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

Optimal. Leaf size=96 \[ \frac{\left (3 a^2+2 a b+3 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{3 \left (a^2-b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0927501, antiderivative size = 96, normalized size of antiderivative = 1., 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 = {3190, 413, 385, 203} \[ \frac{\left (3 a^2+2 a b+3 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{3 \left (a^2-b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 d} \]

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 3190

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]

Rule 413

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 385

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{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{\operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 5.94609, size = 303, normalized size = 3.16 \[ -\frac{\text{csch}^3(c+d x) \left (128 \sinh ^6(c+d x) \left (7 a^2+12 a b \sinh ^2(c+d x)+5 b^2 \sinh ^4(c+d x)\right ) \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2\right \},\left \{1,1,\frac{9}{2}\right \},-\sinh ^2(c+d x)\right )+128 \sinh ^6(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2\right \},\left \{1,1,1,\frac{9}{2}\right \},-\sinh ^2(c+d x)\right )-\frac{105 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right ) \left (\left (37 a^2+988 a b+649 b^2\right ) \sinh ^4(c+d x)+1125 a^2+2 b (11 a+189 b) \sinh ^6(c+d x)+2 a (297 a+875 b) \sinh ^2(c+d x)+9 b^2 \sinh ^8(c+d x)\right )}{\sqrt{-\sinh ^2(c+d x)}}+35 \left (3375 a^2+a \sinh ^2(c+d x) (657 a+607 b \cosh (2 (c+d x))+4643 b)+485 b^2 \sinh ^6(c+d x)+1947 b^2 \sinh ^4(c+d x)\right )\right )}{6720 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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]))/(6720*d)

________________________________________________________________________________________

Maple [B]  time = 0.051, size = 237, normalized size = 2.5 \begin{align*}{\frac{{a}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}-{\frac{2\,ab\sinh \left ( dx+c \right ) }{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{ab\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{6\,d}}+{\frac{ab{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{4\,d}}+{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{2\,d}}-{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}-{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}} \end{align*}

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)

[Out]

1/4/d*a^2*tanh(d*x+c)*sech(d*x+c)^3+3/8/d*a^2*sech(d*x+c)*tanh(d*x+c)+3/4/d*a^2*arctan(exp(d*x+c))-2/3/d*a*b*s
inh(d*x+c)/cosh(d*x+c)^4+1/6/d*a*b*tanh(d*x+c)*sech(d*x+c)^3+1/4/d*a*b*sech(d*x+c)*tanh(d*x+c)+1/2/d*a*b*arcta
n(exp(d*x+c))-1/d*b^2*sinh(d*x+c)^3/cosh(d*x+c)^4-1/d*b^2*sinh(d*x+c)/cosh(d*x+c)^4+1/4/d*b^2*tanh(d*x+c)*sech
(d*x+c)^3+3/8/d*b^2*sech(d*x+c)*tanh(d*x+c)+3/4/d*b^2*arctan(exp(d*x+c))

________________________________________________________________________________________

Maxima [B]  time = 1.72498, size = 468, normalized size = 4.88 \begin{align*} -\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{align*}

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)))

________________________________________________________________________________________

Fricas [B]  time = 1.5918, size = 3672, normalized size = 38.25 \begin{align*} \text{result too large to display} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.16042, size = 297, normalized size = 3.09 \begin{align*} \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 )}}{16 \, d} + \frac{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 )}}{4 \,{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2} d} \end{align*}

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)/d + 1/4*(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)