3.1037 \(\int \frac {\tanh (c+d x)}{\sqrt {a \sinh ^2(c+d x)}} \, dx\)

Optimal. Leaf size=30 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a \sinh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]

[Out]

arctan((a*sinh(d*x+c)^2)^(1/2)/a^(1/2))/d/a^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3205, 63, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a \sinh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]/Sqrt[a*Sinh[c + d*x]^2],x]

[Out]

ArcTan[Sqrt[a*Sinh[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rule 3205

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 (c+d x)}{\sqrt {a \sinh ^2(c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a x} (1+x)} \, dx,x,\sinh ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1+\frac {x^2}{a}} \, dx,x,\sqrt {a \sinh ^2(c+d x)}\right )}{a d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a \sinh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 31, normalized size = 1.03 \[ \frac {\sinh (c+d x) \tan ^{-1}(\sinh (c+d x))}{d \sqrt {a \sinh ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]/Sqrt[a*Sinh[c + d*x]^2],x]

[Out]

(ArcTan[Sinh[c + d*x]]*Sinh[c + d*x])/(d*Sqrt[a*Sinh[c + d*x]^2])

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 335, normalized size = 11.17 \[ \left [-\frac {\sqrt {-a} \log \left (-\frac {a \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} {\left (\cosh \left (d x + c\right ) e^{\left (d x + c\right )} + e^{\left (d x + c\right )} \sinh \left (d x + c\right )\right )} \sqrt {-a} e^{\left (-d x - c\right )} - {\left (a e^{\left (2 \, d x + 2 \, c\right )} - a\right )} \sinh \left (d x + c\right )^{2} - {\left (a \cosh \left (d x + c\right )^{2} - a\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (a \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - a}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \sinh \left (d x + c\right )^{2} - \cosh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{2} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (\cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 1}\right )}{a d}, \frac {2 \, \sqrt {a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a*sinh(d*x+c)^2)^(1/2),x, algorithm="fricas")

[Out]

[-sqrt(-a)*log(-(a*cosh(d*x + c)^2 + 2*sqrt(a*e^(4*d*x + 4*c) - 2*a*e^(2*d*x + 2*c) + a)*(cosh(d*x + c)*e^(d*x
+ c) + e^(d*x + c)*sinh(d*x + c))*sqrt(-a)*e^(-d*x - c) - (a*e^(2*d*x + 2*c) - a)*sinh(d*x + c)^2 - (a*cosh(d
*x + c)^2 - a)*e^(2*d*x + 2*c) - 2*(a*cosh(d*x + c)*e^(2*d*x + 2*c) - a*cosh(d*x + c))*sinh(d*x + c) - a)/((e^
(2*d*x + 2*c) - 1)*sinh(d*x + c)^2 - cosh(d*x + c)^2 + (cosh(d*x + c)^2 + 1)*e^(2*d*x + 2*c) + 2*(cosh(d*x + c
)*e^(2*d*x + 2*c) - cosh(d*x + c))*sinh(d*x + c) - 1))/(a*d), 2*sqrt(a*e^(4*d*x + 4*c) - 2*a*e^(2*d*x + 2*c) +
a)*arctan(cosh(d*x + c) + sinh(d*x + c))/(a*d*e^(2*d*x + 2*c) - a*d)]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a*sinh(d*x+c)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(exp(c)^3*t_nostep^3-exp(c)*t_nostep)]index.cc index_m operator + Error: Bad Argument Value

________________________________________________________________________________________

maple [C]  time = 0.36, size = 39, normalized size = 1.30 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2} \sqrt {a \left (\sinh ^{2}\left (d x +c \right )\right )}}, \sinh \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)/(a*sinh(d*x+c)^2)^(1/2),x)

[Out]

`int/indef0`(sinh(d*x+c)/cosh(d*x+c)^2/(a*sinh(d*x+c)^2)^(1/2),sinh(d*x+c))/d

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 18, normalized size = 0.60 \[ \frac {2 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{\sqrt {a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a*sinh(d*x+c)^2)^(1/2),x, algorithm="maxima")

[Out]

2*arctan(e^(-d*x - c))/(sqrt(a)*d)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {tanh}\left (c+d\,x\right )}{\sqrt {a\,{\mathrm {sinh}\left (c+d\,x\right )}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)/(a*sinh(c + d*x)^2)^(1/2),x)

[Out]

int(tanh(c + d*x)/(a*sinh(c + d*x)^2)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh {\left (c + d x \right )}}{\sqrt {a \sinh ^{2}{\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a*sinh(d*x+c)**2)**(1/2),x)

[Out]

Integral(tanh(c + d*x)/sqrt(a*sinh(c + d*x)**2), x)

________________________________________________________________________________________