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

Optimal. Leaf size=175 \[ \frac {(c+d x)^2 \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {d^2 \text {ArcTan}(\sinh (a+b x))}{b^3}-\frac {i d (c+d x) \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {i d (c+d x) \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {i d^2 \text {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {i d^2 \text {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b} \]

[Out]

(d*x+c)^2*arctan(exp(b*x+a))/b-d^2*arctan(sinh(b*x+a))/b^3-I*d*(d*x+c)*polylog(2,-I*exp(b*x+a))/b^2+I*d*(d*x+c
)*polylog(2,I*exp(b*x+a))/b^2+I*d^2*polylog(3,-I*exp(b*x+a))/b^3-I*d^2*polylog(3,I*exp(b*x+a))/b^3+d*(d*x+c)*s
ech(b*x+a)/b^2+1/2*(d*x+c)^2*sech(b*x+a)*tanh(b*x+a)/b

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Rubi [A]
time = 0.10, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4271, 3855, 4265, 2611, 2320, 6724} \begin {gather*} -\frac {d^2 \text {ArcTan}(\sinh (a+b x))}{b^3}+\frac {(c+d x)^2 \text {ArcTan}\left (e^{a+b x}\right )}{b}+\frac {i d^2 \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}-\frac {i d^2 \text {Li}_3\left (i e^{a+b x}\right )}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac {i d (c+d x) \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \tanh (a+b x) \text {sech}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Sech[a + b*x]^3,x]

[Out]

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*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 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int (c+d x)^2 \text {sech}^3(a+b x) \, dx &=\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^2 \text {sech}(a+b x) \, dx-\frac {d^2 \int \text {sech}(a+b x) \, dx}{b^2}\\ &=\frac {(c+d x)^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d^2 \tan ^{-1}(\sinh (a+b x))}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b}-\frac {(i d) \int (c+d x) \log \left (1-i e^{a+b x}\right ) \, dx}{b}+\frac {(i d) \int (c+d x) \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=\frac {(c+d x)^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d^2 \tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac {i d (c+d x) \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {\left (i d^2\right ) \int \text {Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (i d^2\right ) \int \text {Li}_2\left (i e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac {(c+d x)^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d^2 \tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac {i d (c+d x) \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {\left (i d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\left (i d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=\frac {(c+d x)^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {d^2 \tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {i d (c+d x) \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}+\frac {i d (c+d x) \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {i d^2 \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}-\frac {i d^2 \text {Li}_3\left (i e^{a+b x}\right )}{b^3}+\frac {d (c+d x) \text {sech}(a+b x)}{b^2}+\frac {(c+d x)^2 \text {sech}(a+b x) \tanh (a+b x)}{2 b}\\ \end {align*}

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Mathematica [A]
time = 1.37, size = 270, normalized size = 1.54 \begin {gather*} \frac {i \left (-2 i b^2 c^2 \text {ArcTan}\left (e^{a+b x}\right )+4 i d^2 \text {ArcTan}\left (e^{a+b x}\right )+2 b^2 c d x \log \left (1-i e^{a+b x}\right )+b^2 d^2 x^2 \log \left (1-i e^{a+b x}\right )-2 b^2 c d x \log \left (1+i e^{a+b x}\right )-b^2 d^2 x^2 \log \left (1+i e^{a+b x}\right )-2 b d (c+d x) \text {PolyLog}\left (2,-i e^{a+b x}\right )+2 b d (c+d x) \text {PolyLog}\left (2,i e^{a+b x}\right )+2 d^2 \text {PolyLog}\left (3,-i e^{a+b x}\right )-2 d^2 \text {PolyLog}\left (3,i e^{a+b x}\right )\right )+b^2 (c+d x)^2 \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)+b (c+d x) \text {sech}(a+b x) (2 d+b (c+d x) \tanh (a))}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*((-2*I)*b^2*c^2*ArcTan[E^(a + b*x)] + (4*I)*d^2*ArcTan[E^(a + b*x)] + 2*b^2*c*d*x*Log[1 - I*E^(a + b*x)] +
b^2*d^2*x^2*Log[1 - I*E^(a + b*x)] - 2*b^2*c*d*x*Log[1 + I*E^(a + b*x)] - b^2*d^2*x^2*Log[1 + I*E^(a + b*x)] -
2*b*d*(c + d*x)*PolyLog[2, (-I)*E^(a + b*x)] + 2*b*d*(c + d*x)*PolyLog[2, I*E^(a + b*x)] + 2*d^2*PolyLog[3, (
-I)*E^(a + b*x)] - 2*d^2*PolyLog[3, I*E^(a + b*x)]) + b^2*(c + d*x)^2*Sech[a]*Sech[a + b*x]^2*Sinh[b*x] + b*(c
+ d*x)*Sech[a + b*x]*(2*d + b*(c + d*x)*Tanh[a]))/(2*b^3)

