3.303 \(\int \text{sech}^7(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=131 \[ \frac{\left (5 a^2+2 a b+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \tanh (c+d x) \text{sech}^3(c+d x)}{24 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{6 d} \]

[Out]

((5*a^2 + 2*a*b + b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((5*a^2 + 2*a*b + b^2)*Sech[c + d*x]*Tanh[c + d*x])/(16
*d) + ((a - b)*(5*a + 3*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + ((a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*
x]^2)*Tanh[c + d*x])/(6*d)

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Rubi [A]  time = 0.132593, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3190, 413, 385, 199, 203} \[ \frac{\left (5 a^2+2 a b+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \tanh (c+d x) \text{sech}^3(c+d x)}{24 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((5*a^2 + 2*a*b + b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((5*a^2 + 2*a*b + b^2)*Sech[c + d*x]*Tanh[c + d*x])/(16
*d) + ((a - b)*(5*a + 3*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + ((a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*
x]^2)*Tanh[c + d*x])/(6*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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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}^7(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 )^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac{\operatorname{Subst}\left (\int \frac{a (5 a+b)+3 b (a+b) x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac{(a-b) (5 a+3 b) \text{sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac{\left (5 a^2+2 a b+b^2\right ) \text{sech}(c+d x) \tanh (c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \text{sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac{\left (5 a^2+2 a b+b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{\left (5 a^2+2 a b+b^2\right ) \text{sech}(c+d x) \tanh (c+d x)}{16 d}+\frac{(a-b) (5 a+3 b) \text{sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}\\ \end{align*}

Mathematica [C]  time = 10.1684, size = 715, normalized size = 5.46 \[ \frac{\text{csch}^3(c+d x) \left (32 \left (-\sinh ^2(c+d x)\right )^{3/2} \sinh ^4(c+d x) \left (5 a^2+9 a b \sinh ^2(c+d x)+4 b^2 \sinh ^4(c+d x)\right ) \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 )+4 \left (-\sinh ^2(c+d x)\right )^{3/2} \sinh ^4(c+d x) \left (155 a^2+242 a b \sinh ^2(c+d x)+95 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 )+16 a^2 \left (-\sinh ^2(c+d x)\right )^{3/2} \sinh ^4(c+d x) \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{9}{2}\right \},-\sinh ^2(c+d x)\right )+32 a b \left (-\sinh ^2(c+d x)\right )^{3/2} \sinh ^6(c+d x) \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{9}{2}\right \},-\sinh ^2(c+d x)\right )+16 b^2 \left (-\sinh ^2(c+d x)\right )^{3/2} \sinh ^8(c+d x) \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{9}{2}\right \},-\sinh ^2(c+d x)\right )+14980 a^2 \left (-\sinh ^2(c+d x)\right )^{3/2}-65625 a^2 \sqrt{-\sinh ^2(c+d x)}+1680 a^2 \sinh ^4(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+36855 a^2 \sinh ^2(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+65625 a^2 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )-23555 a b \left (-\sinh ^2(c+d x)\right )^{5/2}+91875 a b \left (-\sinh ^2(c+d x)\right )^{3/2}+1365 a b \sinh ^6(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+54180 a b \sinh ^4(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+91875 a b \sinh ^2(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+8855 b^2 \left (-\sinh ^2(c+d x)\right )^{3/2} \sinh ^4(c+d x)-32970 b^2 \left (-\sinh ^2(c+d x)\right )^{5/2}+525 b^2 \sinh ^8(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+19845 b^2 \sinh ^6(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )+32970 b^2 \sinh ^4(c+d x) \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right )\right )}{2520 d \sqrt{-\sinh ^2(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

(Csch[c + d*x]^3*(65625*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] + 36855*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c
+ d*x]^2 + 91875*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 + 1680*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2
]]*Sinh[c + d*x]^4 + 54180*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 32970*b^2*ArcTanh[Sqrt[-Sinh[
c + d*x]^2]]*Sinh[c + d*x]^4 + 1365*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 + 19845*b^2*ArcTanh[Sq
rt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 + 525*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 65625*a^2*Sq
rt[-Sinh[c + d*x]^2] + 14980*a^2*(-Sinh[c + d*x]^2)^(3/2) + 91875*a*b*(-Sinh[c + d*x]^2)^(3/2) + 8855*b^2*Sinh
[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2) + 16*a^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh
[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2) + 32*a*b*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1,
1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2) + 16*b^2*HypergeometricPFQ[{3/2, 2, 2,
2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(-Sinh[c + d*x]^2)^(3/2) - 23555*a*b*(-Sinh[c +
d*x]^2)^(5/2) - 32970*b^2*(-Sinh[c + d*x]^2)^(5/2) + 32*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 9/2},
-Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2)*(5*a^2 + 9*a*b*Sinh[c + d*x]^2 + 4*b^2*Sinh[c + d*x
]^4) + 4*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(
3/2)*(155*a^2 + 242*a*b*Sinh[c + d*x]^2 + 95*b^2*Sinh[c + d*x]^4)))/(2520*d*Sqrt[-Sinh[c + d*x]^2])

