3.314 \(\int \text{sech}^7(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\)
Optimal. Leaf size=154 \[ \frac{(a+b) \left (5 a^2-2 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{(a-b) \left (15 a^2+14 a b+15 b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{48 d}+\frac{5 \left (a^2-b^2\right ) \tanh (c+d x) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{24 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]
[Out]
((a + b)*(5*a^2 - 2*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((a - b)*(15*a^2 + 14*a*b + 15*b^2)*Sech[c +
d*x]*Tanh[c + d*x])/(48*d) + (5*(a^2 - b^2)*Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) + ((
a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(6*d)
________________________________________________________________________________________
Rubi [A] time = 0.152918, antiderivative size = 154, 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, 526, 385, 203} \[ \frac{(a+b) \left (5 a^2-2 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{(a-b) \left (15 a^2+14 a b+15 b^2\right ) \tanh (c+d x) \text{sech}(c+d x)}{48 d}+\frac{5 \left (a^2-b^2\right ) \tanh (c+d x) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{24 d}+\frac{(a-b) \tanh (c+d x) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
((a + b)*(5*a^2 - 2*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((a - b)*(15*a^2 + 14*a*b + 15*b^2)*Sech[c +
d*x]*Tanh[c + d*x])/(48*d) + (5*(a^2 - b^2)*Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) + ((
a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^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 526
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(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 - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*(p + 1) + (b*e - a*f)*(n*q + 1))*x
^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]
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}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{\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 )^2 \tanh (c+d x)}{6 d}+\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right ) \left (a (5 a+b)+b (a+5 b) x^2\right )}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac{5 \left (a^2-b^2\right ) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}-\frac{\operatorname{Subst}\left (\int \frac{-a \left (15 a^2+4 a b+5 b^2\right )-b \left (5 a^2+4 a b+15 b^2\right ) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{24 d}\\ &=\frac{(a-b) \left (15 a^2+14 a b+15 b^2\right ) \text{sech}(c+d x) \tanh (c+d x)}{48 d}+\frac{5 \left (a^2-b^2\right ) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}+\frac{\left ((a+b) \left (5 a^2-2 a b+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac{(a+b) \left (5 a^2-2 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac{(a-b) \left (15 a^2+14 a b+15 b^2\right ) \text{sech}(c+d x) \tanh (c+d x)}{48 d}+\frac{5 \left (a^2-b^2\right ) \text{sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac{(a-b) \text{sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}\\ \end{align*}
Mathematica [C] time = 14.2621, size = 1192, normalized size = 7.74 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]
[Out]
(Csch[c + d*x]^5*(-117228825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] - 109265625*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2
]]*Sinh[c + d*x]^2 - 274542345*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 17069535*a^3*ArcTanh[Sq
rt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 260465625*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 2155
49775*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 142065*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[
c + d*x]^6 - 41427855*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 207173295*a*b^2*ArcTanh[Sqrt[-Si
nh[c + d*x]^2]]*Sinh[c + d*x]^6 - 58009455*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 210735*a^2*b*
ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 33756345*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]
^8 - 56109375*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 174825*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2
]]*Sinh[c + d*x]^10 - 9261945*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10 - 48825*b^3*ArcTanh[Sqrt[-S
inh[c + d*x]^2]]*Sinh[c + d*x]^12 + 117228825*a^3*Sqrt[-Sinh[c + d*x]^2] + 4093425*a^3*Sinh[c + d*x]^4*Sqrt[-S
inh[c + d*x]^2] + 168951510*a^2*b*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 215549775*a*b^2*Sinh[c + d*x]^4*Sqr
t[-Sinh[c + d*x]^2] + 9514449*a^2*b*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 135323370*a*b^2*Sinh[c + d*x]^6*S
qrt[-Sinh[c + d*x]^2] + 58009455*b^3*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 7808535*a*b^2*Sinh[c + d*x]^8*Sq
rt[-Sinh[c + d*x]^2] + 36772890*b^3*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 2160711*b^3*Sinh[c + d*x]^10*Sqrt
