3.28 \(\int \frac{\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=109 \[ \frac{\left (a^2+a b+b^2\right ) \cosh (c+d x)}{b^3 d}-\frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{b^{7/2} d \sqrt{a-b}}-\frac{(a+2 b) \cosh ^3(c+d x)}{3 b^2 d}+\frac{\cosh ^5(c+d x)}{5 b d} \]

[Out]

-((a^3*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(7/2)*d)) + ((a^2 + a*b + b^2)*Cosh[c + d*x
])/(b^3*d) - ((a + 2*b)*Cosh[c + d*x]^3)/(3*b^2*d) + Cosh[c + d*x]^5/(5*b*d)

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Rubi [A]  time = 0.149371, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3186, 390, 205} \[ \frac{\left (a^2+a b+b^2\right ) \cosh (c+d x)}{b^3 d}-\frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{b^{7/2} d \sqrt{a-b}}-\frac{(a+2 b) \cosh ^3(c+d x)}{3 b^2 d}+\frac{\cosh ^5(c+d x)}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(7/2)*d)) + ((a^2 + a*b + b^2)*Cosh[c + d*x
])/(b^3*d) - ((a + 2*b)*Cosh[c + d*x]^3)/(3*b^2*d) + Cosh[c + d*x]^5/(5*b*d)

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.868052, size = 165, normalized size = 1.51 \[ \frac{30 \sqrt{b} \left (8 a^2+6 a b+5 b^2\right ) \cosh (c+d x)-\frac{240 a^3 \left (\tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )+\tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )\right )}{\sqrt{a-b}}-5 b^{3/2} (4 a+5 b) \cosh (3 (c+d x))+3 b^{5/2} \cosh (5 (c+d x))}{240 b^{7/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-240*a^3*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c
+ d*x)/2])/Sqrt[a - b]]))/Sqrt[a - b] + 30*Sqrt[b]*(8*a^2 + 6*a*b + 5*b^2)*Cosh[c + d*x] - 5*b^(3/2)*(4*a + 5*
b)*Cosh[3*(c + d*x)] + 3*b^(5/2)*Cosh[5*(c + d*x)])/(240*b^(7/2)*d)

