### 3.59 $$\int \sinh ^2(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx$$

Optimal. Leaf size=129 $\frac{\sinh (c+d x) \cosh (c+d x) \left (a^2+2 a b \tanh (c+d x)+b^2\right )}{2 d}-\frac{1}{2} x \left (a^2+7 b^2\right )+\frac{a b \tanh ^2(c+d x)}{d}-\frac{4 a b \log (\cosh (c+d x))}{d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}+\frac{2 b^2 \tanh ^3(c+d x)}{3 d}+\frac{3 b^2 \tanh (c+d x)}{d}$

[Out]

-((a^2 + 7*b^2)*x)/2 - (4*a*b*Log[Cosh[c + d*x]])/d + (3*b^2*Tanh[c + d*x])/d + (a*b*Tanh[c + d*x]^2)/d + (2*b
^2*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a^2 + b^2 + 2*a*b*Tanh
[c + d*x]))/(2*d)

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Rubi [A]  time = 0.205109, antiderivative size = 159, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 5, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.217, Rules used = {3663, 1804, 1802, 633, 31} $\frac{\left (a^2+7 b^2\right ) \tanh (c+d x)}{2 d}+\frac{\sinh ^2(c+d x) \left (\left (a^2+b^2\right ) \tanh (c+d x)+2 a b\right )}{2 d}+\frac{a b \tanh ^2(c+d x)}{d}+\frac{(a+b) (a+7 b) \log (1-\tanh (c+d x))}{4 d}-\frac{(a-7 b) (a-b) \log (\tanh (c+d x)+1)}{4 d}+\frac{b^2 \tanh ^5(c+d x)}{5 d}+\frac{2 b^2 \tanh ^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a + b)*(a + 7*b)*Log[1 - Tanh[c + d*x]])/(4*d) - ((a - 7*b)*(a - b)*Log[1 + Tanh[c + d*x]])/(4*d) + ((a^2 +
7*b^2)*Tanh[c + d*x])/(2*d) + (a*b*Tanh[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/
(5*d) + (Sinh[c + d*x]^2*(2*a*b + (a^2 + b^2)*Tanh[c + d*x]))/(2*d)

Rule 3663

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 1.54989, size = 137, normalized size = 1.06 $\frac{15 a^2 \sinh (2 (c+d x))-30 a^2 c-30 a^2 d x+30 a b \cosh (2 (c+d x))-240 a b \log (\cosh (c+d x))-4 b \text{sech}^2(c+d x) (15 a+16 b \tanh (c+d x))+15 b^2 \sinh (2 (c+d x))+232 b^2 \tanh (c+d x)+12 b^2 \tanh (c+d x) \text{sech}^4(c+d x)-210 b^2 c-210 b^2 d x}{60 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-30*a^2*c - 210*b^2*c - 30*a^2*d*x - 210*b^2*d*x + 30*a*b*Cosh[2*(c + d*x)] - 240*a*b*Log[Cosh[c + d*x]] + 15
*a^2*Sinh[2*(c + d*x)] + 15*b^2*Sinh[2*(c + d*x)] + 232*b^2*Tanh[c + d*x] + 12*b^2*Sech[c + d*x]^4*Tanh[c + d*
x] - 4*b*Sech[c + d*x]^2*(15*a + 16*b*Tanh[c + d*x]))/(60*d)

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Maple [A]  time = 0.056, size = 173, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}\sinh \left ( dx+c \right ) \cosh \left ( dx+c \right ) }{2\,d}}-{\frac{{a}^{2}x}{2}}-{\frac{{a}^{2}c}{2\,d}}+{\frac{ab \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{ab\ln \left ( \cosh \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{ab \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{2\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-{\frac{7\,{b}^{2}x}{2}}-{\frac{7\,c{b}^{2}}{2\,d}}+{\frac{7\,{b}^{2}\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{7\,{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{6\,d}}+{\frac{7\,{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{10\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x)

[Out]

