3.312 \(\int \text{sech}^5(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\)

Optimal. Leaf size=103 \[ \frac{3 (a-b) \left ((a+b)^2+4 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(a-b)^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{3 (a-b)^2 (a+3 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{b^3 \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.132009, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3190, 390, 1157, 385, 203} \[ \frac{3 (a-b) \left ((a+b)^2+4 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{(a-b)^3 \tanh (c+d x) \text{sech}^3(c+d x)}{4 d}+\frac{3 (a-b)^2 (a+3 b) \tanh (c+d x) \text{sech}(c+d x)}{8 d}+\frac{b^3 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(3*(a - b)*(4*b^2 + (a + b)^2)*ArcTan[Sinh[c + d*x]])/(8*d) + (b^3*Sinh[c + d*x])/d + (3*(a - b)^2*(a + 3*b)*S
ech[c + d*x]*Tanh[c + d*x])/(8*d) + ((a - b)^3*Sech[c + d*x]^3*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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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 )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^3+\frac{a^3-b^3+3 b \left (a^2-b^2\right ) x^2+3 (a-b) b^2 x^4}{\left (1+x^2\right )^3}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{b^3 \sinh (c+d x)}{d}+\frac{\operatorname{Subst}\left (\int \frac{a^3-b^3+3 b \left (a^2-b^2\right ) x^2+3 (a-b) b^2 x^4}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{b^3 \sinh (c+d x)}{d}+\frac{(a-b)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-3 (a-b) (a+b)^2-12 (a-b) b^2 x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d}\\ &=\frac{b^3 \sinh (c+d x)}{d}+\frac{3 (a-b)^2 (a+3 b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}+\frac{(a-b)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac{\left (3 (a-b) \left (4 b^2+(a+b)^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac{3 (a-b) \left (4 b^2+(a+b)^2\right ) \tan ^{-1}(\sinh (c+d x))}{8 d}+\frac{b^3 \sinh (c+d x)}{d}+\frac{3 (a-b)^2 (a+3 b) \text{sech}(c+d x) \tanh (c+d x)}{8 d}+\frac{(a-b)^3 \text{sech}^3(c+d x) \tanh (c+d x)}{4 d}\\ \end{align*}

Mathematica [C]  time = 9.93774, size = 472, normalized size = 4.58 \[ -\frac{\text{csch}^5(c+d x) \left (256 \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{11}{2}\right \},-\sinh ^2(c+d x)\right )+384 \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \left (7 a+5 b \sinh ^2(c+d x)\right ) \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2,2\right \},\left \{1,1,1,\frac{11}{2}\right \},-\sinh ^2(c+d x)\right )-21 \left (9 a^2 b \left (2131 \sinh ^4(c+d x)+41615 \sinh ^2(c+d x)+72030\right ) \sinh ^2(c+d x)+a^3 \left (8226 \sinh ^4(c+d x)+140965 \sinh ^2(c+d x)+252105\right )+15 a b^2 \left (1128 \sinh ^4(c+d x)+21529 \sinh ^2(c+d x)+36015\right ) \sinh ^4(c+d x)+b^3 \left (4887 \sinh ^4(c+d x)+90805 \sinh ^2(c+d x)+149460\right ) \sinh ^6(c+d x)\right )+\frac{315 \tanh ^{-1}\left (\sqrt{-\sinh ^2(c+d x)}\right ) \left (9 a^2 b \left (3 \sinh ^6(c+d x)+640 \sinh ^4(c+d x)+4375 \sinh ^2(c+d x)+4802\right ) \sinh ^2(c+d x)+a^3 \left (-62 \sinh ^6(c+d x)+2187 \sinh ^4(c+d x)+15000 \sinh ^2(c+d x)+16807\right )+3 a b^2 \left (8 \sinh ^6(c+d x)+1701 \sinh ^4(c+d x)+11178 \sinh ^2(c+d x)+12005\right ) \sinh ^4(c+d x)+b^3 \left (7 \sinh ^6(c+d x)+1458 \sinh ^4(c+d x)+9375 \sinh ^2(c+d x)+9964\right ) \sinh ^6(c+d x)\right )}{\sqrt{-\sinh ^2(c+d x)}}\right )}{60480 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

