\(\int x^3 (a+b \text {sech}(c+d x^2))^2 \, dx\) [10]

Optimal result

Integrand size = 18, antiderivative size = 119 \[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^4}{4}+\frac {2 a b x^2 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 \log \left (\cosh \left (c+d x^2\right )\right )}{2 d^2}-\frac {i a b \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i a b \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {b^2 x^2 \tanh \left (c+d x^2\right )}{2 d} \]

[Out]

1/4*a^2*x^4+2*a*b*x^2*arctan(exp(d*x^2+c))/d-1/2*b^2*ln(cosh(d*x^2+c))/d^2-I*a*b*polylog(2,-I*exp(d*x^2+c))/d^
2+I*a*b*polylog(2,I*exp(d*x^2+c))/d^2+1/2*b^2*x^2*tanh(d*x^2+c)/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5544, 4275, 4265, 2317, 2438, 4269, 3556} \[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^4}{4}+\frac {2 a b x^2 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {i a b \operatorname {PolyLog}\left (2,-i e^{d x^2+c}\right )}{d^2}+\frac {i a b \operatorname {PolyLog}\left (2,i e^{d x^2+c}\right )}{d^2}-\frac {b^2 \log \left (\cosh \left (c+d x^2\right )\right )}{2 d^2}+\frac {b^2 x^2 \tanh \left (c+d x^2\right )}{2 d} \]

[In]

Int[x^3*(a + b*Sech[c + d*x^2])^2,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^2*ArcTan[E^(c + d*x^2)])/d - (b^2*Log[Cosh[c + d*x^2]])/(2*d^2) - (I*a*b*PolyLog[2, (-I
)*E^(c + d*x^2)])/d^2 + (I*a*b*PolyLog[2, I*E^(c + d*x^2)])/d^2 + (b^2*x^2*Tanh[c + d*x^2])/(2*d)

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 5544

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x (a+b \text {sech}(c+d x))^2 \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (a^2 x+2 a b x \text {sech}(c+d x)+b^2 x \text {sech}^2(c+d x)\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^4}{4}+(a b) \text {Subst}\left (\int x \text {sech}(c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \text {Subst}\left (\int x \text {sech}^2(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^4}{4}+\frac {2 a b x^2 \arctan \left (e^{c+d x^2}\right )}{d}+\frac {b^2 x^2 \tanh \left (c+d x^2\right )}{2 d}-\frac {(i a b) \text {Subst}\left (\int \log \left (1-i e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {(i a b) \text {Subst}\left (\int \log \left (1+i e^{c+d x}\right ) \, dx,x,x^2\right )}{d}-\frac {b^2 \text {Subst}\left (\int \tanh (c+d x) \, dx,x,x^2\right )}{2 d} \\ & = \frac {a^2 x^4}{4}+\frac {2 a b x^2 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 \log \left (\cosh \left (c+d x^2\right )\right )}{2 d^2}+\frac {b^2 x^2 \tanh \left (c+d x^2\right )}{2 d}-\frac {(i a b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^2}+\frac {(i a b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^2} \\ & = \frac {a^2 x^4}{4}+\frac {2 a b x^2 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {b^2 \log \left (\cosh \left (c+d x^2\right )\right )}{2 d^2}-\frac {i a b \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i a b \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {b^2 x^2 \tanh \left (c+d x^2\right )}{2 d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(324\) vs. \(2(119)=238\).

Time = 1.94 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.72 \[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {4 b^2 d e^{2 c} x^2+a^2 d^2 x^4+a^2 d^2 e^{2 c} x^4+4 i a b d x^2 \log \left (1-i e^{c+d x^2}\right )+4 i a b d e^{2 c} x^2 \log \left (1-i e^{c+d x^2}\right )-4 i a b d x^2 \log \left (1+i e^{c+d x^2}\right )-4 i a b d e^{2 c} x^2 \log \left (1+i e^{c+d x^2}\right )-2 b^2 \log \left (1+e^{2 \left (c+d x^2\right )}\right )-2 b^2 e^{2 c} \log \left (1+e^{2 \left (c+d x^2\right )}\right )-4 i a b \left (1+e^{2 c}\right ) \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )+4 i a b \left (1+e^{2 c}\right ) \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )+2 b^2 d x^2 \text {sech}(c) \text {sech}\left (c+d x^2\right ) \sinh \left (d x^2\right )+2 b^2 d e^{2 c} x^2 \text {sech}(c) \text {sech}\left (c+d x^2\right ) \sinh \left (d x^2\right )}{4 d^2 \left (1+e^{2 c}\right )} \]

