3.2.44 \(\int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) [144]

Optimal. Leaf size=54 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {2}{a d \sqrt {a+b \text {sech}(c+d x)}} \]

[Out]

2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-2/a/d/(a+b*sech(d*x+c))^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3970, 53, 65, 213} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {2}{a d \sqrt {a+b \text {sech}(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]

[Out]

(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - 2/(a*d*Sqrt[a + b*Sech[c + d*x]])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 213

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

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x (a+x)^{3/2}} \, dx,x,b \text {sech}(c+d x)\right )}{d}\\ &=-\frac {2}{a d \sqrt {a+b \text {sech}(c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \text {sech}(c+d x)\right )}{a d}\\ &=-\frac {2}{a d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{a d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {2}{a d \sqrt {a+b \text {sech}(c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 79, normalized size = 1.46 \begin {gather*} \frac {2 \left (-1+\frac {\tanh ^{-1}\left (\frac {\sqrt {b+a \cosh (c+d x)}}{\sqrt {a \cosh (c+d x)}}\right ) \sqrt {b+a \cosh (c+d x)}}{\sqrt {a \cosh (c+d x)}}\right )}{a d \sqrt {a+b \text {sech}(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]

[Out]

(2*(-1 + (ArcTanh[Sqrt[b + a*Cosh[c + d*x]]/Sqrt[a*Cosh[c + d*x]]]*Sqrt[b + a*Cosh[c + d*x]])/Sqrt[a*Cosh[c +
d*x]]))/(a*d*Sqrt[a + b*Sech[c + d*x]])

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Maple [A]
time = 1.54, size = 46, normalized size = 0.85

method	result	size
derivativedivides	\(-\frac {\frac {2}{a \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}-\frac {2 \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}}{d}\)	\(46\)
default	\(-\frac {\frac {2}{a \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}-\frac {2 \arctanh \left (\frac {\sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}}{d}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/d*(2/a/(a+b*sech(d*x+c))^(1/2)-2/a^(3/2)*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2)))

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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(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(tanh(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (46) = 92\).
time = 0.82, size = 917, normalized size = 16.98 \begin {gather*} \left [\frac {{\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (-\frac {2 \, a^{2} \cosh \left (d x + c\right )^{4} + 2 \, a^{2} \sinh \left (d x + c\right )^{4} + 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (2 \, a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right ) + {\left (4 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (12 \, a^{2} \cosh \left (d x + c\right )^{2} + 12 \, a b \cosh \left (d x + c\right ) + 4 \, a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, {\left (a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} + b \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right )^{3} + 2 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} + 3 \, b \cosh \left (d x + c\right ) + 2 \, a\right )} \sinh \left (d x + c\right )^{2} + b \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {a} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}} + 2 \, {\left (4 \, a^{2} \cosh \left (d x + c\right )^{3} + 6 \, a b \cosh \left (d x + c\right )^{2} + 2 \, a b + {\left (4 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 4 \, {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a\right )} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{2 \, {\left (a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d \sinh \left (d x + c\right )^{2} + 2 \, a^{2} b d \cosh \left (d x + c\right ) + a^{3} d + 2 \, {\left (a^{3} d \cosh \left (d x + c\right ) + a^{2} b d\right )} \sinh \left (d x + c\right )\right )}}, -\frac {{\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {-a} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + b \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {-a} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a\right )} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d \sinh \left (d x + c\right )^{2} + 2 \, a^{2} b d \cosh \left (d x + c\right ) + a^{3} d + 2 \, {\left (a^{3} d \cosh \left (d x + c\right ) + a^{2} b d\right )} \sinh \left (d x + c\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[1/2*((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*
sqrt(a)*log(-(2*a^2*cosh(d*x + c)^4 + 2*a^2*sinh(d*x + c)^4 + 4*a*b*cosh(d*x + c)^3 + 4*(2*a^2*cosh(d*x + c) +
a*b)*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + (4*a^2 + b^2)*cosh(d*x + c)^2 + (12*a^2*cosh(d*x + c)^2 + 12*a*b
*cosh(d*x + c) + 4*a^2 + b^2)*sinh(d*x + c)^2 + 2*a^2 + 2*(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 + b*cosh(d*x
+ c)^3 + (4*a*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 + 3*b*cosh(d*x +
c) + 2*a)*sinh(d*x + c)^2 + b*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 4*a*cosh(d*x + c)
+ b)*sinh(d*x + c) + a)*sqrt(a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 2*(4*a^2*cosh(d*x + c)^3 + 6*a*b*c
osh(d*x + c)^2 + 2*a*b + (4*a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d
*x + c) + sinh(d*x + c)^2)) - 4*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a)*
sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)))/(a^3*d*cosh(d*x + c)^2 + a^3*d*sinh(d*x + c)^2 + 2*a^2*b*d*cosh(d*x
+ c) + a^3*d + 2*(a^3*d*cosh(d*x + c) + a^2*b*d)*sinh(d*x + c)), -((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2
*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*sqrt(-a)*arctan((a*cosh(d*x + c)^2 + a*sinh(d*x
+ c)^2 + b*cosh(d*x + c) + (2*a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*sqrt(-a)*sqrt((a*cosh(d*x + c) + b)/cosh
(d*x + c))/(a^2*cosh(d*x + c)^2 + a^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + a^2 + 2*(a^2*cosh(d*x + c) + a*b
)*sinh(d*x + c))) + 2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a)*sqrt((a*co
sh(d*x + c) + b)/cosh(d*x + c)))/(a^3*d*cosh(d*x + c)^2 + a^3*d*sinh(d*x + c)^2 + 2*a^2*b*d*cosh(d*x + c) + a^
3*d + 2*(a^3*d*cosh(d*x + c) + a^2*b*d)*sinh(d*x + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c))**(3/2),x)

[Out]

Integral(tanh(c + d*x)/(a + b*sech(c + d*x))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(tanh(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)

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Mupad [B]
time = 1.77, size = 50, normalized size = 0.93 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}}{\sqrt {a}}\right )}{a^{3/2}\,d}-\frac {2}{a\,d\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)/(a + b/cosh(c + d*x))^(3/2),x)

[Out]

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

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