-(Csch[c + d*x]^5*(256*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c +
d*x]^8*(a + b*Sinh[c + d*x]^2)^3 + 384*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]
*Sinh[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^2*(7*a + 5*b*Sinh[c + d*x]^2) - 21*(15*a*b^2*Sinh[c + d*x]^4*(36015 +
21529*Sinh[c + d*x]^2 + 1128*Sinh[c + d*x]^4) + 9*a^2*b*Sinh[c + d*x]^2*(72030 + 41615*Sinh[c + d*x]^2 + 2131
*Sinh[c + d*x]^4) + b^3*Sinh[c + d*x]^6*(149460 + 90805*Sinh[c + d*x]^2 + 4887*Sinh[c + d*x]^4) + a^3*(252105
+ 140965*Sinh[c + d*x]^2 + 8226*Sinh[c + d*x]^4)) + (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(a^3*(16807 + 15000*S
inh[c + d*x]^2 + 2187*Sinh[c + d*x]^4 - 62*Sinh[c + d*x]^6) + 9*a^2*b*Sinh[c + d*x]^2*(4802 + 4375*Sinh[c + d*
x]^2 + 640*Sinh[c + d*x]^4 + 3*Sinh[c + d*x]^6) + b^3*Sinh[c + d*x]^6*(9964 + 9375*Sinh[c + d*x]^2 + 1458*Sinh
[c + d*x]^4 + 7*Sinh[c + d*x]^6) + 3*a*b^2*Sinh[c + d*x]^4*(12005 + 11178*Sinh[c + d*x]^2 + 1701*Sinh[c + d*x]
^4 + 8*Sinh[c + d*x]^6)))/Sqrt[-Sinh[c + d*x]^2]))/(60480*d)

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Maple [B]  time = 0.062, size = 376, normalized size = 3.7 \begin{align*}{\frac{{a}^{3}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}-{\frac{{a}^{2}b\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}b\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{2}b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}b\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}-3\,{\frac{a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}-3\,{\frac{a{b}^{2}\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}+{\frac{9\,a{b}^{2}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}+{\frac{9\,a{b}^{2}\arctan \left ({{\rm e}^{dx+c}} \right ) }{4\,d}}+{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+5\,{\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}+5\,{\frac{{b}^{3}\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,{b}^{3}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{4\,d}}-{\frac{15\,{b}^{3}{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{8\,d}}-{\frac{15\,{b}^{3}\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)^3,x)

[Out]

1/4/d*a^3*tanh(d*x+c)*sech(d*x+c)^3+3/8/d*a^3*sech(d*x+c)*tanh(d*x+c)+3/4/d*a^3*arctan(exp(d*x+c))-1/d*a^2*b*s
inh(d*x+c)/cosh(d*x+c)^4+1/4/d*a^2*b*tanh(d*x+c)*sech(d*x+c)^3+3/8/d*a^2*b*sech(d*x+c)*tanh(d*x+c)+3/4/d*a^2*b
*arctan(exp(d*x+c))-3/d*a*b^2*sinh(d*x+c)^3/cosh(d*x+c)^4-3/d*a*b^2*sinh(d*x+c)/cosh(d*x+c)^4+3/4/d*a*b^2*tanh
(d*x+c)*sech(d*x+c)^3+9/8/d*a*b^2*sech(d*x+c)*tanh(d*x+c)+9/4/d*a*b^2*arctan(exp(d*x+c))+1/d*b^3*sinh(d*x+c)^5
/cosh(d*x+c)^4+5/d*b^3*sinh(d*x+c)^3/cosh(d*x+c)^4+5/d*b^3*sinh(d*x+c)/cosh(d*x+c)^4-5/4/d*b^3*tanh(d*x+c)*sec
h(d*x+c)^3-15/8/d*b^3*sech(d*x+c)*tanh(d*x+c)-15/4/d*b^3*arctan(exp(d*x+c))

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Maxima [B]  time = 1.71048, size = 660, normalized size = 6.41 \begin{align*} \frac{1}{4} \, b^{3}{\left (\frac{15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{2 \, e^{\left (-d x - c\right )}}{d} + \frac{17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13 \, e^{\left (-4 \, d x - 4 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2}{d{\left (e^{\left (-d x - c\right )} + 4 \, e^{\left (-3 \, d x - 3 \, c\right )} + 6 \, e^{\left (-5 \, d x - 5 \, c\right )} + 4 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac{3}{4} \, a 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^{3}{\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{3}{4} \, a^{2} 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)^3,x, algorithm="maxima")

[Out]

1/4*b^3*(15*arctan(e^(-d*x - c))/d - 2*e^(-d*x - c)/d + (17*e^(-2*d*x - 2*c) + 13*e^(-4*d*x - 4*c) + 7*e^(-6*d
*x - 6*c) - 7*e^(-8*d*x - 8*c) + 2)/(d*(e^(-d*x - c) + 4*e^(-3*d*x - 3*c) + 6*e^(-5*d*x - 5*c) + 4*e^(-7*d*x -
7*c) + e^(-9*d*x - 9*c)))) - 3/4*a*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^3*(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))) - 3/4*a^2*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]  time = 1.76507, size = 5520, normalized size = 53.59 \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)^3,x, algorithm="fricas")