1/2/d*a^2*sinh(d*x+c)*cosh(d*x+c)-1/2*a^2*x-1/2/d*a^2*c+1/d*a*b*sinh(d*x+c)^4/cosh(d*x+c)^2-4*a*b*ln(cosh(d*x+
c))/d+2*a*b*tanh(d*x+c)^2/d+1/2/d*b^2*sinh(d*x+c)^7/cosh(d*x+c)^5-7/2*b^2*x-7/2/d*c*b^2+7/2*b^2*tanh(d*x+c)/d+
7/6*b^2*tanh(d*x+c)^3/d+7/10*b^2*tanh(d*x+c)^5/d

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Maxima [B]  time = 1.57803, size = 406, normalized size = 3.15 \begin{align*} -\frac{1}{8} \, a^{2}{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac{1}{120} \, b^{2}{\left (\frac{420 \,{\left (d x + c\right )}}{d} + \frac{15 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{1003 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3350 \, e^{\left (-4 \, d x - 4 \, c\right )} + 5590 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3915 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1455 \, e^{\left (-10 \, d x - 10 \, c\right )} + 15}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 10 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )}\right )}}\right )} - \frac{1}{4} \, a b{\left (\frac{16 \,{\left (d x + c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d} + \frac{16 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

-1/8*a^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/120*b^2*(420*(d*x + c)/d + 15*e^(-2*d*x - 2*c)/d -
(1003*e^(-2*d*x - 2*c) + 3350*e^(-4*d*x - 4*c) + 5590*e^(-6*d*x - 6*c) + 3915*e^(-8*d*x - 8*c) + 1455*e^(-10*
d*x - 10*c) + 15)/(d*(e^(-2*d*x - 2*c) + 5*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 10*e^(-8*d*x - 8*c) + 5*e^
(-10*d*x - 10*c) + e^(-12*d*x - 12*c)))) - 1/4*a*b*(16*(d*x + c)/d - e^(-2*d*x - 2*c)/d + 16*log(e^(-2*d*x - 2
*c) + 1)/d - (2*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 1)/(d*(e^(-2*d*x - 2*c) + 2*e^(-4*d*x - 4*c) + e^(-6*
d*x - 6*c))))

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Fricas [B]  time = 2.87053, size = 9671, normalized size = 74.97 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

