3.229 \(\int \frac {\sinh ^7(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\)

Optimal. Leaf size=148 \[ -\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cosh ^3(c+d x)}{3 b d}+\frac {\cosh (c+d x)}{b d} \]

[Out]

cosh(d*x+c)/b/d-1/3*cosh(d*x+c)^3/b/d-1/2*a*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/b^(7/4)/d/(a^(
1/2)-b^(1/2))^(1/2)+1/2*a*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/b^(7/4)/d/(a^(1/2)+b^(1/2))^(1/
2)

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Rubi [A]  time = 0.25, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3215, 1170, 1166, 205, 208} \[ -\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{7/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cosh ^3(c+d x)}{3 b d}+\frac {\cosh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4),x]

[Out]

-(a*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(7/4)*d) + (a*ArcTan
h[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(7/4)*d) + Cosh[c + d*x]/(b*d
) - Cosh[c + d*x]^3/(3*b*d)

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]

Rule 208

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(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 - 2*b*ff^2*x^2 + b*ff^4*x^4
)^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 ^4(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{b}+\frac {x^2}{b}+\frac {a-a x^2}{b \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x)}{b d}-\frac {\cosh ^3(c+d x)}{3 b d}-\frac {\operatorname {Subst}\left (\int \frac {a-a x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{b d}\\ &=\frac {\cosh (c+d x)}{b d}-\frac {\cosh ^3(c+d x)}{3 b d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 b d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 b d}\\ &=-\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{7/4} d}+\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{7/4} d}+\frac {\cosh (c+d x)}{b d}-\frac {\cosh ^3(c+d x)}{3 b d}\\ \end {align*}

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Mathematica [C]  time = 0.35, size = 390, normalized size = 2.64 \[ \frac {-3 a \text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {$\#$1}^6 c+\text {$\#$1}^6 d x-6 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-3 \text {$\#$1}^4 c-3 \text {$\#$1}^4 d x+6 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+3 \text {$\#$1}^2 c+3 \text {$\#$1}^2 d x-2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-c-d x}{\text {$\#$1}^7 b-3 \text {$\#$1}^5 b-8 \text {$\#$1}^3 a+3 \text {$\#$1}^3 b-\text {$\#$1} b}\& \right ]+18 \cosh (c+d x)-2 \cosh (3 (c+d x))}{24 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4),x]

[Out]

(18*Cosh[c + d*x] - 2*Cosh[3*(c + d*x)] - 3*a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8
& , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] +
3*c*#1^2 + 3*d*x*#1^2 + 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2
]*#1]*#1^2 - 3*c*#1^4 - 3*d*x*#1^4 - 6*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin
h[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1^6 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]
*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(24*b*d)

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fricas [B]  time = 0.75, size = 1617, normalized size = 10.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/24*(cosh(d*x + c)^6 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6 + 3*(5*cosh(d*x + c)^2 - 3)*sinh(d*
x + c)^4 - 9*cosh(d*x + c)^4 + 4*(5*cosh(d*x + c)^3 - 9*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*cosh(d*x + c)^4
- 18*cosh(d*x + c)^2 - 3)*sinh(d*x + c)^2 - 6*(b*d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b
*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x + c)^3)*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8
+ b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cosh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sin
h(d*x + c)^2 + a^3 + 2*(a^2*b^2*d*cosh(d*x + c) + a^2*b^2*d*sinh(d*x + c) - ((a*b^5 - b^6)*d^3*cosh(d*x + c) +
(a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5
/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + 6*(b*d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^
2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x + c)^3)*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5
/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cosh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*si
nh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3 - 2*(a^2*b^2*d*cosh(d*x + c) + a^2*b^2*d*sinh(d*x + c) - ((a*b^5 - b^6
)*d^3*cosh(d*x + c) + (a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^
3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - 6*(b*d*cosh(d*x + c)^3 +
3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x + c)^3)*sqrt(((a*b^3
- b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cosh(d*x + c)^2 + 2*
a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3 + 2*(a^2*b^2*d*cosh(d*x + c) + a^2*b^2*d*sinh(d*x
+ c) + ((a*b^5 - b^6)*d^3*cosh(d*x + c) + (a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)
*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) + 6*(b*
d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d*x +
c)^3)*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^3*c
osh(d*x + c)^2 + 2*a^3*cosh(d*x + c)*sinh(d*x + c) + a^3*sinh(d*x + c)^2 + a^3 - 2*(a^2*b^2*d*cosh(d*x + c) +
a^2*b^2*d*sinh(d*x + c) + ((a*b^5 - b^6)*d^3*cosh(d*x + c) + (a*b^5 - b^6)*d^3*sinh(d*x + c))*sqrt(a^5/((a^2*b
^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 -
b^4)*d^2))) - 9*cosh(d*x + c)^2 + 6*(cosh(d*x + c)^5 - 6*cosh(d*x + c)^3 - 3*cosh(d*x + c))*sinh(d*x + c) + 1)
/(b*d*cosh(d*x + c)^3 + 3*b*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b*d*cosh(d*x + c)*sinh(d*x + c)^2 + b*d*sinh(d
*x + c)^3)