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{2} \mathrm {sech}\left (b x +a \right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2651 vs. \(2 (150) = 300\).
time = 0.44, size = 2651, normalized size = 15.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(b^2*d^2*x^2 + b^2*c^2 + 2*b*c*d + 2*(b^2*c*d + b*d^2)*x)*cosh(b*x + a)^3 + 6*(b^2*d^2*x^2 + b^2*c^2 +
2*b*c*d + 2*(b^2*c*d + b*d^2)*x)*cosh(b*x + a)*sinh(b*x + a)^2 + 2*(b^2*d^2*x^2 + b^2*c^2 + 2*b*c*d + 2*(b^2*c
*d + b*d^2)*x)*sinh(b*x + a)^3 - 2*(b^2*d^2*x^2 + b^2*c^2 - 2*b*c*d + 2*(b^2*c*d - b*d^2)*x)*cosh(b*x + a) - 2
*((-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)^4 + 4*(-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b*d^2*
x - I*b*c*d)*sinh(b*x + a)^4 - I*b*d^2*x - I*b*c*d + 2*(-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)^2 + 2*(-I*b*d^2*x
- I*b*c*d + 3*(-I*b*d^2*x - I*b*c*d)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 4*((-I*b*d^2*x - I*b*c*d)*cosh(b*x + a
)^3 + (-I*b*d^2*x - I*b*c*d)*cosh(b*x + a))*sinh(b*x + a))*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 2*((I*b*
d^2*x + I*b*c*d)*cosh(b*x + a)^4 + 4*(I*b*d^2*x + I*b*c*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (I*b*d^2*x + I*b*c*
d)*sinh(b*x + a)^4 + I*b*d^2*x + I*b*c*d + 2*(I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^2 + 2*(I*b*d^2*x + I*b*c*d +
3*(I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 4*((I*b*d^2*x + I*b*c*d)*cosh(b*x + a)^3 + (I*b*d^2
*x + I*b*c*d)*cosh(b*x + a))*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + ((I*b^2*c^2 - 2*I*a*b*
c*d + I*(a^2 - 2)*d^2)*cosh(b*x + a)^4 - 4*(-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x + a)*sinh(b*x
+ a)^3 + (I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2)*sinh(b*x + a)^4 + I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*
d^2 - 2*(-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x + a)^2 - 2*(-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 -
2)*d^2 + 3*(-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 4*((-I*b^2*c^2 + 2*
I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x + a)^3 + (-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x + a))*sin
h(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + I) + ((-I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2)*cosh(b*x +
a)^4 - 4*(I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b^2*c^2 + 2*I*a*b*c*d
- I*(a^2 - 2)*d^2)*sinh(b*x + a)^4 - I*b^2*c^2 + 2*I*a*b*c*d - I*(a^2 - 2)*d^2 - 2*(I*b^2*c^2 - 2*I*a*b*c*d +
I*(a^2 - 2)*d^2)*cosh(b*x + a)^2 - 2*(I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2 + 3*(I*b^2*c^2 - 2*I*a*b*c*d
+ I*(a^2 - 2)*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 4*((I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2)*cosh(b*x
+ a)^3 + (I*b^2*c^2 - 2*I*a*b*c*d + I*(a^2 - 2)*d^2)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*
