3.1.38 \(\int (c+d x) \text {sech}^3(a+b x) \, dx\) [38]

Optimal. Leaf size=102 \[ \frac {(c+d x) \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i d \text {PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b} \]

[Out]

(d*x+c)*arctan(exp(b*x+a))/b-1/2*I*d*polylog(2,-I*exp(b*x+a))/b^2+1/2*I*d*polylog(2,I*exp(b*x+a))/b^2+1/2*d*se
ch(b*x+a)/b^2+1/2*(d*x+c)*sech(b*x+a)*tanh(b*x+a)/b

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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4270, 4265, 2317, 2438} \begin {gather*} \frac {(c+d x) \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i d \text {Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {Li}_2\left (i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Sech[a + b*x]^3,x]

[Out]

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

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 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 4270

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

Rubi steps

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

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Mathematica [A]
time = 2.41, size = 178, normalized size = 1.75 \begin {gather*} \frac {b c \text {ArcTan}(\sinh (a+b x))+\frac {1}{2} d \left (-\left ((-2 i a+\pi -2 i b x) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )\right )+(-2 i a+\pi ) \log \left (\cot \left (\frac {1}{4} (2 i a+\pi +2 i b x)\right )\right )-2 i \left (\text {PolyLog}\left (2,-i e^{a+b x}\right )-\text {PolyLog}\left (2,i e^{a+b x}\right )\right )\right )+b d x \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)+d \text {sech}(a+b x) (1+b x \tanh (a))+b c \text {sech}(a+b x) \tanh (a+b x)}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (87 ) = 174\).
time = 1.19, size = 216, normalized size = 2.12

method	result	size
risch	\(\frac {{\mathrm e}^{b x +a} \left (b d x \,{\mathrm e}^{2 b x +2 a}+b c \,{\mathrm e}^{2 b x +2 a}-b d x +{\mathrm e}^{2 b x +2 a} d -b c +d \right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}+\frac {c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {i d \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {i d \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*sech(b*x+a)^3,x, algorithm="maxima")

[Out]

d*(((b*x*e^(3*a) + e^(3*a))*e^(3*b*x) - (b*x*e^a - e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a)
+ b^2) + 8*integrate(1/8*x*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)) - c*(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)))

