3.3.91 \(\int \text {sech}^5(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [291]

Optimal. Leaf size=70 \[ \frac {(3 a+b) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {(3 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \]

[Out]

1/8*(3*a+b)*arctan(sinh(d*x+c))/d+1/8*(3*a+b)*sech(d*x+c)*tanh(d*x+c)/d+1/4*(a-b)*sech(d*x+c)^3*tanh(d*x+c)/d

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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3269, 393, 205, 209} \begin {gather*} \frac {(3 a+b) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {(a-b) \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}+\frac {(3 a+b) \tanh (c+d x) \text {sech}(c+d x)}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2),x]

[Out]

((3*a + b)*ArcTan[Sinh[c + d*x]])/(8*d) + ((3*a + b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + ((a - b)*Sech[c + d*
x]^3*Tanh[c + d*x])/(4*d)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 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}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b) \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {(3 a+b) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac {(3 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {(3 a+b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {(3 a+b) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac {(3 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 60, normalized size = 0.86 \begin {gather*} \frac {(3 a+b) \text {ArcTan}(\sinh (c+d x))+(3 a+b) \text {sech}(c+d x) \tanh (c+d x)+2 (a-b) \text {sech}^3(c+d x) \tanh (c+d x)}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2),x]

[Out]

((3*a + b)*ArcTan[Sinh[c + d*x]] + (3*a + b)*Sech[c + d*x]*Tanh[c + d*x] + 2*(a - b)*Sech[c + d*x]^3*Tanh[c +
d*x])/(8*d)

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Maple [C] Result contains complex when optimal does not.
time = 1.29, size = 172, normalized size = 2.46

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 a \,{\mathrm e}^{6 d x +6 c}+b \,{\mathrm e}^{6 d x +6 c}+11 a \,{\mathrm e}^{4 d x +4 c}-7 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{2 d x +2 c}+7 b \,{\mathrm e}^{2 d x +2 c}-3 a -b \right )}{4 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{8 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{8 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{8 d}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(d*x+c)*(3*a*exp(6*d*x+6*c)+b*exp(6*d*x+6*c)+11*a*exp(4*d*x+4*c)-7*b*exp(4*d*x+4*c)-11*a*exp(2*d*x+2*c)
+7*b*exp(2*d*x+2*c)-3*a-b)/d/(1+exp(2*d*x+2*c))^4+3/8*I/d*ln(exp(d*x+c)+I)*a+1/8*I/d*ln(exp(d*x+c)+I)*b-3/8*I/
d*ln(exp(d*x+c)-I)*a-1/8*I/d*ln(exp(d*x+c)-I)*b