[-Sinh[c + d*x]^2] - 70189350*a^3*(-Sinh[c + d*x]^2)^(3/2) - 274542345*a^2*b*(-Sinh[c + d*x]^2)^(3/2) + 1024*a
^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[
c + d*x]^2)^(3/2) + 3072*a^2*b*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x
]^2]*Sinh[c + d*x]^8*(-Sinh[c + d*x]^2)^(3/2) + 3072*a*b^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1
, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10*(-Sinh[c + d*x]^2)^(3/2) + 1024*b^3*HypergeometricPFQ[{3/2,
2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12*(-Sinh[c + d*x]^2)^(3/2) + 1536*H
ypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2
)^(3/2)*(a + b*Sinh[c + d*x]^2)^2*(9*a + 7*b*Sinh[c + d*x]^2) + 256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1
, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2)*(295*a^3 + 741*a^2*b*Sinh[c + d*x]^2 +
621*a*b^2*Sinh[c + d*x]^4 + 175*b^3*Sinh[c + d*x]^6)))/(725760*d*Sqrt[-Sinh[c + d*x]^2])
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Maple [B] time = 0.062, size = 467, normalized size = 3. \begin{align*} \text{result too large to display} \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)^3,x)
[Out]
1/6/d*a^3*tanh(d*x+c)*sech(d*x+c)^5+5/24/d*a^3*tanh(d*x+c)*sech(d*x+c)^3+5/16/d*a^3*sech(d*x+c)*tanh(d*x+c)+5/
8/d*a^3*arctan(exp(d*x+c))-3/5/d*a^2*b*sinh(d*x+c)/cosh(d*x+c)^6+1/10/d*a^2*b*tanh(d*x+c)*sech(d*x+c)^5+1/8/d*
a^2*b*tanh(d*x+c)*sech(d*x+c)^3+3/16/d*a^2*b*sech(d*x+c)*tanh(d*x+c)+3/8/d*a^2*b*arctan(exp(d*x+c))-1/d*a*b^2*
sinh(d*x+c)^3/cosh(d*x+c)^6-3/5/d*a*b^2*sinh(d*x+c)/cosh(d*x+c)^6+1/10/d*a*b^2*tanh(d*x+c)*sech(d*x+c)^5+1/8/d
*a*b^2*tanh(d*x+c)*sech(d*x+c)^3+3/16/d*a*b^2*sech(d*x+c)*tanh(d*x+c)+3/8/d*a*b^2*arctan(exp(d*x+c))-1/d*b^3*s
inh(d*x+c)^5/cosh(d*x+c)^6-5/3/d*b^3*sinh(d*x+c)^3/cosh(d*x+c)^6-1/d*b^3*sinh(d*x+c)/cosh(d*x+c)^6+1/6/d*b^3*t
anh(d*x+c)*sech(d*x+c)^5+5/24/d*b^3*tanh(d*x+c)*sech(d*x+c)^3+5/16/d*b^3*sech(d*x+c)*tanh(d*x+c)+5/8/d*b^3*arc
tan(exp(d*x+c))
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Maxima [B] time = 1.83213, size = 872, normalized size = 5.66 \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)^3,x, algorithm="maxima")
[Out]
-1/24*b^3*(15*arctan(e^(-d*x - c))/d + (33*e^(-d*x - c) - 5*e^(-3*d*x - 3*c) + 90*e^(-5*d*x - 5*c) - 90*e^(-7*
d*x - 7*c) + 5*e^(-9*d*x - 9*c) - 33*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))) - 1/24*a^3*(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))) - 1/8*a^2*b*(3*arctan(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) + 15*e^(-8*d*x - 8*c) +
6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 1/8*a*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 = 1.81393, size = 9044, normalized size = 58.73 \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)^3,x, algorithm="fricas")
[Out]
1/24*(3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^11 + 33*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d
*x + c)*sinh(d*x + c)^10 + 3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*sinh(d*x + c)^11 + (85*a^3 + 51*a^2*b - 141*
a*b^2 + 5*b^3)*cosh(d*x + c)^9 + (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3 + 165*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^3 + (85*a^3 +
51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 18*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh
(d*x + c)^7 + 18*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^4 + 11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b
^3 + 2*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 42*(33*(5*a^3 + 3*a^2*b + 3*
a*b^2 - 11*b^3)*cosh(d*x + c)^5 + 2*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(11*a^3 - 19*a
^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 18*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x +
c)^5 + 18*(77*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^6 + 7*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3
)*cosh(d*x + c)^4 - 11*a^3 + 19*a^2*b - 13*a*b^2 + 5*b^3 + 21*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^5 + 18*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^7 + 7*(85*a^3 + 51*a^2*b -
141*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^3 - 5*(11*a^3 -
19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x
+ c)^3 + (495*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^8 + 84*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^
3)*cosh(d*x + c)^6 + 630*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^4 - 85*a^3 - 51*a^2*b + 141*a*b^
2 - 5*b^3 - 180*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 3*(55*(5*a^3 + 3*a^2
*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^9 + 12*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 126*(11*
a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^5 - 60*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^3
- (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((5*a^3 + 3*a^2*b + 3*a*b^2 + 5*
b^3)*cosh(d*x + c)^12 + 12*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^3 + 3*a^2
*b + 3*a*b^2 + 5*b^3)*sinh(d*x + c)^12 + 6*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 6*(5*a^3 + 3
*a^2*b + 3*a*b^2 + 5*b^3 + 11*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(
5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh
(d*x + c)^9 + 15*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 15*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b
^3)*cosh(d*x + c)^4 + 5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3 + 18*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^8 + 24*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 30*(5*a^3 + 3*a^2*b + 3*a*b
^2 + 5*b^3)*cosh(d*x + c)^3 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(5*a^3
+ 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 4*(231*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 3
15*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 25*a^3 + 15*a^2*b + 15*a*b^2 + 25*b^3 + 105*(5*a^3 +
3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(
d*x + c)^7 + 63*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*c
osh(d*x + c)^3 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^3 + 3*a^2*b +
3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 15*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 84*(5*a^3 + 3*
a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 70*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 5*a^3 + 3*
a^2*b + 3*a*b^2 + 5*b^3 + 20*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 20*(11*(5*
a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 36*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 42
*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 20*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^3
+ 3*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3 + 6
*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2 + 6*(11*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)
^10 + 45*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 70*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x
+ c)^6 + 50*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3 + 15*(5*a
^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*((5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cos
h(d*x + c)^11 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 10*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)
*cosh(d*x + c)^7 + 10*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b
^3)*cosh(d*x + c)^3 + (5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) +
sinh(d*x + c)) - 3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c) + 3*(11*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11
*b^3)*cosh(d*x + c)^10 + 3*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 42*(11*a^3 - 19*a^2*b + 1
3*a*b^2 - 5*b^3)*cosh(d*x + c)^6 - 30*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^4 - 5*a^3 - 3*a^2*b
- 3*a*b^2 + 11*b^3 - (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)
^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + 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*cos
h(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*cosh(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)*sinh(d*x + c)^2 + 12*(d*cosh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 + 10*d*c
osh(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)
________________________________________________________________________________________
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)**3,x)
[Out]
Timed out
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Giac [B] time = 1.29618, size = 518, normalized size = 3.36 \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^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )}}{32 \, d} + \frac{15 \, a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 33 \, b^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 160 \, a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 96 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 96 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 160 \, b^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 240 \, b^{3}{\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)^3,x, algorithm="giac")
[Out]
1/32*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)/d + 1/24*(15*
a^3*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a^2*b*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a*b^2*(e^(d*x + c) - e^(-d*x -
c))^5 - 33*b^3*(e^(d*x + c) - e^(-d*x - c))^5 + 160*a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 96*a^2*b*(e^(d*x + c
) - e^(-d*x - c))^3 - 96*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 - 160*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 528*a
^3*(e^(d*x + c) - e^(-d*x - c)) - 144*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 144*a*b^2*(e^(d*x + c) - e^(-d*x -
c)) - 240*b^3*(e^(d*x + c) - e^(-d*x - c)))/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3*d)