3.5.49 \(\int \frac {\tanh (e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\) [449]

Optimal. Leaf size=21 \[ -\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]

[Out]

-1/3/f/(a*cosh(f*x+e)^2)^(3/2)

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Rubi [A]
time = 0.06, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3255, 3284, 16, 32} \begin {gather*} -\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[e + f*x]/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-1/3*1/(f*(a*Cosh[e + f*x]^2)^(3/2))

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 3284

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 21, normalized size = 1.00 \begin {gather*} -\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[e + f*x]/(a + a*Sinh[e + f*x]^2)^(3/2),x]

[Out]

-1/3*1/(f*(a*Cosh[e + f*x]^2)^(3/2))

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Maple [A]
time = 0.62, size = 20, normalized size = 0.95

method	result	size
derivativedivides	\(-\frac {1}{3 \left (a +a \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} f}\)	\(20\)
default	\(-\frac {1}{3 \left (a +a \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} f}\)	\(20\)
risch	\(-\frac {8 \,{\mathrm e}^{2 f x +2 e}}{3 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(f*x+e)/(a+a*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/(a+a*sinh(f*x+e)^2)^(3/2)/f

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (18) = 36\).
time = 0.56, size = 65, normalized size = 3.10 \begin {gather*} -\frac {8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (3 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 3 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

-8/3*e^(-3*f*x - 3*e)/((3*a^(3/2)*e^(-2*f*x - 2*e) + 3*a^(3/2)*e^(-4*f*x - 4*e) + a^(3/2)*e^(-6*f*x - 6*e) + a
^(3/2))*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (17) = 34\).
time = 0.41, size = 608, normalized size = 28.95 \begin {gather*} -\frac {8 \, {\left (\cosh \left (f x + e\right )^{3} e^{\left (f x + e\right )} + 3 \, \cosh \left (f x + e\right )^{2} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + 3 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3}\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{3 \, {\left (a^{2} f \cosh \left (f x + e\right )^{6} + 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{6} + 6 \, {\left (a^{2} f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{5} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} + 3 \, a^{2} f \cosh \left (f x + e\right ) + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} + 3 \, a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{3} + a^{2} f + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} + 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} + 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} f \cosh \left (f x + e\right )^{6} + 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 6 \, {\left (a^{2} f \cosh \left (f x + e\right )^{5} + 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right ) + {\left (a^{2} f \cosh \left (f x + e\right )^{5} + 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

-8/3*(cosh(f*x + e)^3*e^(f*x + e) + 3*cosh(f*x + e)^2*e^(f*x + e)*sinh(f*x + e) + 3*cosh(f*x + e)*e^(f*x + e)*
sinh(f*x + e)^2 + e^(f*x + e)*sinh(f*x + e)^3)*sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/
(a^2*f*cosh(f*x + e)^6 + 3*a^2*f*cosh(f*x + e)^4 + (a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^6 + 6*(a^2*f*
cosh(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e)^5 + 3*a^2*f*cosh(f*x + e)^2 + 3*(5*a^2*f*co
sh(f*x + e)^2 + a^2*f + (5*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^4 + 4*(5*a^2*f*cosh(f
*x + e)^3 + 3*a^2*f*cosh(f*x + e) + (5*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*
x + e)^3 + a^2*f + 3*(5*a^2*f*cosh(f*x + e)^4 + 6*a^2*f*cosh(f*x + e)^2 + a^2*f + (5*a^2*f*cosh(f*x + e)^4 + 6
*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^2 + (a^2*f*cosh(f*x + e)^6 + 3*a^2*f*cosh(f*x +
e)^4 + 3*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2*e) + 6*(a^2*f*cosh(f*x + e)^5 + 2*a^2*f*cosh(f*x + e)^3
+ a^2*f*cosh(f*x + e) + (a^2*f*cosh(f*x + e)^5 + 2*a^2*f*cosh(f*x + e)^3 + a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e
))*sinh(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)/(a+a*sinh(f*x+e)**2)**(3/2),x)

[Out]

Integral(tanh(e + f*x)/(a*(sinh(e + f*x)**2 + 1))**(3/2), x)

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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(tanh(f*x+e)/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [B]
time = 0.88, size = 58, normalized size = 2.76 \begin {gather*} -\frac {16\,{\mathrm {e}}^{4\,e+4\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)/(a + a*sinh(e + f*x)^2)^(3/2),x)

[Out]

-(16*exp(4*e + 4*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(3*a^2*f*(exp(2*e + 2*f*x) + 1)^4)

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