3.1.28 \(\int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [28]

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

[Out]

(a^2+a*b+b^2)*cosh(d*x+c)/b^3/d-1/3*(a+2*b)*cosh(d*x+c)^3/b^2/d+1/5*cosh(d*x+c)^5/b/d-a^3*arctan(cosh(d*x+c)*b
^(1/2)/(a-b)^(1/2))/b^(7/2)/d/(a-b)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 398, 211} \begin {gather*} -\frac {a^3 \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{7/2} d \sqrt {a-b}}+\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 {gather*}

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 211

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

Rule 398

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 3265

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]

Rubi steps

\begin {align*} \int \frac {\sinh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {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 \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.61, size = 165, normalized size = 1.51 \begin {gather*} \frac {-\frac {240 a^3 \left (\text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{\sqrt {a-b}}+30 \sqrt {b} \left (8 a^2+6 a b+5 b^2\right ) \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} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(295\) vs. \(2(97)=194\).
time = 1.18, size = 296, normalized size = 2.72

method	result	size
derivativedivides	\(\frac {-\frac {a^{3} \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b^{3} \sqrt {a b -b^{2}}}+\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {4 a -b}{12 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-8 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-4 a +b}{12 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {8 a^{2}+4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(296\)
default	\(\frac {-\frac {a^{3} \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b^{3} \sqrt {a b -b^{2}}}+\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {4 a -b}{12 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-8 a^{2}-4 a b -3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{5 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-4 a -3 b}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-4 a +b}{12 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {8 a^{2}+4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(296\)
risch	\(\frac {{\mathrm e}^{5 d x +5 c}}{160 b d}-\frac {5 \,{\mathrm e}^{3 d x +3 c}}{96 b d}-\frac {{\mathrm e}^{3 d x +3 c} a}{24 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}+\frac {3 a \,{\mathrm e}^{d x +c}}{8 b^{2} d}+\frac {5 \,{\mathrm e}^{d x +c}}{16 b d}+\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}+\frac {3 \,{\mathrm e}^{-d x -c} a}{8 b^{2} d}+\frac {5 \,{\mathrm e}^{-d x -c}}{16 b d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c}}{96 b d}-\frac {{\mathrm e}^{-3 d x -3 c} a}{24 b^{2} d}+\frac {{\mathrm e}^{-5 d x -5 c}}{160 b d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d \,b^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d \,b^{3}}\)	\(319\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (97) = 194\).
time = 0.53, size = 3242, normalized size = 29.74 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 1.64, size = 415, normalized size = 3.81 \begin {gather*} \frac {{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,b\,d}-\frac {\sqrt {a^6}\,\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^7\,d^2\,\left (a-b\right )}}{2\,b^3\,d\,\left (a-b\right )\,\sqrt {a^6}}\right )+2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^7}{b^{11}\,d\,{\left (a-b\right )}^2\,\sqrt {a^6}}-\frac {4\,\left (2\,a^4\,b^4\,d\,\sqrt {a^6}-2\,a^5\,b^3\,d\,\sqrt {a^6}\right )}{a^3\,b^8\,\left (a-b\right )\,\sqrt {a\,b^7\,d^2-b^8\,d^2}\,\sqrt {b^7\,d^2\,\left (a-b\right )}}\right )+\frac {2\,a^7\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{b^{11}\,d\,{\left (a-b\right )}^2\,\sqrt {a^6}}\right )\,\left (b^9\,\sqrt {a\,b^7\,d^2-b^8\,d^2}-a\,b^8\,\sqrt {a\,b^7\,d^2-b^8\,d^2}\right )}{4\,a^4}\right )\right )}{2\,\sqrt {a\,b^7\,d^2-b^8\,d^2}}+\frac {{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,b\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2+6\,a\,b+5\,b^2\right )}{16\,b^3\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a^2+6\,a\,b+5\,b^2\right )}{16\,b^3\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (4\,a+5\,b\right )}{96\,b^2\,d}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (4\,a+5\,b\right )}{96\,b^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(- 5*c - 5*d*x)/(160*b*d) - ((a^6)^(1/2)*(2*atan((a^3*exp(d*x)*exp(c)*(b^7*d^2*(a - b))^(1/2))/(2*b^3*d*(a
- b)*(a^6)^(1/2))) + 2*atan(((exp(d*x)*exp(c)*((2*a^7)/(b^11*d*(a - b)^2*(a^6)^(1/2)) - (4*(2*a^4*b^4*d*(a^6)^
(1/2) - 2*a^5*b^3*d*(a^6)^(1/2)))/(a^3*b^8*(a - b)*(a*b^7*d^2 - b^8*d^2)^(1/2)*(b^7*d^2*(a - b))^(1/2))) + (2*
a^7*exp(3*c)*exp(3*d*x))/(b^11*d*(a - b)^2*(a^6)^(1/2)))*(b^9*(a*b^7*d^2 - b^8*d^2)^(1/2) - a*b^8*(a*b^7*d^2 -
b^8*d^2)^(1/2)))/(4*a^4))))/(2*(a*b^7*d^2 - b^8*d^2)^(1/2)) + exp(5*c + 5*d*x)/(160*b*d) + (exp(c + d*x)*(6*a
*b + 8*a^2 + 5*b^2))/(16*b^3*d) + (exp(- c - d*x)*(6*a*b + 8*a^2 + 5*b^2))/(16*b^3*d) - (exp(- 3*c - 3*d*x)*(4
*a + 5*b))/(96*b^2*d) - (exp(3*c + 3*d*x)*(4*a + 5*b))/(96*b^2*d)

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