3.5.31 \(\int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^6(e+f x) \, dx\) [431]

Optimal. Leaf size=120 \[ -\frac {15 \text {ArcTan}(\sinh (e+f x)) \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)}{8 f}+\frac {15 \sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{8 f}-\frac {5 \sqrt {a \cosh ^2(e+f x)} \tanh ^3(e+f x)}{8 f}-\frac {\sqrt {a \cosh ^2(e+f x)} \tanh ^5(e+f x)}{4 f} \]

[Out]

-15/8*arctan(sinh(f*x+e))*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f+15/8*(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f-5/8
*(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)^3/f-1/4*(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)^5/f

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Rubi [A]
time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3255, 3286, 2672, 294, 327, 209} \begin {gather*} -\frac {15 \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \text {ArcTan}(\sinh (e+f x))}{8 f}-\frac {\tanh ^5(e+f x) \sqrt {a \cosh ^2(e+f x)}}{4 f}-\frac {5 \tanh ^3(e+f x) \sqrt {a \cosh ^2(e+f x)}}{8 f}+\frac {15 \tanh (e+f x) \sqrt {a \cosh ^2(e+f x)}}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^6,x]

[Out]

(-15*ArcTan[Sinh[e + f*x]]*Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x])/(8*f) + (15*Sqrt[a*Cosh[e + f*x]^2]*Tanh[e +
f*x])/(8*f) - (5*Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x]^3)/(8*f) - (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x]^5)/(
4*f)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2672

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 3255

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \sqrt {a+a \sinh ^2(e+f x)} \tanh ^6(e+f x) \, dx &=\int \sqrt {a \cosh ^2(e+f x)} \tanh ^6(e+f x) \, dx\\ &=\left (\sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \int \sinh (e+f x) \tanh ^5(e+f x) \, dx\\ &=\frac {\left (\sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\sqrt {a \cosh ^2(e+f x)} \tanh ^5(e+f x)}{4 f}+\frac {\left (5 \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (e+f x)\right )}{4 f}\\ &=-\frac {5 \sqrt {a \cosh ^2(e+f x)} \tanh ^3(e+f x)}{8 f}-\frac {\sqrt {a \cosh ^2(e+f x)} \tanh ^5(e+f x)}{4 f}+\frac {\left (15 \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (e+f x)\right )}{8 f}\\ &=\frac {15 \sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{8 f}-\frac {5 \sqrt {a \cosh ^2(e+f x)} \tanh ^3(e+f x)}{8 f}-\frac {\sqrt {a \cosh ^2(e+f x)} \tanh ^5(e+f x)}{4 f}-\frac {\left (15 \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (e+f x)\right )}{8 f}\\ &=-\frac {15 \tan ^{-1}(\sinh (e+f x)) \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)}{8 f}+\frac {15 \sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{8 f}-\frac {5 \sqrt {a \cosh ^2(e+f x)} \tanh ^3(e+f x)}{8 f}-\frac {\sqrt {a \cosh ^2(e+f x)} \tanh ^5(e+f x)}{4 f}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 75, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {a \cosh ^2(e+f x)} \text {sech}^5(e+f x) \left (60 \text {ArcTan}(\sinh (e+f x)) \cosh ^4(e+f x)-5 \sinh (e+f x)-15 \sinh (3 (e+f x))-2 \sinh (5 (e+f x))\right )}{32 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^6,x]

[Out]

-1/32*(Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x]^5*(60*ArcTan[Sinh[e + f*x]]*Cosh[e + f*x]^4 - 5*Sinh[e + f*x] - 1
5*Sinh[3*(e + f*x)] - 2*Sinh[5*(e + f*x)]))/f

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Maple [A]
time = 1.66, size = 85, normalized size = 0.71

method	result	size
default	\(-\frac {a \left (15 \arctan \left (\sinh \left (f x +e \right )\right ) \left (\cosh ^{4}\left (f x +e \right )\right )-8 \left (\cosh ^{4}\left (f x +e \right )\right ) \sinh \left (f x +e \right )-9 \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{8 \cosh \left (f x +e \right )^{3} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f}\)	\(85\)
risch	\(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {\left (9 \,{\mathrm e}^{6 f x +6 e}+{\mathrm e}^{4 f x +4 e}-{\mathrm e}^{2 f x +2 e}-9\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{4 f \left ({\mathrm e}^{2 f x +2 e}+1\right )^{5}}+\frac {15 i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{8 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {15 i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{8 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\)	\(313\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^6,x,method=_RETURNVERBOSE)

[Out]