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Maple [B]  time = 0.044, size = 448, normalized size = 4.1 \begin{align*}{\frac{1}{5\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-{\frac{1}{2\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}+{\frac{a}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3}{8\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{a}{3\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{12\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{{a}^{2}}{d{b}^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{a}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3}{8\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{5\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-5}}-{\frac{1}{2\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}+{\frac{a}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3}{8\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{a}{3\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{12\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{{a}^{2}}{d{b}^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{a}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{3}{8\,bd} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{{a}^{3}}{d{b}^{3}}\arctan \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,a+4\,b \right ){\frac{1}{\sqrt{ab-{b}^{2}}}}} \right ){\frac{1}{\sqrt{ab-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/d/b/(tanh(1/2*d*x+1/2*c)+1)^5-1/2/d/b/(tanh(1/2*d*x+1/2*c)+1)^4+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^2*a+3/8/
d/b/(tanh(1/2*d*x+1/2*c)+1)^2-1/3/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^3*a+1/12/d/b/(tanh(1/2*d*x+1/2*c)+1)^3+1/d/b^3
/(tanh(1/2*d*x+1/2*c)+1)*a^2+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)+1)*a+3/8/d/b/(tanh(1/2*d*x+1/2*c)+1)-1/5/d/b/(tanh
(1/2*d*x+1/2*c)-1)^5-1/2/d/b/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^2*a+3/8/d/b/(tanh(1/2
*d*x+1/2*c)-1)^2+1/3/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^3*a-1/12/d/b/(tanh(1/2*d*x+1/2*c)-1)^3-1/d/b^3/(tanh(1/2*d*
x+1/2*c)-1)*a^2-1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)*a-3/8/d/b/(tanh(1/2*d*x+1/2*c)-1)-1/d*a^3/b^3/(a*b-b^2)^(1/2
)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 3 \, b^{2} - 5 \,{\left (4 \, a b e^{\left (8 \, c\right )} + 5 \, b^{2} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 30 \,{\left (8 \, a^{2} e^{\left (6 \, c\right )} + 6 \, a b e^{\left (6 \, c\right )} + 5 \, b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 30 \,{\left (8 \, a^{2} e^{\left (4 \, c\right )} + 6 \, a b e^{\left (4 \, c\right )} + 5 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 5 \,{\left (4 \, a b e^{\left (2 \, c\right )} + 5 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, b^{3} d} - \frac{1}{128} \, \int \frac{256 \,{\left (a^{3} e^{\left (3 \, d x + 3 \, c\right )} - a^{3} e^{\left (d x + c\right )}\right )}}{b^{4} e^{\left (4 \, d x + 4 \, c\right )} + b^{4} + 2 \,{\left (2 \, a b^{3} e^{\left (2 \, c\right )} - b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(3*b^2*e^(10*d*x + 10*c) + 3*b^2 - 5*(4*a*b*e^(8*c) + 5*b^2*e^(8*c))*e^(8*d*x) + 30*(8*a^2*e^(6*c) + 6*a
*b*e^(6*c) + 5*b^2*e^(6*c))*e^(6*d*x) + 30*(8*a^2*e^(4*c) + 6*a*b*e^(4*c) + 5*b^2*e^(4*c))*e^(4*d*x) - 5*(4*a*
b*e^(2*c) + 5*b^2*e^(2*c))*e^(2*d*x))*e^(-5*d*x - 5*c)/(b^3*d) - 1/128*integrate(256*(a^3*e^(3*d*x + 3*c) - a^
3*e^(d*x + c))/(b^4*e^(4*d*x + 4*c) + b^4 + 2*(2*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.28412, size = 7640, normalized size = 70.09 \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)^7/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/480*(3*(a*b^3 - b^4)*cosh(d*x + c)^10 + 30*(a*b^3 - b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + 3*(a*b^3 - b^4)*si
nh(d*x + c)^10 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^8 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4 - 27*(a*b^3 - b^
4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 40*(9*(a*b^3 - b^4)*cosh(d*x + c)^3 - (4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d
*x + c))*sinh(d*x + c)^7 + 30*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 10*(63*(a*b^3 - b^4)*cos
h(d*x + c)^4 + 24*a^3*b - 6*a^2*b^2 - 3*a*b^3 - 15*b^4 - 14*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(
d*x + c)^6 + 4*(189*(a*b^3 - b^4)*cosh(d*x + c)^5 - 70*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 45*(8*a^3
*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 30*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh
(d*x + c)^4 + 10*(63*(a*b^3 - b^4)*cosh(d*x + c)^6 - 35*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^4 + 24*a^3*b
