3.125 \(\int (a+b \text{sech}^2(c+d x))^3 \tanh ^3(c+d x) \, dx\)
Optimal. Leaf size=103 \[ -\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}+\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}+\frac{\text{sech}^8(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{8 b d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d} \]
[Out]
(a^3*Log[Cosh[c + d*x]])/d - (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (3*a*b^2*Sech[c + d*x]^4)/(4*d) - (b^3*Sech[c +
d*x]^6)/(6*d) + ((b + a*Cosh[c + d*x]^2)^4*Sech[c + d*x]^8)/(8*b*d)
________________________________________________________________________________________
Rubi [A] time = 0.0976315, antiderivative size = 103, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{4138, 446, 78, 43} \[ -\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}+\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}+\frac{\text{sech}^8(c+d x) \left (a \cosh ^2(c+d x)+b\right )^4}{8 b d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^3,x]
[Out]
(a^3*Log[Cosh[c + d*x]])/d - (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (3*a*b^2*Sech[c + d*x]^4)/(4*d) - (b^3*Sech[c +
d*x]^6)/(6*d) + ((b + a*Cosh[c + d*x]^2)^4*Sech[c + d*x]^8)/(8*b*d)
Rule 4138
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*
(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^3 \tanh ^3(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (b+a x^2\right )^3}{x^9} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x) (b+a x)^3}{x^5} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{\left (b+a \cosh ^2(c+d x)\right )^4 \text{sech}^8(c+d x)}{8 b d}+\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{\left (b+a \cosh ^2(c+d x)\right )^4 \text{sech}^8(c+d x)}{8 b d}+\frac{\operatorname{Subst}\left (\int \left (\frac{b^3}{x^4}+\frac{3 a b^2}{x^3}+\frac{3 a^2 b}{x^2}+\frac{a^3}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{a^3 \log (\cosh (c+d x))}{d}-\frac{3 a^2 b \text{sech}^2(c+d x)}{2 d}-\frac{3 a b^2 \text{sech}^4(c+d x)}{4 d}-\frac{b^3 \text{sech}^6(c+d x)}{6 d}+\frac{\left (b+a \cosh ^2(c+d x)\right )^4 \text{sech}^8(c+d x)}{8 b d}\\ \end{align*}
Mathematica [A] time = 0.885562, size = 128, normalized size = 1.24 \[ \frac{\cosh ^6(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \left (12 a^2 (a-3 b) \text{sech}^2(c+d x)+24 a^3 \log (\cosh (c+d x))+4 b^2 (3 a-b) \text{sech}^6(c+d x)+18 a b (a-b) \text{sech}^4(c+d x)+3 b^3 \text{sech}^8(c+d x)\right )}{3 d (a \cosh (2 c+2 d x)+a+2 b)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^3,x]
[Out]
(Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*(24*a^3*Log[Cosh[c + d*x]] + 12*a^2*(a - 3*b)*Sech[c + d*x]^2 + 18*
a*(a - b)*b*Sech[c + d*x]^4 + 4*(3*a - b)*b^2*Sech[c + d*x]^6 + 3*b^3*Sech[c + d*x]^8))/(3*d*(a + 2*b + a*Cosh
[2*c + 2*d*x])^3)
________________________________________________________________________________________
Maple [B] time = 0.051, size = 253, normalized size = 2.5 \begin{align*}{\frac{{a}^{3}\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{2}{a}^{3}}{2\,d}}-{\frac{3\,{a}^{2}b \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{2}b \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{8\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{8}}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{24\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{6}}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{24\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{24\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^3,x)
[Out]
a^3*ln(cosh(d*x+c))/d-1/2/d*tanh(d*x+c)^2*a^3-3/4/d*a^2*b*sinh(d*x+c)^2/cosh(d*x+c)^4+3/4/d*a^2*b*sinh(d*x+c)^
2/cosh(d*x+c)^2-1/2/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^6+1/4/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^4+1/4/d*a*b^2*si
nh(d*x+c)^2/cosh(d*x+c)^2-1/8/d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^8+1/24/d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^6+1/24/d*
b^3*sinh(d*x+c)^2/cosh(d*x+c)^4+1/24/d*b^3*sinh(d*x+c)^2/cosh(d*x+c)^2
________________________________________________________________________________________
Maxima [B] time = 1.56819, size = 880, normalized size = 8.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^3,x, algorithm="maxima")
[Out]
3/4*a^2*b*tanh(d*x + c)^4/d + a^3*(x + c/d + log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x
- 2*c) + e^(-4*d*x - 4*c) + 1))) - 4*a*b^2*(3*e^(-4*d*x - 4*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)) - 2*e^(-6*d*x - 6*
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 - 1
0*c) + e^(-12*d*x - 12*c) + 1)) + 3*e^(-8*d*x - 8*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))) - 32/3*b^3*(e^(-6*d*x - 6*c
)/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 1
0*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) - e^(-8*d*x - 8*c)/(d*(8*e^(-2*
d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-
12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)) + e^(-10*d*x - 10*c)/(d*(8*e^(-2*d*x - 2*c) +
28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*
c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1)))
________________________________________________________________________________________
Fricas [B] time = 2.69868, size = 12142, normalized size = 117.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^3,x, algorithm="fricas")
[Out]
-1/3*(3*a^3*d*x*cosh(d*x + c)^16 + 48*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^15 + 3*a^3*d*x*sinh(d*x + c)^16 + 6*
(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^14 + 6*(60*a^3*d*x*cosh(d*x + c)^2 + 4*a^3*d*x - a^3 + 3*a^2*b)*sinh
(d*x + c)^14 + 84*(20*a^3*d*x*cosh(d*x + c)^3 + (4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^13 +
12*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^12 + 6*(910*a^3*d*x*cosh(d*x + c)^4 + 14*a^3*d*x - 6*
a^3 + 12*a^2*b + 6*a*b^2 + 91*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 24*(546*a^3*d*x*
cosh(d*x + c)^5 + 91*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^3 + 6*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*c
osh(d*x + c))*sinh(d*x + c)^11 + 2*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^10 + 2*(
12012*a^3*d*x*cosh(d*x + c)^6 + 84*a^3*d*x + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^4 - 45*a^3 + 63*a^
2*b + 24*a*b^2 + 16*b^3 + 396*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 4*(8
580*a^3*d*x*cosh(d*x + c)^7 + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^5 + 660*(7*a^3*d*x - 3*a^3 + 6*a^
2*b + 3*a*b^2)*cosh(d*x + c)^3 + 5*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c))*sinh(d*
x + c)^9 + 2*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^8 + 2*(19305*a^3*d*x*cosh(d*x
+ c)^8 + 9009*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^6 + 105*a^3*d*x + 2970*(7*a^3*d*x - 3*a^3 + 6*a^2*b +
3*a*b^2)*cosh(d*x + c)^4 - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3 + 45*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*
b^2 + 16*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(2145*a^3*d*x*cosh(d*x + c)^9 + 1287*(4*a^3*d*x - a^3 + 3*
a^2*b)*cosh(d*x + c)^7 + 594*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 15*(84*a^3*d*x - 45*a^3
+ 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^3 + (105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(
d*x + c))*sinh(d*x + c)^7 + 2*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^6 + 2*(12012*
a^3*d*x*cosh(d*x + c)^10 + 9009*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^8 + 5544*(7*a^3*d*x - 3*a^3 + 6*a^2*
b + 3*a*b^2)*cosh(d*x + c)^6 + 84*a^3*d*x + 210*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x
+ c)^4 - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3 + 28*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh
(d*x + c)^2)*sinh(d*x + c)^6 + 4*(3276*a^3*d*x*cosh(d*x + c)^11 + 3003*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x +
c)^9 + 2376*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^7 + 126*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24
*a*b^2 + 16*b^3)*cosh(d*x + c)^5 + 28*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^3 +
3*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*a^3*d*x + 12*(7*a^3*
d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 2*(2730*a^3*d*x*cosh(d*x + c)^12 + 3003*(4*a^3*d*x - a^3 +
3*a^2*b)*cosh(d*x + c)^10 + 2970*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^8 + 210*(84*a^3*d*x - 4
5*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^6 + 42*a^3*d*x + 70*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*
a*b^2 - 16*b^3)*cosh(d*x + c)^4 - 18*a^3 + 36*a^2*b + 18*a*b^2 + 15*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2
+ 16*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(210*a^3*d*x*cosh(d*x + c)^13 + 273*(4*a^3*d*x - a^3 + 3*a^2*b
)*cosh(d*x + c)^11 + 330*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^9 + 30*(84*a^3*d*x - 45*a^3 + 6
3*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^7 + 14*(105*a^3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d
*x + c)^5 + 5*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^3 + 6*(7*a^3*d*x - 3*a^3 + 6*
a^2*b + 3*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^2 + 2*(180*a^3*d
*x*cosh(d*x + c)^14 + 273*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^12 + 396*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*
a*b^2)*cosh(d*x + c)^10 + 45*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^8 + 28*(105*a^
3*d*x - 60*a^3 + 72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^6 + 12*a^3*d*x + 15*(84*a^3*d*x - 45*a^3 + 63*a^2
*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^4 - 3*a^3 + 9*a^2*b + 36*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 - 3*(a^3*cosh(d*x + c)^16 + 16*a^3*cosh(d*x + c)*sinh(d*x + c)^15 + a^3*sinh(d*x + c
)^16 + 8*a^3*cosh(d*x + c)^14 + 28*a^3*cosh(d*x + c)^12 + 8*(15*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^14 +
112*(5*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^13 + 56*a^3*cosh(d*x + c)^10 + 28*(65*a^3*cosh(d
*x + c)^4 + 26*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^12 + 112*(39*a^3*cosh(d*x + c)^5 + 26*a^3*cosh(d*x + c
)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^11 + 70*a^3*cosh(d*x + c)^8 + 56*(143*a^3*cosh(d*x + c)^6 + 143*a^3*c
osh(d*x + c)^4 + 33*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^10 + 16*(715*a^3*cosh(d*x + c)^7 + 1001*a^3*cosh(
d*x + c)^5 + 385*a^3*cosh(d*x + c)^3 + 35*a^3*cosh(d*x + c))*sinh(d*x + c)^9 + 56*a^3*cosh(d*x + c)^6 + 2*(643
5*a^3*cosh(d*x + c)^8 + 12012*a^3*cosh(d*x + c)^6 + 6930*a^3*cosh(d*x + c)^4 + 1260*a^3*cosh(d*x + c)^2 + 35*a
^3)*sinh(d*x + c)^8 + 16*(715*a^3*cosh(d*x + c)^9 + 1716*a^3*cosh(d*x + c)^7 + 1386*a^3*cosh(d*x + c)^5 + 420*
a^3*cosh(d*x + c)^3 + 35*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 28*a^3*cosh(d*x + c)^4 + 56*(143*a^3*cosh(d*x +
c)^10 + 429*a^3*cosh(d*x + c)^8 + 462*a^3*cosh(d*x + c)^6 + 210*a^3*cosh(d*x + c)^4 + 35*a^3*cosh(d*x + c)^2 +
a^3)*sinh(d*x + c)^6 + 112*(39*a^3*cosh(d*x + c)^11 + 143*a^3*cosh(d*x + c)^9 + 198*a^3*cosh(d*x + c)^7 + 126
*a^3*cosh(d*x + c)^5 + 35*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 8*a^3*cosh(d*x + c)^2 +
28*(65*a^3*cosh(d*x + c)^12 + 286*a^3*cosh(d*x + c)^10 + 495*a^3*cosh(d*x + c)^8 + 420*a^3*cosh(d*x + c)^6 +
175*a^3*cosh(d*x + c)^4 + 30*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 112*(5*a^3*cosh(d*x + c)^13 + 26*a^3
*cosh(d*x + c)^11 + 55*a^3*cosh(d*x + c)^9 + 60*a^3*cosh(d*x + c)^7 + 35*a^3*cosh(d*x + c)^5 + 10*a^3*cosh(d*x
+ c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 8*(15*a^3*cosh(d*x + c)^14 + 91*a^3*cosh(d*x + c)^12 + 23
1*a^3*cosh(d*x + c)^10 + 315*a^3*cosh(d*x + c)^8 + 245*a^3*cosh(d*x + c)^6 + 105*a^3*cosh(d*x + c)^4 + 21*a^3*
cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^2 + 16*(a^3*cosh(d*x + c)^15 + 7*a^3*cosh(d*x + c)^13 + 21*a^3*cosh(d*x +
c)^11 + 35*a^3*cosh(d*x + c)^9 + 35*a^3*cosh(d*x + c)^7 + 21*a^3*cosh(d*x + c)^5 + 7*a^3*cosh(d*x + c)^3 + a^
3*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(12*a^3*d*x*cosh(d*x
+ c)^15 + 21*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*x + c)^13 + 36*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*
x + c)^11 + 5*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d*x + c)^9 + 4*(105*a^3*d*x - 60*a^3 +
72*a^2*b + 12*a*b^2 - 16*b^3)*cosh(d*x + c)^7 + 3*(84*a^3*d*x - 45*a^3 + 63*a^2*b + 24*a*b^2 + 16*b^3)*cosh(d
*x + c)^5 + 12*(7*a^3*d*x - 3*a^3 + 6*a^2*b + 3*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^3*d*x - a^3 + 3*a^2*b)*cosh(d*
x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^16 + 16*d*cosh(d*x + c)*sinh(d*x + c)^15 + d*sinh(d*x + c)^16 + 8*d*co
sh(d*x + c)^14 + 8*(15*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^14 + 112*(5*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*s
inh(d*x + c)^13 + 28*d*cosh(d*x + c)^12 + 28*(65*d*cosh(d*x + c)^4 + 26*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^1
2 + 112*(39*d*cosh(d*x + c)^5 + 26*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^11 + 56*d*cosh(d*x + c
)^10 + 56*(143*d*cosh(d*x + c)^6 + 143*d*cosh(d*x + c)^4 + 33*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^10 + 16*(71
5*d*cosh(d*x + c)^7 + 1001*d*cosh(d*x + c)^5 + 385*d*cosh(d*x + c)^3 + 35*d*cosh(d*x + c))*sinh(d*x + c)^9 + 7
0*d*cosh(d*x + c)^8 + 2*(6435*d*cosh(d*x + c)^8 + 12012*d*cosh(d*x + c)^6 + 6930*d*cosh(d*x + c)^4 + 1260*d*co
sh(d*x + c)^2 + 35*d)*sinh(d*x + c)^8 + 16*(715*d*cosh(d*x + c)^9 + 1716*d*cosh(d*x + c)^7 + 1386*d*cosh(d*x +
c)^5 + 420*d*cosh(d*x + c)^3 + 35*d*cosh(d*x + c))*sinh(d*x + c)^7 + 56*d*cosh(d*x + c)^6 + 56*(143*d*cosh(d*
x + c)^10 + 429*d*cosh(d*x + c)^8 + 462*d*cosh(d*x + c)^6 + 210*d*cosh(d*x + c)^4 + 35*d*cosh(d*x + c)^2 + d)*
sinh(d*x + c)^6 + 112*(39*d*cosh(d*x + c)^11 + 143*d*cosh(d*x + c)^9 + 198*d*cosh(d*x + c)^7 + 126*d*cosh(d*x
+ c)^5 + 35*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^4 + 28*(65*d*cosh(d*x
+ c)^12 + 286*d*cosh(d*x + c)^10 + 495*d*cosh(d*x + c)^8 + 420*d*cosh(d*x + c)^6 + 175*d*cosh(d*x + c)^4 + 30*
d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 112*(5*d*cosh(d*x + c)^13 + 26*d*cosh(d*x + c)^11 + 55*d*cosh(d*x + c
)^9 + 60*d*cosh(d*x + c)^7 + 35*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^3 +
8*d*cosh(d*x + c)^2 + 8*(15*d*cosh(d*x + c)^14 + 91*d*cosh(d*x + c)^12 + 231*d*cosh(d*x + c)^10 + 315*d*cosh(d
*x + c)^8 + 245*d*cosh(d*x + c)^6 + 105*d*cosh(d*x + c)^4 + 21*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 16*(d*
cosh(d*x + c)^15 + 7*d*cosh(d*x + c)^13 + 21*d*cosh(d*x + c)^11 + 35*d*cosh(d*x + c)^9 + 35*d*cosh(d*x + c)^7
+ 21*d*cosh(d*x + c)^5 + 7*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)