[In]

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

[Out]

(4*b^2*d*E^(2*c)*x^2 + a^2*d^2*x^4 + a^2*d^2*E^(2*c)*x^4 + (4*I)*a*b*d*x^2*Log[1 - I*E^(c + d*x^2)] + (4*I)*a*
b*d*E^(2*c)*x^2*Log[1 - I*E^(c + d*x^2)] - (4*I)*a*b*d*x^2*Log[1 + I*E^(c + d*x^2)] - (4*I)*a*b*d*E^(2*c)*x^2*
Log[1 + I*E^(c + d*x^2)] - 2*b^2*Log[1 + E^(2*(c + d*x^2))] - 2*b^2*E^(2*c)*Log[1 + E^(2*(c + d*x^2))] - (4*I)
*a*b*(1 + E^(2*c))*PolyLog[2, (-I)*E^(c + d*x^2)] + (4*I)*a*b*(1 + E^(2*c))*PolyLog[2, I*E^(c + d*x^2)] + 2*b^
2*d*x^2*Sech[c]*Sech[c + d*x^2]*Sinh[d*x^2] + 2*b^2*d*E^(2*c)*x^2*Sech[c]*Sech[c + d*x^2]*Sinh[d*x^2])/(4*d^2*
(1 + E^(2*c)))

Maple [F]

\[\int x^{3} {\left (a +b \,\operatorname {sech}\left (d \,x^{2}+c \right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (100) = 200\).

Time = 0.28 (sec) , antiderivative size = 782, normalized size of antiderivative = 6.57 \[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\frac {a^{2} d^{2} x^{4} + 4 \, b^{2} c + {\left (a^{2} d^{2} x^{4} + 4 \, b^{2} d x^{2} + 4 \, b^{2} c\right )} \cosh \left (d x^{2} + c\right )^{2} + 2 \, {\left (a^{2} d^{2} x^{4} + 4 \, b^{2} d x^{2} + 4 \, b^{2} c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + {\left (a^{2} d^{2} x^{4} + 4 \, b^{2} d x^{2} + 4 \, b^{2} c\right )} \sinh \left (d x^{2} + c\right )^{2} - 4 \, {\left (-i \, a b \cosh \left (d x^{2} + c\right )^{2} - 2 i \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - i \, a b \sinh \left (d x^{2} + c\right )^{2} - i \, a b\right )} {\rm Li}_2\left (i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right )\right ) - 4 \, {\left (i \, a b \cosh \left (d x^{2} + c\right )^{2} + 2 i \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + i \, a b \sinh \left (d x^{2} + c\right )^{2} + i \, a b\right )} {\rm Li}_2\left (-i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right )\right ) - 2 \, {\left (2 i \, a b c + {\left (2 i \, a b c + b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} + 2 \, {\left (2 i \, a b c + b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + {\left (2 i \, a b c + b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} + b^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + i\right ) - 2 \, {\left (-2 i \, a b c + {\left (-2 i \, a b c + b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} + 2 \, {\left (-2 i \, a b c + b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + {\left (-2 i \, a b c + b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} + b^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - i\right ) - 4 \, {\left (i \, a b d x^{2} + i \, a b c + {\left (i \, a b d x^{2} + i \, a b c\right )} \cosh \left (d x^{2} + c\right )^{2} + 2 \, {\left (i \, a b d x^{2} + i \, a b c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + {\left (i \, a b d x^{2} + i \, a b c\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right ) + 1\right ) - 4 \, {\left (-i \, a b d x^{2} - i \, a b c + {\left (-i \, a b d x^{2} - i \, a b c\right )} \cosh \left (d x^{2} + c\right )^{2} + 2 \, {\left (-i \, a b d x^{2} - i \, a b c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + {\left (-i \, a b d x^{2} - i \, a b c\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (-i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right ) + 1\right )}{4 \, {\left (d^{2} \cosh \left (d x^{2} + c\right )^{2} + 2 \, d^{2} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + d^{2} \sinh \left (d x^{2} + c\right )^{2} + d^{2}\right )}} \]