x + a) - I) + (-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x + (-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c*d + I*a^2*d^2)*cos
h(b*x + a)^4 - 4*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + 2*I*a*b*c*d - I*a^2*d^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I
*b^2*d^2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c*d + I*a^2*d^2)*sinh(b*x + a)^4 - 2*I*a*b*c*d + I*a^2*d^2 - 2*(I*b^2*d
^2*x^2 + 2*I*b^2*c*d*x + 2*I*a*b*c*d - I*a^2*d^2)*cosh(b*x + a)^2 - 2*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + 2*I*a*b
*c*d - I*a^2*d^2 + 3*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + 2*I*a*b*c*d - I*a^2*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^
2 - 4*((I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + 2*I*a*b*c*d - I*a^2*d^2)*cosh(b*x + a)^3 + (I*b^2*d^2*x^2 + 2*I*b^2*c*
d*x + 2*I*a*b*c*d - I*a^2*d^2)*cosh(b*x + a))*sinh(b*x + a))*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) + (I*b
^2*d^2*x^2 + 2*I*b^2*c*d*x + (I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + 2*I*a*b*c*d - I*a^2*d^2)*cosh(b*x + a)^4 - 4*(-I
*b^2*d^2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c*d + I*a^2*d^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (I*b^2*d^2*x^2 + 2*I*b
^2*c*d*x + 2*I*a*b*c*d - I*a^2*d^2)*sinh(b*x + a)^4 + 2*I*a*b*c*d - I*a^2*d^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*
d*x - 2*I*a*b*c*d + I*a^2*d^2)*cosh(b*x + a)^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c*d + I*a^2*d^2 +
3*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c*d + I*a^2*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 4*((-I*b^2*d^
2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c*d + I*a^2*d^2)*cosh(b*x + a)^3 + (-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - 2*I*a*b*c
*d + I*a^2*d^2)*cosh(b*x + a))*sinh(b*x + a))*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 2*(I*d^2*cosh(b*x
+ a)^4 + 4*I*d^2*cosh(b*x + a)*sinh(b*x + a)^3 + I*d^2*sinh(b*x + a)^4 + 2*I*d^2*cosh(b*x + a)^2 + 2*(3*I*d^2*
cosh(b*x + a)^2 + I*d^2)*sinh(b*x + a)^2 + I*d^2 + 4*(I*d^2*cosh(b*x + a)^3 + I*d^2*cosh(b*x + a))*sinh(b*x +
a))*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) - 2*(-I*d^2*cosh(b*x + a)^4 - 4*I*d^2*cosh(b*x + a)*sinh(b*x
+ a)^3 - I*d^2*sinh(b*x + a)^4 - 2*I*d^2*cosh(b*x + a)^2 + 2*(-3*I*d^2*cosh(b*x + a)^2 - I*d^2)*sinh(b*x + a)
^2 - I*d^2 + 4*(-I*d^2*cosh(b*x + a)^3 - I*d^2*cosh(b*x + a))*sinh(b*x + a))*polylog(3, -I*cosh(b*x + a) - I*s
inh(b*x + a)) - 2*(b^2*d^2*x^2 + b^2*c^2 - 2*b*c*d - 3*(b^2*d^2*x^2 + b^2*c^2 + 2*b*c*d + 2*(b^2*c*d + b*d^2)*
x)*cosh(b*x + a)^2 + 2*(b^2*c*d - b*d^2)*x)*sinh(b*x + a))/(b^3*cosh(b*x + a)^4 + 4*b^3*cosh(b*x + a)*sinh(b*x
+ a)^3 + b^3*sinh(b*x + a)^4 + 2*b^3*cosh(b*x + a)^2 + b^3 + 2*(3*b^3*cosh(b*x + a)^2 + b^3)*sinh(b*x + a)^2
+ 4*(b^3*cosh(b*x + a)^3 + b^3*cosh(b*x + a))*sinh(b*x + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/cosh(a + b*x)^3,x)

[Out]

int((c + d*x)^2/cosh(a + b*x)^3, x)

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