-1/8*a*(15*arctan(sinh(f*x+e))*cosh(f*x+e)^4-8*cosh(f*x+e)^4*sinh(f*x+e)-9*cosh(f*x+e)^2*sinh(f*x+e)+2*sinh(f*
x+e))/cosh(f*x+e)^3/(a*cosh(f*x+e)^2)^(1/2)/f

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 955 vs. \(2 (113) = 226\).
time = 0.51, size = 955, normalized size = 7.96 \begin {gather*} \frac {315 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right )}{128 \, f} + \frac {105 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) + \frac {279 \, \sqrt {a} e^{\left (-f x - e\right )} + 511 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 385 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} + 105 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, e^{\left (-6 \, f x - 6 \, e\right )} + e^{\left (-8 \, f x - 8 \, e\right )} + 1}}{128 \, f} + \frac {105 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) - \frac {105 \, \sqrt {a} e^{\left (-f x - e\right )} + 385 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 511 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} + 279 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, e^{\left (-6 \, f x - 6 \, e\right )} + e^{\left (-8 \, f x - 8 \, e\right )} + 1}}{128 \, f} - \frac {5 \, {\left (15 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) - \frac {15 \, \sqrt {a} e^{\left (-f x - e\right )} + 55 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 73 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 15 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, e^{\left (-6 \, f x - 6 \, e\right )} + e^{\left (-8 \, f x - 8 \, e\right )} + 1}\right )}}{256 \, f} - \frac {5 \, {\left (15 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) - \frac {15 \, \sqrt {a} e^{\left (-f x - e\right )} - 73 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 55 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 15 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, e^{\left (-6 \, f x - 6 \, e\right )} + e^{\left (-8 \, f x - 8 \, e\right )} + 1}\right )}}{256 \, f} + \frac {5 \, {\left (3 \, \sqrt {a} \arctan \left (e^{\left (-f x - e\right )}\right ) - \frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} + 11 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 11 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} - 3 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, e^{\left (-6 \, f x - 6 \, e\right )} + e^{\left (-8 \, f x - 8 \, e\right )} + 1}\right )}}{64 \, f} + \frac {837 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 1533 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 1155 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + 315 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + 128 \, \sqrt {a}}{256 \, f {\left (e^{\left (-f x - e\right )} + 4 \, e^{\left (-3 \, f x - 3 \, e\right )} + 6 \, e^{\left (-5 \, f x - 5 \, e\right )} + 4 \, e^{\left (-7 \, f x - 7 \, e\right )} + e^{\left (-9 \, f x - 9 \, e\right )}\right )}} - \frac {315 \, \sqrt {a} e^{\left (-f x - e\right )} + 1155 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 1533 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )} + 837 \, \sqrt {a} e^{\left (-7 \, f x - 7 \, e\right )} + 128 \, \sqrt {a} e^{\left (-9 \, f x - 9 \, e\right )}}{256 \, f {\left (4 \, e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, e^{\left (-6 \, f x - 6 \, e\right )} + e^{\left (-8 \, f x - 8 \, e\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^6,x, algorithm="maxima")

[Out]