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (64) = 128\).
time = 0.47, size = 228, normalized size = 3.26 \begin {gather*} -\frac {1}{4} \, a {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - \frac {1}{4} \, b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/4*a*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x -
7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - 1/4*b*(arc
tan(e^(-d*x - c))/d - (e^(-d*x - c) - 7*e^(-3*d*x - 3*c) + 7*e^(-5*d*x - 5*c) - e^(-7*d*x - 7*c))/(d*(4*e^(-2*
d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (64) = 128\).
time = 0.38, size = 1046, normalized size = 14.94 \begin {gather*} \frac {{\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + {\left (3 \, a + b\right )} \sinh \left (d x + c\right )^{7} + {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{5} + {\left (21 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 11 \, a - 7 \, b\right )} \sinh \left (d x + c\right )^{5} + 5 \, {\left (7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{3} + {\left (35 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{2} - 11 \, a + 7 \, b\right )} \sinh \left (d x + c\right )^{3} + {\left (21 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{8} + 8 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + {\left (3 \, a + b\right )} \sinh \left (d x + c\right )^{8} + 4 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a + b\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 30 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 9 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{6} + 15 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a + b\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left ({\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{7} + 3 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{5} + 3 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 3 \, a + b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (3 \, a + b\right )} \cosh \left (d x + c\right ) + {\left (7 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{4} - 3 \, {\left (11 \, a - 7 \, b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - b\right )} \sinh \left (d x + c\right )}{4 \, {\left (d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} + 4 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} + 30 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} + 15 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} + 3 \, d \cosh \left (d x + c\right )^{5} + 3 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/4*((3*a + b)*cosh(d*x + c)^7 + 7*(3*a + b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a + b)*sinh(d*x + c)^7 + (11*a
- 7*b)*cosh(d*x + c)^5 + (21*(3*a + b)*cosh(d*x + c)^2 + 11*a - 7*b)*sinh(d*x + c)^5 + 5*(7*(3*a + b)*cosh(d*
x + c)^3 + (11*a - 7*b)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a - 7*b)*cosh(d*x + c)^3 + (35*(3*a + b)*cosh(d*x
+ c)^4 + 10*(11*a - 7*b)*cosh(d*x + c)^2 - 11*a + 7*b)*sinh(d*x + c)^3 + (21*(3*a + b)*cosh(d*x + c)^5 + 10*(
11*a - 7*b)*cosh(d*x + c)^3 - 3*(11*a - 7*b)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a + b)*cosh(d*x + c)^8 + 8*(
3*a + b)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a + b)*sinh(d*x + c)^8 + 4*(3*a + b)*cosh(d*x + c)^6 + 4*(7*(3*a +
b)*cosh(d*x + c)^2 + 3*a + b)*sinh(d*x + c)^6 + 8*(7*(3*a + b)*cosh(d*x + c)^3 + 3*(3*a + b)*cosh(d*x + c))*s
inh(d*x + c)^5 + 6*(3*a + b)*cosh(d*x + c)^4 + 2*(35*(3*a + b)*cosh(d*x + c)^4 + 30*(3*a + b)*cosh(d*x + c)^2
+ 9*a + 3*b)*sinh(d*x + c)^4 + 8*(7*(3*a + b)*cosh(d*x + c)^5 + 10*(3*a + b)*cosh(d*x + c)^3 + 3*(3*a + b)*cos
h(d*x + c))*sinh(d*x + c)^3 + 4*(3*a + b)*cosh(d*x + c)^2 + 4*(7*(3*a + b)*cosh(d*x + c)^6 + 15*(3*a + b)*cosh
(d*x + c)^4 + 9*(3*a + b)*cosh(d*x + c)^2 + 3*a + b)*sinh(d*x + c)^2 + 8*((3*a + b)*cosh(d*x + c)^7 + 3*(3*a +
b)*cosh(d*x + c)^5 + 3*(3*a + b)*cosh(d*x + c)^3 + (3*a + b)*cosh(d*x + c))*sinh(d*x + c) + 3*a + b)*arctan(c
osh(d*x + c) + sinh(d*x + c)) - (3*a + b)*cosh(d*x + c) + (7*(3*a + b)*cosh(d*x + c)^6 + 5*(11*a - 7*b)*cosh(d
*x + c)^4 - 3*(11*a - 7*b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*si
nh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6 + 8*(7*d
*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 + 30*d*c
osh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sin
h(d*x + c)^3 + 4*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 + 15*d*cosh(d*x + c)^4 + 9*d*cosh(d*x + c)^2 + d)*
sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 + 3*d*cosh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x
+ c) + d)

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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 ) \operatorname {sech}^{5}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**5*(a+b*sinh(d*x+c)**2),x)

[Out]

Integral((a + b*sinh(c + d*x)**2)*sech(c + d*x)**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (64) = 128\).
time = 0.44, size = 153, normalized size = 2.19 \begin {gather*} \frac {{\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 (3 \, a + b\right )} + \frac {4 \, {\left (3 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/16*((pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a + b) + 4*(3*a*(e^(d*x + c) - e^(-d*x - c))^
3 + b*(e^(d*x + c) - e^(-d*x - c))^3 + 20*a*(e^(d*x + c) - e^(-d*x - c)) - 4*b*(e^(d*x + c) - e^(-d*x - c)))/(
(e^(d*x + c) - e^(-d*x - c))^2 + 4)^2)/d

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Mupad [B]
time = 0.83, size = 280, normalized size = 4.00 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a\,\sqrt {d^2}+b\,\sqrt {d^2}\right )}{d\,\sqrt {9\,a^2+6\,a\,b+b^2}}\right )\,\sqrt {9\,a^2+6\,a\,b+b^2}}{4\,\sqrt {d^2}}-\frac {\frac {b\,{\mathrm {e}}^{5\,c+5\,d\,x}}{d}+\frac {2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (2\,a-b\right )}{d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+b\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a-3\,b\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a-b\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,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)/cosh(c + d*x)^5,x)

[Out]

(atan((exp(d*x)*exp(c)*(3*a*(d^2)^(1/2) + b*(d^2)^(1/2)))/(d*(6*a*b + 9*a^2 + b^2)^(1/2)))*(6*a*b + 9*a^2 + b^
2)^(1/2))/(4*(d^2)^(1/2)) - ((b*exp(5*c + 5*d*x))/d + (2*exp(3*c + 3*d*x)*(2*a - b))/d + (b*exp(c + d*x))/d)/(
4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) + (exp(c + d*x)*(3*a + b)
)/(4*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(a - 3*b))/(2*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) -
(2*exp(c + d*x)*(a - b))/(d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1))

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