3.4.1 \(\int \text {sech}^5(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\) [301]

Optimal. Leaf size=96 \[ \frac {\left (3 a^2+2 a b+3 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {3 \left (a^2-b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{4 d} \]

[Out]

1/8*(3*a^2+2*a*b+3*b^2)*arctan(sinh(d*x+c))/d+3/8*(a^2-b^2)*sech(d*x+c)*tanh(d*x+c)/d+1/4*(a-b)*sech(d*x+c)^3*
(a+b*sinh(d*x+c)^2)*tanh(d*x+c)/d

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Rubi [A]
time = 0.06, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3269, 424, 393, 209} \begin {gather*} \frac {\left (3 a^2+2 a b+3 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d}+\frac {3 \left (a^2-b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{8 d}+\frac {(a-b) \tanh (c+d x) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2 + 2*a*b + 3*b^2)*ArcTan[Sinh[c + d*x]])/(8*d) + (3*(a^2 - b^2)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + ((
a - b)*Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(4*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 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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 )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{4 d}+\frac {\text {Subst}\left (\int \frac {a (3 a+b)+b (a+3 b) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac {3 \left (a^2-b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{4 d}+\frac {\left (3 a^2+2 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {\left (3 a^2+2 a b+3 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac {3 \left (a^2-b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{4 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 = 4.52, size = 303, normalized size = 3.16 \begin {gather*} -\frac {\text {csch}^3(c+d x) \left (128 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2+128 \, _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 (7 a^2+12 a b \sinh ^2(c+d x)+5 b^2 \sinh ^4(c+d x)\right )+35 \left (3375 a^2+a (657 a+4643 b+607 b \cosh (2 (c+d x))) \sinh ^2(c+d x)+1947 b^2 \sinh ^4(c+d x)+485 b^2 \sinh ^6(c+d x)\right )-\frac {105 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (1125 a^2+2 a (297 a+875 b) \sinh ^2(c+d x)+\left (37 a^2+988 a b+649 b^2\right ) \sinh ^4(c+d x)+2 b (11 a+189 b) \sinh ^6(c+d x)+9 b^2 \sinh ^8(c+d x)\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{6720 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6720*(Csch[c + d*x]^3*(128*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c +
d*x]^6*(a + b*Sinh[c + d*x]^2)^2 + 128*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c
+ d*x]^6*(7*a^2 + 12*a*b*Sinh[c + d*x]^2 + 5*b^2*Sinh[c + d*x]^4) + 35*(3375*a^2 + a*(657*a + 4643*b + 607*b*
Cosh[2*(c + d*x)])*Sinh[c + d*x]^2 + 1947*b^2*Sinh[c + d*x]^4 + 485*b^2*Sinh[c + d*x]^6) - (105*ArcTanh[Sqrt[-
Sinh[c + d*x]^2]]*(1125*a^2 + 2*a*(297*a + 875*b)*Sinh[c + d*x]^2 + (37*a^2 + 988*a*b + 649*b^2)*Sinh[c + d*x]
^4 + 2*b*(11*a + 189*b)*Sinh[c + d*x]^6 + 9*b^2*Sinh[c + d*x]^8))/Sqrt[-Sinh[c + d*x]^2]))/d

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Maple [C] Result contains complex when optimal does not.
time = 1.67, size = 276, normalized size = 2.88

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 a^{2} {\mathrm e}^{6 d x +6 c}+2 a b \,{\mathrm e}^{6 d x +6 c}-5 b^{2} {\mathrm e}^{6 d x +6 c}+11 a^{2} {\mathrm e}^{4 d x +4 c}-14 a b \,{\mathrm e}^{4 d x +4 c}+3 b^{2} {\mathrm e}^{4 d x +4 c}-11 a^{2} {\mathrm e}^{2 d x +2 c}+14 a b \,{\mathrm e}^{2 d x +2 c}-3 b^{2} {\mathrm e}^{2 d x +2 c}-3 a^{2}-2 a b +5 b^{2}\right )}{4 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{8 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a b}{4 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{8 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a b}{4 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{8 d}\)	\(276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*exp(d*x+c)*(3*a^2*exp(6*d*x+6*c)+2*a*b*exp(6*d*x+6*c)-5*b^2*exp(6*d*x+6*c)+11*a^2*exp(4*d*x+4*c)-14*a*b*ex
p(4*d*x+4*c)+3*b^2*exp(4*d*x+4*c)-11*a^2*exp(2*d*x+2*c)+14*a*b*exp(2*d*x+2*c)-3*b^2*exp(2*d*x+2*c)-3*a^2-2*a*b
+5*b^2)/d/(1+exp(2*d*x+2*c))^4+3/8*I/d*ln(exp(d*x+c)+I)*a^2+1/4*I/d*ln(exp(d*x+c)+I)*a*b+3/8*I/d*ln(exp(d*x+c)
+I)*b^2-3/8*I/d*ln(exp(d*x+c)-I)*a^2-1/4*I/d*ln(exp(d*x+c)-I)*a*b-3/8*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. 347 vs. \(2 (90) = 180\).
time = 0.50, size = 347, normalized size = 3.61 \begin {gather*} -\frac {1}{4} \, b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {5 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, 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} \, a^{2} {\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}{2} \, a 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)^2,x, algorithm="maxima")