1/120*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^14 + 210*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^13 + 15*(
a^2 + 2*a*b + b^2)*sinh(d*x + c)^14 - 15*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^
12 - 15*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 91*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - 5*a^2 - 10*a*b - 5*b^2)*sinh(d
*x + c)^12 + 60*(91*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - 3*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*
b^2)*cosh(d*x + c))*sinh(d*x + c)^11 - 15*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x +
c)^10 + 15*(1001*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 - 20*(a^2 - 8*a*b + 7*b^2)*d*x - 66*(4*(a^2 - 8*a*b + 7*b
^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^2 + 9*a^2 - 10*a*b - 87*b^2)*sinh(d*x + c)^10 + 30*(1001*(a^2
+ 2*a*b + b^2)*cosh(d*x + c)^5 - 110*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^3 -
5*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 15*(40*(a^2 - 8*a*
b + 7*b^2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^8 + 15*(3003*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 49
5*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^4 - 40*(a^2 - 8*a*b + 7*b^2)*d*x - 45*(
20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^2 + 5*a^2 - 66*a*b - 251*b^2)*sinh(d*x +
c)^8 + 120*(429*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 99*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^
2)*cosh(d*x + c)^5 - 15*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^3 - (40*(a^2 -
8*a*b + 7*b^2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 5*(120*(a^2 - 8*a*b + 7*b^2)*d
*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)^6 + 5*(9009*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 2772*(4*(a^2
- 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^6 - 630*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 +
10*a*b + 87*b^2)*cosh(d*x + c)^4 - 120*(a^2 - 8*a*b + 7*b^2)*d*x - 84*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 +
66*a*b + 251*b^2)*cosh(d*x + c)^2 - 15*a^2 - 198*a*b - 1103*b^2)*sinh(d*x + c)^6 + 30*(1001*(a^2 + 2*a*b + b^2
)*cosh(d*x + c)^9 - 396*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^7 - 126*(20*(a^2
- 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^5 - 28*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 + 6
6*a*b + 251*b^2)*cosh(d*x + c)^3 - (120*(a^2 - 8*a*b + 7*b^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^5 - 5*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b + 667*b^2)*cosh(d*x + c)^4 + 5*(3003*(a^
2 + 2*a*b + b^2)*cosh(d*x + c)^10 - 1485*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^
8 - 630*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^6 - 210*(40*(a^2 - 8*a*b + 7*b^
2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^4 - 60*(a^2 - 8*a*b + 7*b^2)*d*x - 15*(120*(a^2 - 8*a*b + 7*b
^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)^2 - 27*a^2 - 30*a*b - 667*b^2)*sinh(d*x + c)^4 + 20*(273*
(a^2 + 2*a*b + b^2)*cosh(d*x + c)^11 - 165*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c
)^9 - 90*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^7 - 42*(40*(a^2 - 8*a*b + 7*b^
2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^5 - 5*(120*(a^2 - 8*a*b + 7*b^2)*d*x + 15*a^2 + 198*a*b + 110
3*b^2)*cosh(d*x + c)^3 - (60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b + 667*b^2)*cosh(d*x + c))*sinh(d*x +
c)^3 - (60*(a^2 - 8*a*b + 7*b^2)*d*x + 75*a^2 - 150*a*b + 1003*b^2)*cosh(d*x + c)^2 + (1365*(a^2 + 2*a*b + b^2
)*cosh(d*x + c)^12 - 990*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 - 10*a*b - 5*b^2)*cosh(d*x + c)^10 - 675*(20*(a^
2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c)^8 - 420*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2
+ 66*a*b + 251*b^2)*cosh(d*x + c)^6 - 75*(120*(a^2 - 8*a*b + 7*b^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*
x + c)^4 - 60*(a^2 - 8*a*b + 7*b^2)*d*x - 30*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b + 667*b^2)*cosh(d
*x + c)^2 - 75*a^2 + 150*a*b - 1003*b^2)*sinh(d*x + c)^2 - 15*a^2 + 30*a*b - 15*b^2 - 480*(a*b*cosh(d*x + c)^1
2 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^11 + a*b*sinh(d*x + c)^12 + 5*a*b*cosh(d*x + c)^10 + (66*a*b*cosh(d*x +
c)^2 + 5*a*b)*sinh(d*x + c)^10 + 10*a*b*cosh(d*x + c)^8 + 10*(22*a*b*cosh(d*x + c)^3 + 5*a*b*cosh(d*x + c))*s
inh(d*x + c)^9 + 5*(99*a*b*cosh(d*x + c)^4 + 45*a*b*cosh(d*x + c)^2 + 2*a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x
+ c)^6 + 8*(99*a*b*cosh(d*x + c)^5 + 75*a*b*cosh(d*x + c)^3 + 10*a*b*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(462*
a*b*cosh(d*x + c)^6 + 525*a*b*cosh(d*x + c)^4 + 140*a*b*cosh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^6 + 5*a*b*cosh(
d*x + c)^4 + 4*(198*a*b*cosh(d*x + c)^7 + 315*a*b*cosh(d*x + c)^5 + 140*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x
+ c))*sinh(d*x + c)^5 + 5*(99*a*b*cosh(d*x + c)^8 + 210*a*b*cosh(d*x + c)^6 + 140*a*b*cosh(d*x + c)^4 + 30*a*b
*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 + a*b*cosh(d*x + c)^2 + 20*(11*a*b*cosh(d*x + c)^9 + 30*a*b*cosh(d*x +
c)^7 + 28*a*b*cosh(d*x + c)^5 + 10*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^3 + (66*a*b*cosh(d*
x + c)^10 + 225*a*b*cosh(d*x + c)^8 + 280*a*b*cosh(d*x + c)^6 + 150*a*b*cosh(d*x + c)^4 + 30*a*b*cosh(d*x + c)
^2 + a*b)*sinh(d*x + c)^2 + 2*(6*a*b*cosh(d*x + c)^11 + 25*a*b*cosh(d*x + c)^9 + 40*a*b*cosh(d*x + c)^7 + 30*a
*b*cosh(d*x + c)^5 + 10*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x
+ c) - sinh(d*x + c))) + 2*(105*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^13 - 90*(4*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2
- 10*a*b - 5*b^2)*cosh(d*x + c)^11 - 75*(20*(a^2 - 8*a*b + 7*b^2)*d*x - 9*a^2 + 10*a*b + 87*b^2)*cosh(d*x + c
)^9 - 60*(40*(a^2 - 8*a*b + 7*b^2)*d*x - 5*a^2 + 66*a*b + 251*b^2)*cosh(d*x + c)^7 - 15*(120*(a^2 - 8*a*b + 7*
b^2)*d*x + 15*a^2 + 198*a*b + 1103*b^2)*cosh(d*x + c)^5 - 10*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 27*a^2 + 30*a*b +
667*b^2)*cosh(d*x + c)^3 - (60*(a^2 - 8*a*b + 7*b^2)*d*x + 75*a^2 - 150*a*b + 1003*b^2)*cosh(d*x + c))*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 + 5*d*cosh(d*x + c)^10
+ (66*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^10 + 10*(22*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)
^9 + 10*d*cosh(d*x + c)^8 + 5*(99*d*cosh(d*x + c)^4 + 45*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8*(99*d*co
sh(d*x + c)^5 + 75*d*cosh(d*x + c)^3 + 10*d*cosh(d*x + c))*sinh(d*x + c)^7 + 10*d*cosh(d*x + c)^6 + 2*(462*d*c
osh(d*x + c)^6 + 525*d*cosh(d*x + c)^4 + 140*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 4*(198*d*cosh(d*x + c)
^7 + 315*d*cosh(d*x + c)^5 + 140*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^5 + 5*d*cosh(d*x + c)^4
+ 5*(99*d*cosh(d*x + c)^8 + 210*d*cosh(d*x + c)^6 + 140*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + d)*sinh(d*
x + c)^4 + 20*(11*d*cosh(d*x + c)^9 + 30*d*cosh(d*x + c)^7 + 28*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + d*c
osh(d*x + c))*sinh(d*x + c)^3 + d*cosh(d*x + c)^2 + (66*d*cosh(d*x + c)^10 + 225*d*cosh(d*x + c)^8 + 280*d*cos
h(d*x + c)^6 + 150*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 2*(6*d*cosh(d*x + c)^11 + 2
5*d*cosh(d*x + c)^9 + 40*d*cosh(d*x + c)^7 + 30*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*si
nh(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(sinh(d*x+c)**2*(a+b*tanh(d*x+c)**3)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.83589, size = 402, normalized size = 3.12 \begin{align*} -\frac{60 \,{\left (a^{2} - 8 \, a b + 7 \, b^{2}\right )} d x + 480 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 15 \,{\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 14 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} + 2 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 15 \,{\left (a^{2} e^{\left (2 \, d x + 16 \, c\right )} + 2 \, a b e^{\left (2 \, d x + 16 \, c\right )} + b^{2} e^{\left (2 \, d x + 16 \, c\right )}\right )} e^{\left (-14 \, c\right )} - \frac{8 \,{\left (137 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 625 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 180 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1190 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 480 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1190 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 680 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 625 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 400 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 137 \, a b - 116 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*tanh(d*x+c)^3)^2,x, algorithm="giac")

[Out]

-1/120*(60*(a^2 - 8*a*b + 7*b^2)*d*x + 480*a*b*log(e^(2*d*x + 2*c) + 1) - 15*(2*a^2*e^(2*d*x + 2*c) - 16*a*b*e
^(2*d*x + 2*c) + 14*b^2*e^(2*d*x + 2*c) - a^2 + 2*a*b - b^2)*e^(-2*d*x - 2*c) - 15*(a^2*e^(2*d*x + 16*c) + 2*a
*b*e^(2*d*x + 16*c) + b^2*e^(2*d*x + 16*c))*e^(-14*c) - 8*(137*a*b*e^(10*d*x + 10*c) + 625*a*b*e^(8*d*x + 8*c)
- 180*b^2*e^(8*d*x + 8*c) + 1190*a*b*e^(6*d*x + 6*c) - 480*b^2*e^(6*d*x + 6*c) + 1190*a*b*e^(4*d*x + 4*c) - 6
80*b^2*e^(4*d*x + 4*c) + 625*a*b*e^(2*d*x + 2*c) - 400*b^2*e^(2*d*x + 2*c) + 137*a*b - 116*b^2)/(e^(2*d*x + 2*
c) + 1)^5)/d