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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. 1267 vs. \(2 (81) = 162\).
time = 0.39, size = 1267, normalized size = 12.42 \begin {gather*} \frac {2 \, {\left (b d x + b c + d\right )} \cosh \left (b x + a\right )^{3} + 6 \, {\left (b d x + b c + d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, {\left (b d x + b c + d\right )} \sinh \left (b x + a\right )^{3} - 2 \, {\left (b d x + b c - d\right )} \cosh \left (b x + a\right ) + {\left (i \, d \cosh \left (b x + a\right )^{4} + 4 i \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + i \, d \sinh \left (b x + a\right )^{4} + 2 i \, d \cosh \left (b x + a\right )^{2} - 2 \, {\left (-3 i \, d \cosh \left (b x + a\right )^{2} - i \, d\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (-i \, d \cosh \left (b x + a\right )^{3} - i \, d \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + i \, d\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + {\left (-i \, d \cosh \left (b x + a\right )^{4} - 4 i \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - i \, d \sinh \left (b x + a\right )^{4} - 2 i \, d \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 i \, d \cosh \left (b x + a\right )^{2} + i \, d\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (i \, d \cosh \left (b x + a\right )^{3} + i \, d \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - i \, d\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left ({\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (i \, b c - i \, a d\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{2} - i \, b c + i \, a d\right )} \sinh \left (b x + a\right )^{2} + i \, b c - i \, a d - 4 \, {\left ({\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + {\left ({\left (-i \, b c + i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (-i \, b c + i \, a d\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{2} + i \, b c - i \, a d\right )} \sinh \left (b x + a\right )^{2} - i \, b c + i \, a d - 4 \, {\left ({\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (i \, b c - i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left ({\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (-i \, b d x - i \, a d\right )} \sinh \left (b x + a\right )^{4} - i \, b d x - 2 \, {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (i \, b d x + 3 \, {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{2} + i \, a d\right )} \sinh \left (b x + a\right )^{2} - i \, a d - 4 \, {\left ({\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + {\left ({\left (i \, b d x + i \, a d\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (i \, b d x + i \, a d\right )} \sinh \left (b x + a\right )^{4} + i \, b d x - 2 \, {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (-i \, b d x + 3 \, {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{2} - i \, a d\right )} \sinh \left (b x + a\right )^{2} + i \, a d - 4 \, {\left ({\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )^{3} + {\left (-i \, b d x - i \, a d\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (b d x - 3 \, {\left (b d x + b c + d\right )} \cosh \left (b x + a\right )^{2} + b c - d\right )} \sinh \left (b x + a\right )}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} + 2 \, b^{2} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} + b^{2}\right )} \sinh \left (b x + a\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + b^{2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(b*d*x + b*c + d)*cosh(b*x + a)^3 + 6*(b*d*x + b*c + d)*cosh(b*x + a)*sinh(b*x + a)^2 + 2*(b*d*x + b*c
+ d)*sinh(b*x + a)^3 - 2*(b*d*x + b*c - d)*cosh(b*x + a) + (I*d*cosh(b*x + a)^4 + 4*I*d*cosh(b*x + a)*sinh(b*x
+ a)^3 + I*d*sinh(b*x + a)^4 + 2*I*d*cosh(b*x + a)^2 - 2*(-3*I*d*cosh(b*x + a)^2 - I*d)*sinh(b*x + a)^2 - 4*(
-I*d*cosh(b*x + a)^3 - I*d*cosh(b*x + a))*sinh(b*x + a) + I*d)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) + (-I*
d*cosh(b*x + a)^4 - 4*I*d*cosh(b*x + a)*sinh(b*x + a)^3 - I*d*sinh(b*x + a)^4 - 2*I*d*cosh(b*x + a)^2 - 2*(3*I
*d*cosh(b*x + a)^2 + I*d)*sinh(b*x + a)^2 - 4*(I*d*cosh(b*x + a)^3 + I*d*cosh(b*x + a))*sinh(b*x + a) - I*d)*d
ilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + ((I*b*c - I*a*d)*cosh(b*x + a)^4 - 4*(-I*b*c + I*a*d)*cosh(b*x + a)
*sinh(b*x + a)^3 + (I*b*c - I*a*d)*sinh(b*x + a)^4 - 2*(-I*b*c + I*a*d)*cosh(b*x + a)^2 - 2*(3*(-I*b*c + I*a*d
)*cosh(b*x + a)^2 - I*b*c + I*a*d)*sinh(b*x + a)^2 + I*b*c - I*a*d - 4*((-I*b*c + I*a*d)*cosh(b*x + a)^3 + (-I
*b*c + I*a*d)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + I) + ((-I*b*c + I*a*d)*cosh(b*
x + a)^4 - 4*(I*b*c - I*a*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b*c + I*a*d)*sinh(b*x + a)^4 - 2*(I*b*c - I*a
*d)*cosh(b*x + a)^2 - 2*(3*(I*b*c - I*a*d)*cosh(b*x + a)^2 + I*b*c - I*a*d)*sinh(b*x + a)^2 - I*b*c + I*a*d -
4*((I*b*c - I*a*d)*cosh(b*x + a)^3 + (I*b*c - I*a*d)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*
x + a) - I) + ((-I*b*d*x - I*a*d)*cosh(b*x + a)^4 - 4*(I*b*d*x + I*a*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b*
d*x - I*a*d)*sinh(b*x + a)^4 - I*b*d*x - 2*(I*b*d*x + I*a*d)*cosh(b*x + a)^2 - 2*(I*b*d*x + 3*(I*b*d*x + I*a*d
)*cosh(b*x + a)^2 + I*a*d)*sinh(b*x + a)^2 - I*a*d - 4*((I*b*d*x + I*a*d)*cosh(b*x + a)^3 + (I*b*d*x + I*a*d)*
cosh(b*x + a))*sinh(b*x + a))*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) + ((I*b*d*x + I*a*d)*cosh(b*x + a)^4
- 4*(-I*b*d*x - I*a*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (I*b*d*x + I*a*d)*sinh(b*x + a)^4 + I*b*d*x - 2*(-I*b*d
*x - I*a*d)*cosh(b*x + a)^2 - 2*(-I*b*d*x + 3*(-I*b*d*x - I*a*d)*cosh(b*x + a)^2 - I*a*d)*sinh(b*x + a)^2 + I*
a*d - 4*((-I*b*d*x - I*a*d)*cosh(b*x + a)^3 + (-I*b*d*x - I*a*d)*cosh(b*x + a))*sinh(b*x + a))*log(-I*cosh(b*x
+ a) - I*sinh(b*x + a) + 1) - 2*(b*d*x - 3*(b*d*x + b*c + d)*cosh(b*x + a)^2 + b*c - d)*sinh(b*x + a))/(b^2*c
osh(b*x + a)^4 + 4*b^2*cosh(b*x + a)*sinh(b*x + a)^3 + b^2*sinh(b*x + a)^4 + 2*b^2*cosh(b*x + a)^2 + 2*(3*b^2*
cosh(b*x + a)^2 + b^2)*sinh(b*x + a)^2 + b^2 + 4*(b^2*cosh(b*x + a)^3 + b^2*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 ) \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)*sech(b*x+a)**3,x)

[Out]

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

[Out]

integrate((d*x + c)*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 {c+d\,x}{{\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)/cosh(a + b*x)^3,x)

[Out]

int((c + d*x)/cosh(a + b*x)^3, x)

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