315/128*sqrt(a)*arctan(e^(-f*x - e))/f + 1/128*(105*sqrt(a)*arctan(e^(-f*x - e)) + (279*sqrt(a)*e^(-f*x - e) +
511*sqrt(a)*e^(-3*f*x - 3*e) + 385*sqrt(a)*e^(-5*f*x - 5*e) + 105*sqrt(a)*e^(-7*f*x - 7*e))/(4*e^(-2*f*x - 2*
e) + 6*e^(-4*f*x - 4*e) + 4*e^(-6*f*x - 6*e) + e^(-8*f*x - 8*e) + 1))/f + 1/128*(105*sqrt(a)*arctan(e^(-f*x -
e)) - (105*sqrt(a)*e^(-f*x - e) + 385*sqrt(a)*e^(-3*f*x - 3*e) + 511*sqrt(a)*e^(-5*f*x - 5*e) + 279*sqrt(a)*e^
(-7*f*x - 7*e))/(4*e^(-2*f*x - 2*e) + 6*e^(-4*f*x - 4*e) + 4*e^(-6*f*x - 6*e) + e^(-8*f*x - 8*e) + 1))/f - 5/2
56*(15*sqrt(a)*arctan(e^(-f*x - e)) - (15*sqrt(a)*e^(-f*x - e) + 55*sqrt(a)*e^(-3*f*x - 3*e) + 73*sqrt(a)*e^(-
5*f*x - 5*e) - 15*sqrt(a)*e^(-7*f*x - 7*e))/(4*e^(-2*f*x - 2*e) + 6*e^(-4*f*x - 4*e) + 4*e^(-6*f*x - 6*e) + e^
(-8*f*x - 8*e) + 1))/f - 5/256*(15*sqrt(a)*arctan(e^(-f*x - e)) - (15*sqrt(a)*e^(-f*x - e) - 73*sqrt(a)*e^(-3*
f*x - 3*e) - 55*sqrt(a)*e^(-5*f*x - 5*e) - 15*sqrt(a)*e^(-7*f*x - 7*e))/(4*e^(-2*f*x - 2*e) + 6*e^(-4*f*x - 4*
e) + 4*e^(-6*f*x - 6*e) + e^(-8*f*x - 8*e) + 1))/f + 5/64*(3*sqrt(a)*arctan(e^(-f*x - e)) - (3*sqrt(a)*e^(-f*x
- e) + 11*sqrt(a)*e^(-3*f*x - 3*e) - 11*sqrt(a)*e^(-5*f*x - 5*e) - 3*sqrt(a)*e^(-7*f*x - 7*e))/(4*e^(-2*f*x -
2*e) + 6*e^(-4*f*x - 4*e) + 4*e^(-6*f*x - 6*e) + e^(-8*f*x - 8*e) + 1))/f + 1/256*(837*sqrt(a)*e^(-2*f*x - 2*
e) + 1533*sqrt(a)*e^(-4*f*x - 4*e) + 1155*sqrt(a)*e^(-6*f*x - 6*e) + 315*sqrt(a)*e^(-8*f*x - 8*e) + 128*sqrt(a
))/(f*(e^(-f*x - e) + 4*e^(-3*f*x - 3*e) + 6*e^(-5*f*x - 5*e) + 4*e^(-7*f*x - 7*e) + e^(-9*f*x - 9*e))) - 1/25
6*(315*sqrt(a)*e^(-f*x - e) + 1155*sqrt(a)*e^(-3*f*x - 3*e) + 1533*sqrt(a)*e^(-5*f*x - 5*e) + 837*sqrt(a)*e^(-
7*f*x - 7*e) + 128*sqrt(a)*e^(-9*f*x - 9*e))/(f*(4*e^(-2*f*x - 2*e) + 6*e^(-4*f*x - 4*e) + 4*e^(-6*f*x - 6*e)
+ e^(-8*f*x - 8*e) + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1645 vs. \(2 (104) = 208\).
time = 0.47, size = 1645, normalized size = 13.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^6,x, algorithm="fricas")

[Out]

