Optimal. Leaf size=80 \[ -\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {35 \text {Shi}\left (\tanh ^{-1}(a x)\right )}{64 a}+\frac {63 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )}{64 a}+\frac {35 \text {Shi}\left (5 \tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Shi}\left (7 \tanh ^{-1}(a x)\right )}{64 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6113, 6181,
5556, 3379} \begin {gather*} -\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {35 \text {Shi}\left (\tanh ^{-1}(a x)\right )}{64 a}+\frac {63 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )}{64 a}+\frac {35 \text {Shi}\left (5 \tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Shi}\left (7 \tanh ^{-1}(a x)\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3379
Rule 5556
Rule 6113
Rule 6181
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)^2} \, dx &=-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+(7 a) \int \frac {x}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7 \text {Subst}\left (\int \frac {\cosh ^6(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7 \text {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 x}+\frac {9 \sinh (3 x)}{64 x}+\frac {5 \sinh (5 x)}{64 x}+\frac {\sinh (7 x)}{64 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7 \text {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {35 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {35 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {63 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {35 \text {Shi}\left (\tanh ^{-1}(a x)\right )}{64 a}+\frac {63 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )}{64 a}+\frac {35 \text {Shi}\left (5 \tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Shi}\left (7 \tanh ^{-1}(a x)\right )}{64 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 65, normalized size = 0.81 \begin {gather*} \frac {-\frac {64}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+7 \left (5 \text {Shi}\left (\tanh ^{-1}(a x)\right )+9 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )+5 \text {Shi}\left (5 \tanh ^{-1}(a x)\right )+\text {Shi}\left (7 \tanh ^{-1}(a x)\right )\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs.
\(2(70)=140\).
time = 2.94, size = 232, normalized size = 2.90
method | result | size |
default | \(\frac {7 \arctanh \left (a x \right ) \hyperbolicSineIntegral \left (7 \arctanh \left (a x \right )\right ) a^{2} x^{2}+35 \arctanh \left (a x \right ) \hyperbolicSineIntegral \left (5 \arctanh \left (a x \right )\right ) a^{2} x^{2}+35 \arctanh \left (a x \right ) \hyperbolicSineIntegral \left (\arctanh \left (a x \right )\right ) a^{2} x^{2}+63 \arctanh \left (a x \right ) \hyperbolicSineIntegral \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}-\cosh \left (7 \arctanh \left (a x \right )\right ) a^{2} x^{2}-7 \cosh \left (5 \arctanh \left (a x \right )\right ) a^{2} x^{2}-21 \cosh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}-7 \hyperbolicSineIntegral \left (7 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )-35 \hyperbolicSineIntegral \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )-35 \hyperbolicSineIntegral \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )-63 \hyperbolicSineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+\cosh \left (7 \arctanh \left (a x \right )\right )+7 \cosh \left (5 \arctanh \left (a x \right )\right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cosh \left (3 \arctanh \left (a x \right )\right )}{64 a \arctanh \left (a x \right ) \left (a^{2} x^{2}-1\right )}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [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]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________