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giac [B]  time = 0.53, size = 553, normalized size = 3.74 \[ -\frac {\frac {12 \, {\left ({\left (\sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 8 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b\right )} b^{2} + {\left (\sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{2} + 8 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{3}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b^{4} - \sqrt {b^{8} + {\left (a b^{3} - b^{4}\right )} b^{4}}}{b^{4}}}}\right )}{a^{2} b^{5} + 7 \, a b^{6} - 8 \, b^{7}} + \frac {6 \, {\left (\sqrt {b^{2} + \sqrt {a b} b} a^{2} b^{3} {\left | b \right |} - \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{4} - {\left (\sqrt {b^{2} + \sqrt {a b} b} a^{2} b + \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a^{2}\right )} b^{2} {\left | b \right |} + {\left (\sqrt {b^{2} + \sqrt {a b} b} a^{2} b^{2} + \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{2}\right )} b^{2}\right )} \log \left (2 \, \sqrt {\frac {b^{4} + \sqrt {b^{8} + {\left (a b^{3} - b^{4}\right )} b^{4}}}{b^{4}}} + e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{a^{2} b^{6} - a b^{7}} - \frac {6 \, {\left (\sqrt {b^{2} + \sqrt {a b} b} a b^{2} {\left | b \right |} - \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{2}\right )} \log \left ({\left | -2 \, \sqrt {\frac {b^{4} + \sqrt {b^{8} + {\left (a b^{3} - b^{4}\right )} b^{4}}}{b^{4}}} + e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} \right |}\right )}{a b^{5} - b^{6}} + \frac {b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{b^{3}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

-1/24*(12*((sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2 + 8*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b)*b^2 + (sqrt(-b^
2 + sqrt(a*b)*b)*a^2*b^2 + 8*sqrt(-b^2 + sqrt(a*b)*b)*a*b^3)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/s
qrt(-(b^4 - sqrt(b^8 + (a*b^3 - b^4)*b^4))/b^4))/(a^2*b^5 + 7*a*b^6 - 8*b^7) + 6*(sqrt(b^2 + sqrt(a*b)*b)*a^2*
b^3*abs(b) - sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a*b^4 - (sqrt(b^2 + sqrt(a*b)*b)*a^2*b + sqrt(a*b)*sqrt(b^2 + s
qrt(a*b)*b)*a^2)*b^2*abs(b) + (sqrt(b^2 + sqrt(a*b)*b)*a^2*b^2 + sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a*b^2)*b^2)
*log(2*sqrt((b^4 + sqrt(b^8 + (a*b^3 - b^4)*b^4))/b^4) + e^(d*x + c) + e^(-d*x - c))/(a^2*b^6 - a*b^7) - 6*(sq
rt(b^2 + sqrt(a*b)*b)*a*b^2*abs(b) - sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a*b^2)*log(abs(-2*sqrt((b^4 + sqrt(b^8
+ (a*b^3 - b^4)*b^4))/b^4) + e^(d*x + c) + e^(-d*x - c)))/(a*b^5 - b^6) + (b^2*(e^(d*x + c) + e^(-d*x - c))^3
- 12*b^2*(e^(d*x + c) + e^(-d*x - c)))/b^3)/d