- 6*a^2*b^2 - 3*a*b^3 - 15*b^4 + 45*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
3*a*b^3 - 3*b^4 + 40*(9*(a*b^3 - b^4)*cosh(d*x + c)^7 - 7*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 15*(8*
a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 3*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c))*si
nh(d*x + c)^3 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^2 + 5*(27*(a*b^3 - b^4)*cosh(d*x + c)^8 - 28*(4*a^
2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 90*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^4 - 4*a^2*b^2
- a*b^3 + 5*b^4 + 36*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 240*(a^3*cosh(d*
x + c)^5 + 5*a^3*cosh(d*x + c)^4*sinh(d*x + c) + 10*a^3*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*a^3*cosh(d*x + c)
^2*sinh(d*x + c)^3 + 5*a^3*cosh(d*x + c)*sinh(d*x + c)^4 + a^3*sinh(d*x + c)^5)*sqrt(-a*b + b^2)*log((b*cosh(d
*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cos
h(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) +
4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c)
+ cosh(d*x + c))*sqrt(-a*b + b^2) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x +
c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3
+ (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 10*(3*(a*b^3 - b^4)*cosh(d*x + c)^9 - 4*(4*a^2*b^2 + a*b^3 -
5*b^4)*cosh(d*x + c)^7 + 18*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 12*(8*a^3*b - 2*a^2*b^2 -
a*b^3 - 5*b^4)*cosh(d*x + c)^3 - (4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a*b^4 - b^5)*d*co
sh(d*x + c)^5 + 5*(a*b^4 - b^5)*d*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a*b^4 - b^5)*d*cosh(d*x + c)^3*sinh(d*x
+ c)^2 + 10*(a*b^4 - b^5)*d*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a*b^4 - b^5)*d*cosh(d*x + c)*sinh(d*x + c)^4
+ (a*b^4 - b^5)*d*sinh(d*x + c)^5), 1/480*(3*(a*b^3 - b^4)*cosh(d*x + c)^10 + 30*(a*b^3 - b^4)*cosh(d*x + c)*s
inh(d*x + c)^9 + 3*(a*b^3 - b^4)*sinh(d*x + c)^10 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^8 - 5*(4*a^2*b
^2 + a*b^3 - 5*b^4 - 27*(a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 40*(9*(a*b^3 - b^4)*cosh(d*x + c)^3 -
(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*
x + c)^6 + 10*(63*(a*b^3 - b^4)*cosh(d*x + c)^4 + 24*a^3*b - 6*a^2*b^2 - 3*a*b^3 - 15*b^4 - 14*(4*a^2*b^2 + a*
b^3 - 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(189*(a*b^3 - b^4)*cosh(d*x + c)^5 - 70*(4*a^2*b^2 + a*b^3 -
5*b^4)*cosh(d*x + c)^3 + 45*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 30*(8*a^3*
b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^4 + 10*(63*(a*b^3 - b^4)*cosh(d*x + c)^6 - 35*(4*a^2*b^2 + a*b^3
- 5*b^4)*cosh(d*x + c)^4 + 24*a^3*b - 6*a^2*b^2 - 3*a*b^3 - 15*b^4 + 45*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*
cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a*b^3 - 3*b^4 + 40*(9*(a*b^3 - b^4)*cosh(d*x + c)^7 - 7*(4*a^2*b^2 + a*b^
3 - 5*b^4)*cosh(d*x + c)^5 + 15*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^3 + 3*(8*a^3*b - 2*a^2*b^2
- a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^2 + 5*(27*(a*b^
3 - b^4)*cosh(d*x + c)^8 - 28*(4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c)^6 + 90*(8*a^3*b - 2*a^2*b^2 - a*b^3 -
5*b^4)*cosh(d*x + c)^4 - 4*a^2*b^2 - a*b^3 + 5*b^4 + 36*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^2 - 480*(a^3*cosh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^4*sinh(d*x + c) + 10*a^3*cosh(d*x + c)^3*sin
h(d*x + c)^2 + 10*a^3*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*a^3*cosh(d*x + c)*sinh(d*x + c)^4 + a^3*sinh(d*x + c
)^5)*sqrt(a*b - b^2)*arctan(-1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 +
(4*a - 3*b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))/sqrt(a*b - b^2)) + 480*(a^3*cosh(
d*x + c)^5 + 5*a^3*cosh(d*x + c)^4*sinh(d*x + c) + 10*a^3*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*a^3*cosh(d*x +
c)^2*sinh(d*x + c)^3 + 5*a^3*cosh(d*x + c)*sinh(d*x + c)^4 + a^3*sinh(d*x + c)^5)*sqrt(a*b - b^2)*arctan(-1/2*
sqrt(a*b - b^2)*(cosh(d*x + c) + sinh(d*x + c))/(a - b)) + 10*(3*(a*b^3 - b^4)*cosh(d*x + c)^9 - 4*(4*a^2*b^2
+ a*b^3 - 5*b^4)*cosh(d*x + c)^7 + 18*(8*a^3*b - 2*a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^5 + 12*(8*a^3*b - 2*
a^2*b^2 - a*b^3 - 5*b^4)*cosh(d*x + c)^3 - (4*a^2*b^2 + a*b^3 - 5*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a*b^4 -
b^5)*d*cosh(d*x + c)^5 + 5*(a*b^4 - b^5)*d*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a*b^4 - b^5)*d*cosh(d*x + c)^3
*sinh(d*x + c)^2 + 10*(a*b^4 - b^5)*d*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a*b^4 - b^5)*d*cosh(d*x + c)*sinh(d
*x + c)^4 + (a*b^4 - b^5)*d*sinh(d*x + c)^5)]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError