Optimal. Leaf size=285 \[ -\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {b \sqrt [4]{c} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} d^{3/2}}-\frac {b \sqrt [4]{c} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} d^{3/2}}-\frac {2 b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{d^{3/2}}+\frac {2 b \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6097, 16, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ -\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {b \sqrt [4]{c} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} d^{3/2}}-\frac {b \sqrt [4]{c} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{\sqrt {2} d^{3/2}}-\frac {2 b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{d^{3/2}}+\frac {2 b \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 204
Rule 205
Rule 208
Rule 297
Rule 298
Rule 300
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {(4 b c) \int \frac {x}{\sqrt {d x} \left (1-c^2 x^4\right )} \, dx}{d}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {(4 b c) \int \frac {\sqrt {d x}}{1-c^2 x^4} \, dx}{d^2}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {(8 b c) \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{d^3}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {(4 b c) \operatorname {Subst}\left (\int \frac {x^2}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )}{d}+\frac {(4 b c) \operatorname {Subst}\left (\int \frac {x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{d}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {\left (2 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{d}-\frac {\left (2 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{d}-\frac {\left (2 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{d}+\frac {\left (2 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{d}\\ &=-\frac {2 b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {2 b \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\left (b \sqrt [4]{c}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}+2 x}{-\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{\sqrt {2} d^{3/2}}+\frac {\left (b \sqrt [4]{c}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}-2 x}{-\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{\sqrt {2} d^{3/2}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{d}\\ &=-\frac {2 b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {2 b \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b \sqrt [4]{c} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} d^{3/2}}-\frac {b \sqrt [4]{c} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} d^{3/2}}+\frac {\left (\sqrt {2} b \sqrt [4]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\left (\sqrt {2} b \sqrt [4]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}\\ &=-\frac {2 b \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {2 b \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b \sqrt [4]{c} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} d^{3/2}}-\frac {b \sqrt [4]{c} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} d^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 268, normalized size = 0.94 \[ -\frac {x \left (4 a+4 b \tanh ^{-1}\left (c x^2\right )+2 b \sqrt [4]{c} \sqrt {x} \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \sqrt [4]{c} \sqrt {x} \log \left (\sqrt [4]{c} \sqrt {x}+1\right )-\sqrt {2} b \sqrt [4]{c} \sqrt {x} \log \left (\sqrt {c} x-\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )+\sqrt {2} b \sqrt [4]{c} \sqrt {x} \log \left (\sqrt {c} x+\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )+2 \sqrt {2} b \sqrt [4]{c} \sqrt {x} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-2 \sqrt {2} b \sqrt [4]{c} \sqrt {x} \tan ^{-1}\left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+1\right )+4 b \sqrt [4]{c} \sqrt {x} \tan ^{-1}\left (\sqrt [4]{c} \sqrt {x}\right )\right )}{2 (d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.83, size = 38, normalized size = 0.13 \[ -\frac {\sqrt {d x} {\left (b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.85, size = 505, normalized size = 1.77 \[ -\frac {\frac {2 \, b \log \left (-\frac {c d^{2} x^{2} + d^{2}}{c d^{2} x^{2} - d^{2}}\right )}{\sqrt {d x}} + \frac {4 \, a}{\sqrt {d x}} - \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} + \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x + \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x - \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} + \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x + \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x - \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c^{2} d^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 272, normalized size = 0.95 \[ -\frac {2 a}{d \sqrt {d x}}-\frac {2 b \arctanh \left (c \,x^{2}\right )}{d \sqrt {d x}}-\frac {2 b \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{2 d \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{d \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{d \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 296, normalized size = 1.04 \[ -\frac {b {\left (\frac {4 \, \operatorname {artanh}\left (c x^{2}\right )}{\sqrt {d x}} - \frac {{\left (d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )} - 2 \, d^{2} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}}\right )}\right )} c}{d^{2}}\right )} + \frac {4 \, a}{\sqrt {d x}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atanh}{\left (c x^{2} \right )}}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________