[Out]

-1/4*b^2*(3*arctan(e^(-d*x - c))/d + (5*e^(-d*x - c) - 3*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c) - 5*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*a^2*(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))) - 1/2*a*b*(arctan(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. 1472 vs. \(2 (90) = 180\).
time = 0.39, size = 1472, normalized size = 15.33 \begin {gather*} \text {Too large to display} \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)^2,x, algorithm="fricas")

[Out]

1/4*((3*a^2 + 2*a*b - 5*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 + 2*a*b - 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^
2 + 2*a*b - 5*b^2)*sinh(d*x + c)^7 + (11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + 2*a*b - 5*b^2)*c
osh(d*x + c)^2 + 11*a^2 - 14*a*b + 3*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + 2*a*b - 5*b^2)*cosh(d*x + c)^3 + (11
*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^3 + (35*(3*a^2
+ 2*a*b - 5*b^2)*cosh(d*x + c)^4 + 10*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^2 - 11*a^2 + 14*a*b - 3*b^2)*si
nh(d*x + c)^3 + (21*(3*a^2 + 2*a*b - 5*b^2)*cosh(d*x + c)^5 + 10*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^3 - 3
*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^8 + 8*(3*a^
2 + 2*a*b + 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)^8 + 4*(3*a^2 + 2*a*b
+ 3*b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 3*a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)
^6 + 8*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 +
6*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^4 + 30*(3*a^2 + 2*a*b
+ 3*b^2)*cosh(d*x + c)^2 + 9*a^2 + 6*a*b + 9*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c
)^5 + 10*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^6 + 15*(3*a^2 + 2*a*b +
3*b^2)*cosh(d*x + c)^4 + 9*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 3*a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)^2 +
3*a^2 + 2*a*b + 3*b^2 + 8*((3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^7 + 3*(3*a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^
5 + 3*(3*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))*arctan(c
osh(d*x + c) + sinh(d*x + c)) - (3*a^2 + 2*a*b - 5*b^2)*cosh(d*x + c) + (7*(3*a^2 + 2*a*b - 5*b^2)*cosh(d*x +
c)^6 + 5*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^4 - 3*(11*a^2 - 14*a*b + 3*b^2)*cosh(d*x + c)^2 - 3*a^2 - 2*a
*b + 5*b^2)*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*co
sh(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*cosh(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))*sinh(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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (90) = 180\).
time = 0.44, size = 218, normalized size = 2.27 \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^{2} + 2 \, a b + 3 \, b^{2}\right )} + \frac {4 \, {\left (3 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 2 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 5 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 8 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 12 \, b^{2} {\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)^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^2 + 2*a*b + 3*b^2) + 4*(3*a^2*(e^(d*x + c)
- e^(-d*x - c))^3 + 2*a*b*(e^(d*x + c) - e^(-d*x - c))^3 - 5*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 20*a^2*(e^(d
*x + c) - e^(-d*x - c)) - 8*a*b*(e^(d*x + c) - e^(-d*x - c)) - 12*b^2*(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 = 0.89, size = 327, normalized size = 3.41 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a^2\,\sqrt {d^2}+3\,b^2\,\sqrt {d^2}+2\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {9\,a^4+12\,a^3\,b+22\,a^2\,b^2+12\,a\,b^3+9\,b^4}}\right )\,\sqrt {9\,a^4+12\,a^3\,b+22\,a^2\,b^2+12\,a\,b^3+9\,b^4}}{4\,\sqrt {d^2}}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\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 )}+\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left (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\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2-10\,a\,b+9\,b^2\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+2\,a\,b-5\,b^2\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,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)^5,x)

[Out]

(atan((exp(d*x)*exp(c)*(3*a^2*(d^2)^(1/2) + 3*b^2*(d^2)^(1/2) + 2*a*b*(d^2)^(1/2)))/(d*(12*a*b^3 + 12*a^3*b +
9*a^4 + 9*b^4 + 22*a^2*b^2)^(1/2)))*(12*a*b^3 + 12*a^3*b + 9*a^4 + 9*b^4 + 22*a^2*b^2)^(1/2))/(4*(d^2)^(1/2))
- (6*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) +
(4*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(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)*(a^2 - 10*a*b + 9*b^2))/(2*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (
exp(c + d*x)*(2*a*b + 3*a^2 - 5*b^2))/(4*d*(exp(2*c + 2*d*x) + 1))

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