[In]

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

[Out]

1/4*(a^2*d^2*x^4 + 4*b^2*c + (a^2*d^2*x^4 + 4*b^2*d*x^2 + 4*b^2*c)*cosh(d*x^2 + c)^2 + 2*(a^2*d^2*x^4 + 4*b^2*
d*x^2 + 4*b^2*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) + (a^2*d^2*x^4 + 4*b^2*d*x^2 + 4*b^2*c)*sinh(d*x^2 + c)^2 - 4
*(-I*a*b*cosh(d*x^2 + c)^2 - 2*I*a*b*cosh(d*x^2 + c)*sinh(d*x^2 + c) - I*a*b*sinh(d*x^2 + c)^2 - I*a*b)*dilog(
I*cosh(d*x^2 + c) + I*sinh(d*x^2 + c)) - 4*(I*a*b*cosh(d*x^2 + c)^2 + 2*I*a*b*cosh(d*x^2 + c)*sinh(d*x^2 + c)
+ I*a*b*sinh(d*x^2 + c)^2 + I*a*b)*dilog(-I*cosh(d*x^2 + c) - I*sinh(d*x^2 + c)) - 2*(2*I*a*b*c + (2*I*a*b*c +
b^2)*cosh(d*x^2 + c)^2 + 2*(2*I*a*b*c + b^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c) + (2*I*a*b*c + b^2)*sinh(d*x^2 +
c)^2 + b^2)*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) + I) - 2*(-2*I*a*b*c + (-2*I*a*b*c + b^2)*cosh(d*x^2 + c)^2
+ 2*(-2*I*a*b*c + b^2)*cosh(d*x^2 + c)*sinh(d*x^2 + c) + (-2*I*a*b*c + b^2)*sinh(d*x^2 + c)^2 + b^2)*log(cosh
(d*x^2 + c) + sinh(d*x^2 + c) - I) - 4*(I*a*b*d*x^2 + I*a*b*c + (I*a*b*d*x^2 + I*a*b*c)*cosh(d*x^2 + c)^2 + 2*
(I*a*b*d*x^2 + I*a*b*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) + (I*a*b*d*x^2 + I*a*b*c)*sinh(d*x^2 + c)^2)*log(I*cos
h(d*x^2 + c) + I*sinh(d*x^2 + c) + 1) - 4*(-I*a*b*d*x^2 - I*a*b*c + (-I*a*b*d*x^2 - I*a*b*c)*cosh(d*x^2 + c)^2
+ 2*(-I*a*b*d*x^2 - I*a*b*c)*cosh(d*x^2 + c)*sinh(d*x^2 + c) + (-I*a*b*d*x^2 - I*a*b*c)*sinh(d*x^2 + c)^2)*lo
g(-I*cosh(d*x^2 + c) - I*sinh(d*x^2 + c) + 1))/(d^2*cosh(d*x^2 + c)^2 + 2*d^2*cosh(d*x^2 + c)*sinh(d*x^2 + c)
+ d^2*sinh(d*x^2 + c)^2 + d^2)

Sympy [F]

\[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^{3} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

1/4*a^2*x^4 + 1/2*(2*x^2*e^(2*d*x^2 + 2*c)/(d*e^(2*d*x^2 + 2*c) + d) - log((e^(2*d*x^2 + 2*c) + 1)*e^(-2*c))/d
^2)*b^2 + 4*a*b*integrate(x^3*e^(d*x^2 + c)/(e^(2*d*x^2 + 2*c) + 1), x)

Giac [F]

\[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{3} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^3\,{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2 \,d x \]

[In]

int(x^3*(a + b/cosh(c + d*x^2))^2,x)

[Out]

int(x^3*(a + b/cosh(c + d*x^2))^2, x)