1/4*(20*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^9 + 2*e^(f*x + e)*sinh(f*x + e)^10 + 15*(6*cosh(f*x + e)^2 + 1
)*e^(f*x + e)*sinh(f*x + e)^8 + 120*(2*cosh(f*x + e)^3 + cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^7 + 5*(84*co
sh(f*x + e)^4 + 84*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^6 + 6*(84*cosh(f*x + e)^5 + 140*cosh(f*x + e
)^3 + 5*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^5 + 5*(84*cosh(f*x + e)^6 + 210*cosh(f*x + e)^4 + 15*cosh(f*x
+ e)^2 - 1)*e^(f*x + e)*sinh(f*x + e)^4 + 20*(12*cosh(f*x + e)^7 + 42*cosh(f*x + e)^5 + 5*cosh(f*x + e)^3 - c
osh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^3 + 15*(6*cosh(f*x + e)^8 + 28*cosh(f*x + e)^6 + 5*cosh(f*x + e)^4 - 2
*cosh(f*x + e)^2 - 1)*e^(f*x + e)*sinh(f*x + e)^2 + 10*(2*cosh(f*x + e)^9 + 12*cosh(f*x + e)^7 + 3*cosh(f*x +
e)^5 - 2*cosh(f*x + e)^3 - 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) - 15*(9*cosh(f*x + e)*e^(f*x + e)*sinh(f
*x + e)^8 + e^(f*x + e)*sinh(f*x + e)^9 + 4*(9*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^7 + 28*(3*cosh(f
*x + e)^3 + cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^6 + 6*(21*cosh(f*x + e)^4 + 14*cosh(f*x + e)^2 + 1)*e^(f*
x + e)*sinh(f*x + e)^5 + 2*(63*cosh(f*x + e)^5 + 70*cosh(f*x + e)^3 + 15*cosh(f*x + e))*e^(f*x + e)*sinh(f*x +
e)^4 + 4*(21*cosh(f*x + e)^6 + 35*cosh(f*x + e)^4 + 15*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^3 + 12*
(3*cosh(f*x + e)^7 + 7*cosh(f*x + e)^5 + 5*cosh(f*x + e)^3 + cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + (9*c
osh(f*x + e)^8 + 28*cosh(f*x + e)^6 + 30*cosh(f*x + e)^4 + 12*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e) +
(cosh(f*x + e)^9 + 4*cosh(f*x + e)^7 + 6*cosh(f*x + e)^5 + 4*cosh(f*x + e)^3 + cosh(f*x + e))*e^(f*x + e))*ar
ctan(cosh(f*x + e) + sinh(f*x + e)) + (2*cosh(f*x + e)^10 + 15*cosh(f*x + e)^8 + 5*cosh(f*x + e)^6 - 5*cosh(f*
x + e)^4 - 15*cosh(f*x + e)^2 - 2)*e^(f*x + e))*sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)
/(f*cosh(f*x + e)^9 + (f*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^9 + 9*(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*
x + e))*sinh(f*x + e)^8 + 4*f*cosh(f*x + e)^7 + 4*(9*f*cosh(f*x + e)^2 + (9*f*cosh(f*x + e)^2 + f)*e^(2*f*x +
2*e) + f)*sinh(f*x + e)^7 + 28*(3*f*cosh(f*x + e)^3 + f*cosh(f*x + e) + (3*f*cosh(f*x + e)^3 + f*cosh(f*x + e)
)*e^(2*f*x + 2*e))*sinh(f*x + e)^6 + 6*f*cosh(f*x + e)^5 + 6*(21*f*cosh(f*x + e)^4 + 14*f*cosh(f*x + e)^2 + (2
1*f*cosh(f*x + e)^4 + 14*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^5 + 2*(63*f*cosh(f*x + e)^5
+ 70*f*cosh(f*x + e)^3 + 15*f*cosh(f*x + e) + (63*f*cosh(f*x + e)^5 + 70*f*cosh(f*x + e)^3 + 15*f*cosh(f*x +
e))*e^(2*f*x + 2*e))*sinh(f*x + e)^4 + 4*f*cosh(f*x + e)^3 + 4*(21*f*cosh(f*x + e)^6 + 35*f*cosh(f*x + e)^4 +
15*f*cosh(f*x + e)^2 + (21*f*cosh(f*x + e)^6 + 35*f*cosh(f*x + e)^4 + 15*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2*e
) + f)*sinh(f*x + e)^3 + 12*(3*f*cosh(f*x + e)^7 + 7*f*cosh(f*x + e)^5 + 5*f*cosh(f*x + e)^3 + f*cosh(f*x + e)
+ (3*f*cosh(f*x + e)^7 + 7*f*cosh(f*x + e)^5 + 5*f*cosh(f*x + e)^3 + f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f
*x + e)^2 + f*cosh(f*x + e) + (f*cosh(f*x + e)^9 + 4*f*cosh(f*x + e)^7 + 6*f*cosh(f*x + e)^5 + 4*f*cosh(f*x +
e)^3 + f*cosh(f*x + e))*e^(2*f*x + 2*e) + (9*f*cosh(f*x + e)^8 + 28*f*cosh(f*x + e)^6 + 30*f*cosh(f*x + e)^4 +
12*f*cosh(f*x + e)^2 + (9*f*cosh(f*x + e)^8 + 28*f*cosh(f*x + e)^6 + 30*f*cosh(f*x + e)^4 + 12*f*cosh(f*x + e
)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \tanh ^{6}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sinh(f*x+e)**2)**(1/2)*tanh(f*x+e)**6,x)

[Out]

Integral(sqrt(a*(sinh(e + f*x)**2 + 1))*tanh(e + f*x)**6, x)

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Giac [A]
time = 0.45, size = 124, normalized size = 1.03 \begin {gather*} -\frac {{\left (15 \, \pi - \frac {4 \, {\left (9 \, {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{3} + 28 \, e^{\left (f x + e\right )} - 28 \, e^{\left (-f x - e\right )}\right )}}{{\left ({\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{2} + 4\right )}^{2}} + 30 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )} e^{\left (-f x - e\right )}\right ) - 8 \, e^{\left (f x + e\right )} + 8 \, e^{\left (-f x - e\right )}\right )} \sqrt {a}}{16 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)^6,x, algorithm="giac")

[Out]

-1/16*(15*pi - 4*(9*(e^(f*x + e) - e^(-f*x - e))^3 + 28*e^(f*x + e) - 28*e^(-f*x - e))/((e^(f*x + e) - e^(-f*x
- e))^2 + 4)^2 + 30*arctan(1/2*(e^(2*f*x + 2*e) - 1)*e^(-f*x - e)) - 8*e^(f*x + e) + 8*e^(-f*x - e))*sqrt(a)/
f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tanh}\left (e+f\,x\right )}^6\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^6*(a + a*sinh(e + f*x)^2)^(1/2),x)

[Out]

int(tanh(e + f*x)^6*(a + a*sinh(e + f*x)^2)^(1/2), x)

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