Optimal. Leaf size=41 \[ \frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6115, 3393,
3382} \begin {gather*} \frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3382
Rule 3393
Rule 6115
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^5(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 x}+\frac {5 \cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {5 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {5 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=\frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 31, normalized size = 0.76 \begin {gather*} \frac {10 \text {Chi}\left (\tanh ^{-1}(a x)\right )+5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )+\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.89, size = 30, normalized size = 0.73
method | result | size |
default | \(\frac {10 \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right )+5 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right )+\hyperbolicCosineIntegral \left (5 \arctanh \left (a x \right )\right )}{16 a}\) | \(30\) |
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 {7}{2}} \operatorname {atanh}{\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 = -0.02 \begin {gather*} \int \frac {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________