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maple [B]  time = 0.13, size = 270, normalized size = 1.82 \[ -\frac {a \sqrt {a b}\, \arctan \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{2 d \,b^{2} \sqrt {-a b +\sqrt {a b}\, a}}-\frac {a \sqrt {a b}\, \arctan \left (\frac {-2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{2 d \,b^{2} \sqrt {-a b -\sqrt {a b}\, a}}+\frac {1}{3 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (e^{\left (6 \, d x + 6 \, c\right )} - 9 \, e^{\left (4 \, d x + 4 \, c\right )} - 9 \, e^{\left (2 \, d x + 2 \, c\right )} + 1\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, b d} - \frac {1}{128} \, \int \frac {256 \, {\left (a e^{\left (7 \, d x + 7 \, c\right )} - 3 \, a e^{\left (5 \, d x + 5 \, c\right )} + 3 \, a e^{\left (3 \, d x + 3 \, c\right )} - a e^{\left (d x + c\right )}\right )}}{b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 4 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 4 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2} - 2 \, {\left (8 \, a b e^{\left (4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/24*(e^(6*d*x + 6*c) - 9*e^(4*d*x + 4*c) - 9*e^(2*d*x + 2*c) + 1)*e^(-3*d*x - 3*c)/(b*d) - 1/128*integrate(2
56*(a*e^(7*d*x + 7*c) - 3*a*e^(5*d*x + 5*c) + 3*a*e^(3*d*x + 3*c) - a*e^(d*x + c))/(b^2*e^(8*d*x + 8*c) - 4*b^
2*e^(6*d*x + 6*c) - 4*b^2*e^(2*d*x + 2*c) + b^2 - 2*(8*a*b*e^(4*c) - 3*b^2*e^(4*c))*e^(4*d*x)), x)

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mupad [B]  time = 9.80, size = 1124, normalized size = 7.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4),x)

[Out]

log((((((4194304*a^8*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a - b)^2) - (8388608*a^7*d^3*exp(c + d*x)*(a
+ b)*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(a - b)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b
^7*d^2*(a - b)))^(1/2))/4 + (2097152*a^9*d*exp(c + d*x))/(b^13*(a - b)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^
2*(a - b)))^(1/2))/4 - (262144*a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^15*(a - b)^2))*(((a^5*b^7)^(1/2) + a^2*
b^4)/(16*(b^8*d^2 - a*b^7*d^2)))^(1/2) - log((((((4194304*a^8*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a -
b)^2) + (8388608*a^7*d^3*exp(c + d*x)*(a + b)*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(
a - b)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (2097152*a^9*d*exp(c + d*x))/(b^13*(a - b
)))*(-((a^5*b^7)^(1/2) + a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (262144*a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(
b^15*(a - b)^2))*(((a^5*b^7)^(1/2) + a^2*b^4)/(16*(b^8*d^2 - a*b^7*d^2)))^(1/2) + log((((((4194304*a^8*d^2*(ex
p(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a - b)^2) - (8388608*a^7*d^3*exp(c + d*x)*(a + b)*(((a^5*b^7)^(1/2) - a^
2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(a - b)))*(((a^5*b^7)^(1/2) - a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 + (2
097152*a^9*d*exp(c + d*x))/(b^13*(a - b)))*(((a^5*b^7)^(1/2) - a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (262144*
a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^15*(a - b)^2))*(-((a^5*b^7)^(1/2) - a^2*b^4)/(16*(b^8*d^2 - a*b^7*d^2)
))^(1/2) - log((((((4194304*a^8*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^11*(a - b)^2) + (8388608*a^7*d^3*exp(
c + d*x)*(a + b)*(((a^5*b^7)^(1/2) - a^2*b^4)/(b^7*d^2*(a - b)))^(1/2))/(b^10*(a - b)))*(((a^5*b^7)^(1/2) - a^
2*b^4)/(b^7*d^2*(a - b)))^(1/2))/4 - (2097152*a^9*d*exp(c + d*x))/(b^13*(a - b)))*(((a^5*b^7)^(1/2) - a^2*b^4)
/(b^7*d^2*(a - b)))^(1/2))/4 - (262144*a^10*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^15*(a - b)^2))*(-((a^5*b^7)^(1/
2) - a^2*b^4)/(16*(b^8*d^2 - a*b^7*d^2)))^(1/2) + (3*exp(c + d*x))/(8*b*d) + (3*exp(- c - d*x))/(8*b*d) - exp(
- 3*c - 3*d*x)/(24*b*d) - exp(3*c + 3*d*x)/(24*b*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**7/(a-b*sinh(d*x+c)**4),x)

[Out]

Timed out

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