________________________________________________________________________________________
Sympy [A] time = 22.6259, size = 178, normalized size = 1.73 \begin{align*} \begin{cases} a^{3} x - \frac{a^{3} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{a^{3} \tanh ^{2}{\left (c + d x \right )}}{2 d} - \frac{3 a^{2} b \tanh ^{2}{\left (c + d x \right )} \operatorname{sech}^{2}{\left (c + d x \right )}}{4 d} - \frac{3 a^{2} b \operatorname{sech}^{2}{\left (c + d x \right )}}{4 d} - \frac{a b^{2} \tanh ^{2}{\left (c + d x \right )} \operatorname{sech}^{4}{\left (c + d x \right )}}{2 d} - \frac{a b^{2} \operatorname{sech}^{4}{\left (c + d x \right )}}{4 d} - \frac{b^{3} \tanh ^{2}{\left (c + d x \right )} \operatorname{sech}^{6}{\left (c + d x \right )}}{8 d} - \frac{b^{3} \operatorname{sech}^{6}{\left (c + d x \right )}}{24 d} & \text{for}\: d \neq 0 \\x \left (a + b \operatorname{sech}^{2}{\left (c \right )}\right )^{3} \tanh ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)**2)**3*tanh(d*x+c)**3,x)
[Out]
Piecewise((a**3*x - a**3*log(tanh(c + d*x) + 1)/d - a**3*tanh(c + d*x)**2/(2*d) - 3*a**2*b*tanh(c + d*x)**2*se
ch(c + d*x)**2/(4*d) - 3*a**2*b*sech(c + d*x)**2/(4*d) - a*b**2*tanh(c + d*x)**2*sech(c + d*x)**4/(2*d) - a*b*
*2*sech(c + d*x)**4/(4*d) - b**3*tanh(c + d*x)**2*sech(c + d*x)**6/(8*d) - b**3*sech(c + d*x)**6/(24*d), Ne(d,
0)), (x*(a + b*sech(c)**2)**3*tanh(c)**3, True))
________________________________________________________________________________________
Giac [B] time = 1.35544, size = 518, normalized size = 5.03 \begin{align*} -\frac{840 \, a^{3} d x - 840 \, a^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac{2283 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 16584 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 53844 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 20160 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 102648 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 35280 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 13440 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 8960 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 126210 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40320 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 8960 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 102648 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 35280 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 13440 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8960 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 53844 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 20160 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 10080 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16584 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5040 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2283 \, a^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^3,x, algorithm="giac")
[Out]
-1/840*(840*a^3*d*x - 840*a^3*log(e^(2*d*x + 2*c) + 1) + (2283*a^3*e^(16*d*x + 16*c) + 16584*a^3*e^(14*d*x + 1
4*c) + 5040*a^2*b*e^(14*d*x + 14*c) + 53844*a^3*e^(12*d*x + 12*c) + 20160*a^2*b*e^(12*d*x + 12*c) + 10080*a*b^
2*e^(12*d*x + 12*c) + 102648*a^3*e^(10*d*x + 10*c) + 35280*a^2*b*e^(10*d*x + 10*c) + 13440*a*b^2*e^(10*d*x + 1
0*c) + 8960*b^3*e^(10*d*x + 10*c) + 126210*a^3*e^(8*d*x + 8*c) + 40320*a^2*b*e^(8*d*x + 8*c) + 6720*a*b^2*e^(8
*d*x + 8*c) - 8960*b^3*e^(8*d*x + 8*c) + 102648*a^3*e^(6*d*x + 6*c) + 35280*a^2*b*e^(6*d*x + 6*c) + 13440*a*b^
2*e^(6*d*x + 6*c) + 8960*b^3*e^(6*d*x + 6*c) + 53844*a^3*e^(4*d*x + 4*c) + 20160*a^2*b*e^(4*d*x + 4*c) + 10080
*a*b^2*e^(4*d*x + 4*c) + 16584*a^3*e^(2*d*x + 2*c) + 5040*a^2*b*e^(2*d*x + 2*c) + 2283*a^3)/(e^(2*d*x + 2*c) +
1)^8)/d