3.3.99 \(\int \text {sech}^3(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\) [299]

Optimal. Leaf size=64 \[ \frac {(a-b) (a+3 b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {b^2 \sinh (c+d x)}{d}+\frac {(a-b)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 64, 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+3 b) (a-b) \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {(a-b)^2 \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {b^2 \sinh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*(a + 3*b)*ArcTan[Sinh[c + d*x]])/(2*d) + (b^2*Sinh[c + d*x])/d + ((a - b)^2*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 )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b^2+\frac {a^2-b^2+2 (a-b) b x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b^2 \sinh (c+d x)}{d}+\frac {\text {Subst}\left (\int \frac {a^2-b^2+2 (a-b) b x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b^2 \sinh (c+d x)}{d}+\frac {(a-b)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {((a-b) (a+3 b)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}\\ &=\frac {(a-b) (a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b^2 \sinh (c+d x)}{d}+\frac {(a-b)^2 \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 = 11.26, size = 253, normalized size = 3.95 \begin {gather*} \frac {\text {csch}^3(c+d x) \left (-64 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2-35 \left (a^2 \left (375+37 \sinh ^2(c+d x)\right )+b^2 \sinh ^4(c+d x) \left (303+61 \sinh ^2(c+d x)\right )+2 a b \sinh ^2(c+d x) \left (375+61 \sinh ^2(c+d x)\right )\right )+\frac {105 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (b^2 \sinh ^4(c+d x) \left (101+54 \sinh ^2(c+d x)+\sinh ^4(c+d x)\right )+2 a b \sinh ^2(c+d x) \left (125+62 \sinh ^2(c+d x)+\sinh ^4(c+d x)\right )+a^2 \left (125+54 \sinh ^2(c+d x)+9 \sinh ^4(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{1680 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Csch[c + d*x]^3*(-64*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(a + b*
Sinh[c + d*x]^2)^2 - 35*(a^2*(375 + 37*Sinh[c + d*x]^2) + b^2*Sinh[c + d*x]^4*(303 + 61*Sinh[c + d*x]^2) + 2*a
*b*Sinh[c + d*x]^2*(375 + 61*Sinh[c + d*x]^2)) + (105*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(b^2*Sinh[c + d*x]^4*(10
1 + 54*Sinh[c + d*x]^2 + Sinh[c + d*x]^4) + 2*a*b*Sinh[c + d*x]^2*(125 + 62*Sinh[c + d*x]^2 + Sinh[c + d*x]^4)
+ a^2*(125 + 54*Sinh[c + d*x]^2 + 9*Sinh[c + d*x]^4)))/Sqrt[-Sinh[c + d*x]^2]))/(1680*d)

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Maple [C] Result contains complex when optimal does not.
time = 1.47, size = 190, normalized size = 2.97

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} b^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {{\mathrm e}^{d x +c} \left (a^{2}-2 a b +b^{2}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a b}{d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a b}{d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{2 d}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (60) = 120\).
time = 0.51, size = 234, normalized size = 3.66 \begin {gather*} \frac {1}{2} \, 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 )} - 2 \, a 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^{2} {\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)^2,x, algorithm="maxima")

[Out]

1/2*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)))) - 2*a*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^2*(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. 759 vs. \(2 (60) = 120\).
time = 0.38, size = 759, normalized size = 11.86 \begin {gather*} \frac {b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 4 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 2 \, {\left ({\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 6 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5} + 2 \, d \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{4} + 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\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)^2,x, algorithm="fricas")

[Out]

1/2*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + (2*a^2 - 4*a*b + 3*b^2)
*cosh(d*x + c)^4 + (15*b^2*cosh(d*x + c)^2 + 2*a^2 - 4*a*b + 3*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3
+ (2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c)^2 + (15*b^2*
cosh(d*x + c)^4 + 6*(2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c)^2 - 2*a^2 + 4*a*b - 3*b^2)*sinh(d*x + c)^2 - b^2 + 2
*((a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^5 + 5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^4 + (a^2 + 2*a*b
- 3*b^2)*sinh(d*x + c)^5 + 2*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^3 + 2*(5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)
^2 + a^2 + 2*a*b - 3*b^2)*sinh(d*x + c)^3 + 2*(5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b - 3*b^
2)*cosh(d*x + c))*sinh(d*x + c)^2 + (a^2 + 2*a*b - 3*b^2)*cosh(d*x + c) + (5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x +
c)^4 + 6*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b - 3*b^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + si
nh(d*x + c)) + 2*(3*b^2*cosh(d*x + c)^5 + 2*(2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c)^3 - (2*a^2 - 4*a*b + 3*b^2)*
cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5 + 2*d
*cosh(d*x + c)^3 + 2*(5*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + 2*(5*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*s
inh(d*x + c)^2 + d*cosh(d*x + c) + (5*d*cosh(d*x + c)^4 + 6*d*cosh(d*x + c)^2 + d)*sinh(d*x + c))

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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 )^{2} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \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)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (60) = 120\).
time = 0.43, size = 163, normalized size = 2.55 \begin {gather*} \frac {2 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + {\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^{2} + 2 \, a b - 3 \, b^{2}\right )} + \frac {4 \, {\left (a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + b^{2} {\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}}{4 \, 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)^2,x, algorithm="giac")

[Out]

1/4*(2*b^2*(e^(d*x + c) - e^(-d*x - c)) + (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a^2 + 2*a*b
- 3*b^2) + 4*(a^2*(e^(d*x + c) - e^(-d*x - c)) - 2*a*b*(e^(d*x + c) - e^(-d*x - c)) + b^2*(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 = 0.88, size = 220, normalized size = 3.44 \begin {gather*} \frac {b^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {d^2}-3\,b^2\,\sqrt {d^2}+2\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^4+4\,a^3\,b-2\,a^2\,b^2-12\,a\,b^3+9\,b^4}}\right )\,\sqrt {a^4+4\,a^3\,b-2\,a^2\,b^2-12\,a\,b^3+9\,b^4}}{\sqrt {d^2}}-\frac {b^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,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)^2/cosh(c + d*x)^3,x)

[Out]

(b^2*exp(c + d*x))/(2*d) + (atan((exp(d*x)*exp(c)*(a^2*(d^2)^(1/2) - 3*b^2*(d^2)^(1/2) + 2*a*b*(d^2)^(1/2)))/(
d*(4*a^3*b - 12*a*b^3 + a^4 + 9*b^4 - 2*a^2*b^2)^(1/2)))*(4*a^3*b - 12*a*b^3 + a^4 + 9*b^4 - 2*a^2*b^2)^(1/2))
/(d^2)^(1/2) - (b^2*exp(- c - d*x))/(2*d) + (exp(c + d*x)*(a^2 - 2*a*b + b^2))/(d*(exp(2*c + 2*d*x) + 1)) - (2
*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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