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Maple [B]  time = 0.082, size = 302, normalized size = 2.3 \begin{align*}{\frac{{a}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{6\,d}}+{\frac{5\,{a}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{24\,d}}+{\frac{5\,{a}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{16\,d}}+{\frac{5\,{a}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{8\,d}}-{\frac{2\,ab\sinh \left ( dx+c \right ) }{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ab\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{15\,d}}+{\frac{ab\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{12\,d}}+{\frac{ab{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}-{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{5\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{5}}{30\,d}}+{\frac{{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{24\,d}}+{\frac{{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{16\,d}}+{\frac{{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^2,x)

[Out]

1/6/d*a^2*tanh(d*x+c)*sech(d*x+c)^5+5/24/d*a^2*tanh(d*x+c)*sech(d*x+c)^3+5/16/d*a^2*sech(d*x+c)*tanh(d*x+c)+5/
8/d*a^2*arctan(exp(d*x+c))-2/5/d*a*b*sinh(d*x+c)/cosh(d*x+c)^6+1/15/d*a*b*tanh(d*x+c)*sech(d*x+c)^5+1/12/d*a*b
*tanh(d*x+c)*sech(d*x+c)^3+1/8/d*a*b*sech(d*x+c)*tanh(d*x+c)+1/4/d*a*b*arctan(exp(d*x+c))-1/3/d*b^2*sinh(d*x+c
)^3/cosh(d*x+c)^6-1/5/d*b^2*sinh(d*x+c)/cosh(d*x+c)^6+1/30/d*b^2*tanh(d*x+c)*sech(d*x+c)^5+1/24/d*b^2*tanh(d*x
+c)*sech(d*x+c)^3+1/16/d*b^2*sech(d*x+c)*tanh(d*x+c)+1/8/d*b^2*arctan(exp(d*x+c))

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Maxima [B]  time = 1.79108, size = 652, normalized size = 4.98 \begin{align*} -\frac{1}{24} \, a^{2}{\left (\frac{15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac{1}{12} \, a b{\left (\frac{3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{3 \, e^{\left (-d x - c\right )} + 17 \, e^{\left (-3 \, d x - 3 \, c\right )} - 114 \, e^{\left (-5 \, d x - 5 \, c\right )} + 114 \, e^{\left (-7 \, d x - 7 \, c\right )} - 17 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac{1}{24} \, b^{2}{\left (\frac{3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{3 \, e^{\left (-d x - c\right )} - 47 \, e^{\left (-3 \, d x - 3 \, c\right )} + 78 \, e^{\left (-5 \, d x - 5 \, c\right )} - 78 \, e^{\left (-7 \, d x - 7 \, c\right )} + 47 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

-1/24*a^2*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) + 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) - 198*e^(
-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) - 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20
*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 1/12*a*b*(3*arcta
n(e^(-d*x - c))/d - (3*e^(-d*x - c) + 17*e^(-3*d*x - 3*c) - 114*e^(-5*d*x - 5*c) + 114*e^(-7*d*x - 7*c) - 17*e
^(-9*d*x - 9*c) - 3*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 1
5*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 1/24*b^2*(3*arctan(e^(-d*x - c))/d - (
3*e^(-d*x - c) - 47*e^(-3*d*x - 3*c) + 78*e^(-5*d*x - 5*c) - 78*e^(-7*d*x - 7*c) + 47*e^(-9*d*x - 9*c) - 3*e^(
-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*
e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1)))

