Optimal. Leaf size=106 \[ \frac {4 b \sqrt {d x}}{3 c}-\frac {2 b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6049, 327, 335,
218, 214, 211} \begin {gather*} \frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {2 b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}+\frac {4 b \sqrt {d x}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 327
Rule 335
Rule 6049
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(2 b c) \int \frac {(d x)^{3/2}}{1-c^2 x^2} \, dx}{3 d}\\ &=\frac {4 b \sqrt {d x}}{3 c}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(2 b d) \int \frac {1}{\sqrt {d x} \left (1-c^2 x^2\right )} \, dx}{3 c}\\ &=\frac {4 b \sqrt {d x}}{3 c}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{1-\frac {c^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{3 c}\\ &=\frac {4 b \sqrt {d x}}{3 c}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {(2 b d) \text {Subst}\left (\int \frac {1}{d-c x^2} \, dx,x,\sqrt {d x}\right )}{3 c}-\frac {(2 b d) \text {Subst}\left (\int \frac {1}{d+c x^2} \, dx,x,\sqrt {d x}\right )}{3 c}\\ &=\frac {4 b \sqrt {d x}}{3 c}-\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}(c x)\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 114, normalized size = 1.08 \begin {gather*} \frac {\sqrt {d x} \left (4 b \sqrt {c} \sqrt {x}+2 a c^{3/2} x^{3/2}-2 b \text {ArcTan}\left (\sqrt {c} \sqrt {x}\right )+2 b c^{3/2} x^{3/2} \tanh ^{-1}(c x)+b \log \left (1-\sqrt {c} \sqrt {x}\right )-b \log \left (1+\sqrt {c} \sqrt {x}\right )\right )}{3 c^{3/2} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 93, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+\frac {2 b \left (d x \right )^{\frac {3}{2}} \arctanh \left (c x \right )}{3}+\frac {4 b d \sqrt {d x}}{3 c}-\frac {2 b \,d^{2} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 c \sqrt {d c}}-\frac {2 b \,d^{2} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 c \sqrt {d c}}}{d}\) | \(93\) |
default | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+\frac {2 b \left (d x \right )^{\frac {3}{2}} \arctanh \left (c x \right )}{3}+\frac {4 b d \sqrt {d x}}{3 c}-\frac {2 b \,d^{2} \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 c \sqrt {d c}}-\frac {2 b \,d^{2} \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{3 c \sqrt {d c}}}{d}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 119, normalized size = 1.12 \begin {gather*} \frac {2 \, \left (d x\right )^{\frac {3}{2}} a + {\left (2 \, \left (d x\right )^{\frac {3}{2}} \operatorname {artanh}\left (c x\right ) - \frac {{\left (\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} c^{2}} - \frac {d^{3} \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d} c^{2}} - \frac {4 \, \sqrt {d x} d^{2}}{c^{2}}\right )} c}{d}\right )} b}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 223, normalized size = 2.10 \begin {gather*} \left [-\frac {2 \, b \sqrt {\frac {d}{c}} \arctan \left (\frac {\sqrt {d x} c \sqrt {\frac {d}{c}}}{d}\right ) - b \sqrt {\frac {d}{c}} \log \left (\frac {c d x - 2 \, \sqrt {d x} c \sqrt {\frac {d}{c}} + d}{c x - 1}\right ) - {\left (b c x \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a c x + 4 \, b\right )} \sqrt {d x}}{3 \, c}, \frac {2 \, b \sqrt {-\frac {d}{c}} \arctan \left (\frac {\sqrt {d x} c \sqrt {-\frac {d}{c}}}{d}\right ) + b \sqrt {-\frac {d}{c}} \log \left (\frac {c d x - 2 \, \sqrt {d x} c \sqrt {-\frac {d}{c}} - d}{c x + 1}\right ) + {\left (b c x \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a c x + 4 \, b\right )} \sqrt {d x}}{3 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 622 vs.
\(2 (100) = 200\).
time = 5.39, size = 636, normalized size = 6.00 \begin {gather*} \frac {2 a \left (d x\right )^{\frac {3}{2}}}{3 d} + \frac {2 b \left (\begin {cases} \frac {c^{2} \left (\frac {d}{c}\right )^{\frac {3}{2}} \sqrt {- \frac {d}{c}} \log {\left (\sqrt {d x} + \sqrt {- \frac {d}{c}} \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {4 c^{2} \sqrt {\frac {d}{c}} \left (d x\right )^{\frac {3}{2}} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} - \frac {c^{2} \sqrt {\frac {d}{c}} \left (- \frac {d}{c}\right )^{\frac {3}{2}} \log {\left (\sqrt {d x} + \sqrt {- \frac {d}{c}} \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {4 c^{2} \left (d x\right )^{\frac {3}{2}} \sqrt {- \frac {d}{c}} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {8 c d \sqrt {\frac {d}{c}} \sqrt {d x}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {4 c d \sqrt {\frac {d}{c}} \sqrt {- \frac {d}{c}} \log {\left (- \sqrt {\frac {d}{c}} + \sqrt {d x} \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} - \frac {6 c d \sqrt {\frac {d}{c}} \sqrt {- \frac {d}{c}} \log {\left (\sqrt {d x} + \sqrt {- \frac {d}{c}} \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {4 c d \sqrt {\frac {d}{c}} \sqrt {- \frac {d}{c}} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {8 c d \sqrt {d x} \sqrt {- \frac {d}{c}}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {4 d^{2} \log {\left (- \sqrt {\frac {d}{c}} + \sqrt {d x} \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} - \frac {4 d^{2} \log {\left (\sqrt {d x} - \sqrt {- \frac {d}{c}} \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} + \frac {4 d^{2} \operatorname {atanh}{\left (c x \right )}}{12 c^{2} \sqrt {\frac {d}{c}} + 12 c^{2} \sqrt {- \frac {d}{c}}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{d} \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 \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\sqrt {d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________