Integrand size = 20, antiderivative size = 83 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {1}{3 a c^2 (c-a c x)^{3/2}}-\frac {1}{2 a c^3 \sqrt {c-a c x}}+\frac {\text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{2 \sqrt {2} a c^{7/2}} \] Output:
-1/3/a/c^2/(-a*c*x+c)^(3/2)-1/2/a/c^3/(-a*c*x+c)^(1/2)+1/4*arctanh(1/2*(-a *c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)/a/c^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{3 a c^2 (c-a c x)^{3/2}} \] Input:
Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(7/2)),x]
Output:
-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (1 - a*x)/2]/(a*c^2*(c - a*c*x)^(3/2 ))
Time = 0.57 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6717, 6680, 35, 61, 61, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)}}{(c-a c x)^{7/2}}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {1-a x}{(a x+1) (c-a c x)^{7/2}}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\int \frac {1}{(a x+1) (c-a c x)^{5/2}}dx}{c}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {\frac {\int \frac {1}{(a x+1) (c-a c x)^{3/2}}dx}{2 c}+\frac {1}{3 a c (c-a c x)^{3/2}}}{c}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {\frac {\frac {\int \frac {1}{(a x+1) \sqrt {c-a c x}}dx}{2 c}+\frac {1}{a c \sqrt {c-a c x}}}{2 c}+\frac {1}{3 a c (c-a c x)^{3/2}}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\frac {\frac {1}{a c \sqrt {c-a c x}}-\frac {\int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a c^2}}{2 c}+\frac {1}{3 a c (c-a c x)^{3/2}}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {\frac {1}{a c \sqrt {c-a c x}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{3/2}}}{2 c}+\frac {1}{3 a c (c-a c x)^{3/2}}}{c}\) |
Input:
Int[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(7/2)),x]
Output:
-((1/(3*a*c*(c - a*c*x)^(3/2)) + (1/(a*c*Sqrt[c - a*c*x]) - ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(Sqrt[2]*a*c^(3/2)))/(2*c))/c)
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
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] - Simp[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] && 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 c^{\frac {5}{2}}}+\frac {1}{4 c^{2} \sqrt {-a c x +c}}+\frac {1}{6 c \left (-a c x +c \right )^{\frac {3}{2}}}\right )}{c a}\) | \(64\) |
default | \(\frac {\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 c^{\frac {5}{2}}}-\frac {1}{2 c^{2} \sqrt {-a c x +c}}-\frac {1}{3 c \left (-a c x +c \right )^{\frac {3}{2}}}}{a c}\) | \(64\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \sqrt {-c \left (a x -1\right )}\, \left (a x -1\right ) \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\sqrt {c}\, \left (-2 a x +\frac {10}{3}\right )}{4 \sqrt {-c \left (a x -1\right )}\, c^{\frac {7}{2}} \left (a x -1\right ) a}\) | \(75\) |
Input:
int((a*x-1)/(a*x+1)/(-a*c*x+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2/c/a*(-1/8/c^(5/2)*2^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2)) +1/4/c^2/(-a*c*x+c)^(1/2)+1/6/c/(-a*c*x+c)^(3/2))
Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.43 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 4 \, \sqrt {-a c x + c} {\left (3 \, a x - 5\right )}}{24 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}, \frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + 2 \, \sqrt {-a c x + c} {\left (3 \, a x - 5\right )}}{12 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:
integrate((a*x-1)/(a*x+1)/(-a*c*x+c)^(7/2),x, algorithm="fricas")
Output:
[1/24*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt (-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 4*sqrt(-a*c*x + c)*(3*a*x - 5))/( a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4), 1/12*(3*sqrt(2)*(a^2*x^2 - 2*a*x + 1)* sqrt(-c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + 2*sqrt(-a *c*x + c)*(3*a*x - 5))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)]
Time = 2.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {1}{6 c \left (- a c x + c\right )^{\frac {3}{2}}} + \frac {1}{4 c^{2} \sqrt {- a c x + c}} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{8 c^{2} \sqrt {- c}}\right )}{a c} & \text {for}\: a c \neq 0 \\\frac {\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x - 2 \log {\left (a x + 1 \right )} + 1}{a} & \text {otherwise} \end {cases}}{c^{\frac {7}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((a*x-1)/(a*x+1)/(-a*c*x+c)**(7/2),x)
Output:
Piecewise((-2*(1/(6*c*(-a*c*x + c)**(3/2)) + 1/(4*c**2*sqrt(-a*c*x + c)) + sqrt(2)*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(8*c**2*sqrt(-c)))/(a *c), Ne(a*c, 0)), (Piecewise((-x, Eq(a, 0)), ((a*x - 2*log(a*x + 1) + 1)/a , True))/c**(7/2), True))
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {5}{2}}} - \frac {4 \, {\left (3 \, a c x - 5 \, c\right )}}{{\left (-a c x + c\right )}^{\frac {3}{2}} c^{2}}}{24 \, a c} \] Input:
integrate((a*x-1)/(a*x+1)/(-a*c*x+c)^(7/2),x, algorithm="maxima")
Output:
-1/24*(3*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c ) + sqrt(-a*c*x + c)))/c^(5/2) - 4*(3*a*c*x - 5*c)/((-a*c*x + c)^(3/2)*c^2 ))/(a*c)
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{4 \, a \sqrt {-c} c^{3}} - \frac {3 \, a c x - 5 \, c}{6 \, {\left (a c x - c\right )} \sqrt {-a c x + c} a c^{3}} \] Input:
integrate((a*x-1)/(a*x+1)/(-a*c*x+c)^(7/2),x, algorithm="giac")
Output:
-1/4*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)*c^3 ) - 1/6*(3*a*c*x - 5*c)/((a*c*x - c)*sqrt(-a*c*x + c)*a*c^3)
Time = 13.82 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{4\,a\,c^{7/2}}-\frac {\frac {c-a\,c\,x}{2\,c^2}+\frac {1}{3\,c}}{a\,c\,{\left (c-a\,c\,x\right )}^{3/2}} \] Input:
int((a*x - 1)/((c - a*c*x)^(7/2)*(a*x + 1)),x)
Output:
(2^(1/2)*atanh((2^(1/2)*(c - a*c*x)^(1/2))/(2*c^(1/2))))/(4*a*c^(7/2)) - ( (c - a*c*x)/(2*c^2) + 1/(3*c))/(a*c*(c - a*c*x)^(3/2))
Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.54 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\sqrt {c}\, \left (-3 \sqrt {-a x +1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a x +3 \sqrt {-a x +1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )+3 \sqrt {-a x +1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a x -3 \sqrt {-a x +1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )-12 a x +20\right )}{24 \sqrt {-a x +1}\, a \,c^{4} \left (a x -1\right )} \] Input:
int((a*x-1)/(a*x+1)/(-a*c*x+c)^(7/2),x)
Output:
(sqrt(c)*( - 3*sqrt( - a*x + 1)*sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2))*a* x + 3*sqrt( - a*x + 1)*sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2)) + 3*sqrt( - a*x + 1)*sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2))*a*x - 3*sqrt( - a*x + 1) *sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2)) - 12*a*x + 20))/(24*sqrt( - a*x + 1)*a*c**4*(a*x - 1))