, b^{3} - 36 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 960 \, {\left ({\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sinh \left (d x + c\right )^{5}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 10 \, {\left (3 \, b^{3} \cosh \left (d x + c\right )^{9} + 4 \, {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 18 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 12 \, {\left (24 \, a^{2} b - 30 \, a b^{2} + 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{480 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right )^{4} \sinh \left (d x + c\right ) + 10 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(3*b^3*cosh(d*x + c)^10 + 30*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 3*b^3*sinh(d*x + c)^10 + 5*(12*a*b^2 -
7*b^3)*cosh(d*x + c)^8 + 5*(27*b^3*cosh(d*x + c)^2 + 12*a*b^2 - 7*b^3)*sinh(d*x + c)^8 + 40*(9*b^3*cosh(d*x +
c)^3 + (12*a*b^2 - 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(24*a^2*b - 30*a*b^2 + 11*b^3)*cosh(d*x + c)^6 +
10*(63*b^3*cosh(d*x + c)^4 + 72*a^2*b - 90*a*b^2 + 33*b^3 + 14*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^6 + 4*(189*b^3*cosh(d*x + c)^5 + 70*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^3 + 45*(24*a^2*b - 30*a*b^2 + 11*b^3)
*cosh(d*x + c))*sinh(d*x + c)^5 - 30*(24*a^2*b - 30*a*b^2 + 11*b^3)*cosh(d*x + c)^4 + 10*(63*b^3*cosh(d*x + c)
^6 + 35*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^4 - 72*a^2*b + 90*a*b^2 - 33*b^3 + 45*(24*a^2*b - 30*a*b^2 + 11*b^3)*
cosh(d*x + c)^2)*sinh(d*x + c)^4 + 40*(9*b^3*cosh(d*x + c)^7 + 7*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^5 + 15*(24*a
^2*b - 30*a*b^2 + 11*b^3)*cosh(d*x + c)^3 - 3*(24*a^2*b - 30*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 -
3*b^3 - 5*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^2 + 5*(27*b^3*cosh(d*x + c)^8 + 28*(12*a*b^2 - 7*b^3)*cosh(d*x + c)
^6 + 90*(24*a^2*b - 30*a*b^2 + 11*b^3)*cosh(d*x + c)^4 - 12*a*b^2 + 7*b^3 - 36*(24*a^2*b - 30*a*b^2 + 11*b^3)*
cosh(d*x + c)^2)*sinh(d*x + c)^2 + 960*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(a^3 - 3*a^2*b + 3
*a*b^2 - b^3)*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^3*sinh(d*x + c)
^2 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*co
sh(d*x + c)*sinh(d*x + c)^4 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sinh(d*x + c)^5)*arctan(cosh(d*x + c) + sinh(d*x
+ c)) + 10*(3*b^3*cosh(d*x + c)^9 + 4*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^7 + 18*(24*a^2*b - 30*a*b^2 + 11*b^3)*
cosh(d*x + c)^5 - 12*(24*a^2*b - 30*a*b^2 + 11*b^3)*cosh(d*x + c)^3 - (12*a*b^2 - 7*b^3)*cosh(d*x + c))*sinh(d
*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^4*sinh(d*x + c) + 10*d*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*d*
cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (82) = 164\).
time = 0.44, size = 204, normalized size = 2.37 \begin {gather*} \frac {3 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 35 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 720 \, a^{2} b e^{\left (d x + c\right )} - 900 \, a b^{2} e^{\left (d x + c\right )} + 330 \, b^{3} e^{\left (d x + c\right )} + 960 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - {\left (720 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 900 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 330 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(3*b^3*e^(5*d*x + 5*c) + 60*a*b^2*e^(3*d*x + 3*c) - 35*b^3*e^(3*d*x + 3*c) + 720*a^2*b*e^(d*x + c) - 900
*a*b^2*e^(d*x + c) + 330*b^3*e^(d*x + c) + 960*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*arctan(e^(d*x + c)) - (720*a^2*
b*e^(4*d*x + 4*c) - 900*a*b^2*e^(4*d*x + 4*c) + 330*b^3*e^(4*d*x + 4*c) + 60*a*b^2*e^(2*d*x + 2*c) - 35*b^3*e^
(2*d*x + 2*c) + 3*b^3)*e^(-5*d*x - 5*c))/d

________________________________________________________________________________________

Mupad [B]
time = 1.04, size = 294, normalized size = 3.42 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-30\,a\,b^2+11\,b^3\right )}{16\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (24\,a^2\,b-30\,a\,b^2+11\,b^3\right )}{16\,d}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {d^2}-b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}-3\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}}\right )\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}}{\sqrt {d^2}}-\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}-\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (12\,a-7\,b\right )}{96\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (12\,a-7\,b\right )}{96\,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),x)

[Out]

(exp(c + d*x)*(24*a^2*b - 30*a*b^2 + 11*b^3))/(16*d) - (exp(- c - d*x)*(24*a^2*b - 30*a*b^2 + 11*b^3))/(16*d)
+ (2*atan((exp(d*x)*exp(c)*(a^3*(d^2)^(1/2) - b^3*(d^2)^(1/2) + 3*a*b^2*(d^2)^(1/2) - 3*a^2*b*(d^2)^(1/2)))/(d
*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)^(1/2)))*(a^6 - 6*a^5*b - 6*a*b^5 + b^6
+ 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)^(1/2))/(d^2)^(1/2) - (b^3*exp(- 5*c - 5*d*x))/(160*d) + (b^3*exp(5*c
+ 5*d*x))/(160*d) - (b^2*exp(- 3*c - 3*d*x)*(12*a - 7*b))/(96*d) + (b^2*exp(3*c + 3*d*x)*(12*a - 7*b))/(96*d)

________________________________________________________________________________________