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Fricas [B]  time = 1.71361, size = 7356, normalized size = 56.15 \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)^7*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/24*(3*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^11 + 33*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^10 + 3*(
5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^11 + (85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^9 + (165*(5*a^2 + 2*a*b + b^2
)*cosh(d*x + c)^2 + 85*a^2 + 34*a*b - 47*b^2)*sinh(d*x + c)^9 + 9*(55*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 +
(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 6*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^7 + 6*(
165*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^2 + 33*a^2 - 38*a*b + 1
3*b^2)*sinh(d*x + c)^7 + 42*(33*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x
+ c)^3 + (33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 - 6*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c
)^5 + 6*(231*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 21*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^4 + 21*(33*a^
2 - 38*a*b + 13*b^2)*cosh(d*x + c)^2 - 33*a^2 + 38*a*b - 13*b^2)*sinh(d*x + c)^5 + 6*(165*(5*a^2 + 2*a*b + b^2
)*cosh(d*x + c)^7 + 21*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^5 + 35*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c
)^3 - 5*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^3
+ (495*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 84*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^6 + 210*(33*a^2 -
38*a*b + 13*b^2)*cosh(d*x + c)^4 - 60*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^2 - 85*a^2 - 34*a*b + 47*b^2)*s
inh(d*x + c)^3 + 3*(55*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 + 12*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^7 +
42*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^5 - 20*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^3 - (85*a^2 + 34*
a*b - 47*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^12 + 12*(5*a^2 + 2*a*b +
b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^12 + 6*(5*a^2 + 2*a*b + b^2)*cosh(d
*x + c)^10 + 6*(11*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^10 + 20*(11*(5*a
^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^8 + 15*(33*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 18*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c
)^2 + 5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^8 + 24*(33*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 30*(5*a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^3 + 5*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(5*a^2 + 2*a*b + b^2)*cos
h(d*x + c)^6 + 4*(231*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 315*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 105*
(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 25*a^2 + 10*a*b + 5*b^2)*sinh(d*x + c)^6 + 24*(33*(5*a^2 + 2*a*b + b^2
)*cosh(d*x + c)^7 + 63*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 35*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 5*(5
*a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 15*(33*(5*a^2
+ 2*a*b + b^2)*cosh(d*x + c)^8 + 84*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 70*(5*a^2 + 2*a*b + b^2)*cosh(d*x
+ c)^4 + 20*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 20*(11*(5*a^2 + 2*a
*b + b^2)*cosh(d*x + c)^9 + 36*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 42*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^
5 + 20*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(5*a
^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 6*(11*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 + 45*(5*a^2 + 2*a*b + b^2)*co
sh(d*x + c)^8 + 70*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 50*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 15*(5*a^
2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^2 + 5*a^2 + 2*a*b + b^2 + 12*((5*a^2 + 2
*a*b + b^2)*cosh(d*x + c)^11 + 5*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 + 10*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c
)^7 + 10*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 5*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (5*a^2 + 2*a*b + b^
2)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 3*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)
+ 3*(11*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 + 3*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^8 + 14*(33*a^2 -
38*a*b + 13*b^2)*cosh(d*x + c)^6 - 10*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^4 - (85*a^2 + 34*a*b - 47*b^2)*
cosh(d*x + c)^2 - 5*a^2 - 2*a*b - b^2)*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^1
1 + d*sinh(d*x + c)^12 + 6*d*cosh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^10 + 20*(11*d*cosh(
d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^8 + 15*(33*d*cosh(d*x + c)^4 + 18*d*cosh(
d*x + c)^2 + d)*sinh(d*x + c)^8 + 24*(33*d*cosh(d*x + c)^5 + 30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*
x + c)^7 + 20*d*cosh(d*x + c)^6 + 4*(231*d*cosh(d*x + c)^6 + 315*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 + 5
*d)*sinh(d*x + c)^6 + 24*(33*d*cosh(d*x + c)^7 + 63*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 + 5*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 15*d*cosh(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8 + 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x +
c)^4 + 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x + c)^9 + 36*d*cosh(d*x + c)^7 + 42*d*cos
h(d*x + c)^5 + 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(
d*x + c)^10 + 45*d*cosh(d*x + c)^8 + 70*d*cosh(d*x + c)^6 + 50*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + d)*s
inh(d*x + c)^2 + 12*(d*cosh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 + 10*d*cosh(d*x + c)^7 + 10*d*cosh(d*x + c)^5 +
5*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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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)**7*(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.1794, size = 394, normalized size = 3.01 \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 (5 \, a^{2} + 2 \, a b + b^{2}\right )}}{32 \, d} + \frac{15 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 6 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 3 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 160 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 64 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 32 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 96 \, a b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 48 \, b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{24 \,{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/32*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(5*a^2 + 2*a*b + b^2)/d + 1/24*(15*a^2*(e^(d*x +
c) - e^(-d*x - c))^5 + 6*a*b*(e^(d*x + c) - e^(-d*x - c))^5 + 3*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 160*a^2*(
e^(d*x + c) - e^(-d*x - c))^3 + 64*a*b*(e^(d*x + c) - e^(-d*x - c))^3 - 32*b^2*(e^(d*x + c) - e^(-d*x - c))^3
+ 528*a^2*(e^(d*x + c) - e^(-d*x - c)) - 96*a*b*(e^(d*x + c) - e^(-d*x - c)) - 48*b^2*(e^(d*x + c) - e^(-d*x -
c)))/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3*d)