[Out]

1/4*(2*b^3*cosh(d*x + c)^10 + 20*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 2*b^3*sinh(d*x + c)^10 + 3*(a^3 + a^2*b -
5*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 3*(30*b^3*cosh(d*x + c)^2 + a^3 + a^2*b - 5*a*b^2 + 5*b^3)*sinh(d*x + c)^8
+ 24*(10*b^3*cosh(d*x + c)^3 + (a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + (11*a^3 - 21*
a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + (420*b^3*cosh(d*x + c)^4 + 11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3 + 84
*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 6*(84*b^3*cosh(d*x + c)^5 + 28*(a^3 + a^2*
b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 -
(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + (420*b^3*cosh(d*x + c)^6 + 210*(a^3 + a^2*b - 5*a*b^2
+ 5*b^3)*cosh(d*x + c)^4 - 11*a^3 + 21*a^2*b - 9*a*b^2 - 5*b^3 + 15*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh
(d*x + c)^2)*sinh(d*x + c)^4 + 4*(60*b^3*cosh(d*x + c)^7 + 42*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^5
+ 5*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^3 - (11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c
))*sinh(d*x + c)^3 - 2*b^3 - 3*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^2 + 3*(30*b^3*cosh(d*x + c)^8 + 2
8*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 5*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^4 -
a^3 - a^2*b + 5*a*b^2 - 5*b^3 - 2*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*(
(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^9 + 9*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)*sinh(d*x + c
)^8 + (a^3 + a^2*b + 3*a*b^2 - 5*b^3)*sinh(d*x + c)^9 + 4*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^7 + 4*
(a^3 + a^2*b + 3*a*b^2 - 5*b^3 + 9*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 28*(3*(a
^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + (a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^6
+ 6*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 6*(21*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^4 +
a^3 + a^2*b + 3*a*b^2 - 5*b^3 + 14*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(63*(
a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 70*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + 15*(a^3
+ a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^3
+ 4*(21*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^6 + 35*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^4 +
a^3 + a^2*b + 3*a*b^2 - 5*b^3 + 15*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 12*(3*(
a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^7 + 7*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 5*(a^3 +
a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + (a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (a
^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c) + (9*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^8 + 28*(a^3 + a
^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^6 + 30*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + a^3 + a^2*b + 3
*a*b^2 - 5*b^3 + 12*(a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sin
h(d*x + c)) + 2*(10*b^3*cosh(d*x + c)^9 + 12*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 3*(11*a^3 - 21*
a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^5 - 2*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^3 - 3*(a^3 +
a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8
+ d*sinh(d*x + c)^9 + 4*d*cosh(d*x + c)^7 + 4*(9*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^7 + 28*(3*d*cosh(d*x + c
)^3 + d*cosh(d*x + c))*sinh(d*x + c)^6 + 6*d*cosh(d*x + c)^5 + 6*(21*d*cosh(d*x + c)^4 + 14*d*cosh(d*x + c)^2
+ d)*sinh(d*x + c)^5 + 2*(63*d*cosh(d*x + c)^5 + 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^4 +
4*d*cosh(d*x + c)^3 + 4*(21*d*cosh(d*x + c)^6 + 35*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)
^3 + 12*(3*d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^2 +
d*cosh(d*x + c) + (9*d*cosh(d*x + c)^8 + 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 + 12*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c))

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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)**5*(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.35793, size = 412, normalized size = 4. \begin{align*} \frac{b^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{2 \, d} + \frac{3 \,{\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 (a^{3} + a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\right )}}{16 \, d} + \frac{3 \, a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 15 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 9 \, b^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 12 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 36 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 28 \, b^{3}{\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)^3,x, algorithm="giac")

[Out]

1/2*b^3*(e^(d*x + c) - e^(-d*x - c))/d + 3/16*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a^3 + a
^2*b + 3*a*b^2 - 5*b^3)/d + 1/4*(3*a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 3*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3
- 15*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 9*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 20*a^3*(e^(d*x + c) - e^(-
d*x - c)) - 12*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 36*a*b^2*(e^(d*x + c) - e^(-d*x - c)) + 28*b^3*(e^(d*x + c
) - e^(-d*x - c)))/(((e^(d*x + c) - e^(-d*x - c))^2 + 4)^2*d)