Optimal. Leaf size=398 \[ \frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {c^2 x^2+1}}+\frac {c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {2 b c d \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {3 b c^3 d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}-\frac {b^2 c d \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 c^2 d x \sqrt {c^2 d x^2+d}-\frac {5 b^2 c d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{4 \sqrt {c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.42, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5739, 5682, 5675, 5661, 321, 215, 5726, 5659, 3716, 2190, 2279, 2391, 195} \[ \frac {b^2 c d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {3 b c^3 d x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {c^2 x^2+1}}-\frac {c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {2 b c d \sqrt {c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 c^2 d x \sqrt {c^2 d x^2+d}-\frac {5 b^2 c d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{4 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 195
Rule 215
Rule 321
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5661
Rule 5675
Rule 5682
Rule 5726
Rule 5739
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (3 c^2 d\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{\sqrt {1+c^2 x^2}}\\ &=b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {1}{2} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{2 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}-\frac {\left (4 b c d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c d \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.97, size = 369, normalized size = 0.93 \[ \frac {36 a^2 c d^{3/2} x \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+12 a^2 d \left (c^2 x^2-2\right ) \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+24 a b d \sqrt {c^2 d x^2+d} \left (-2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right )-6 a b c d x \sqrt {c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-8 b^2 d \sqrt {c^2 d x^2+d} \left (\sinh ^{-1}(c x) \left (3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-c x \left (\sinh ^{-1}(c x)+3\right ) \sinh ^{-1}(c x)-6 c x \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )+3 c x \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )\right )+b^2 c d x \sqrt {c^2 d x^2+d} \left (4 \sinh ^{-1}(c x)^3+\left (6 \sinh ^{-1}(c x)^2+3\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )-6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 x \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{2} d x^{2} + a^{2} d + {\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} + a b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.36, size = 954, normalized size = 2.40 \[ -\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c}{4 \sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} x^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) d c}{\sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \arcsinh \left (c x \right )}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} \arcsinh \left (c x \right ) x^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} \arcsinh \left (c x \right )^{2} x}{2 \left (c^{2} x^{2}+1\right )}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) d c}{\sqrt {c^{2} x^{2}+1}}+\frac {3 a^{2} c^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} \arcsinh \left (c x \right )^{2} x^{3}}{2 c^{2} x^{2}+2}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) d c}{\sqrt {c^{2} x^{2}+1}}+\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} d c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) d}{x \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} x^{3}}{4 c^{2} x^{2}+4}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} x}{4 c^{2} x^{2}+4}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{2} \arcsinh \left (c x \right ) x}{c^{2} x^{2}+1}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{4} \arcsinh \left (c x \right ) x^{3}}{c^{2} x^{2}+1}+\frac {3 a^{2} c^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) d c}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} d}{x \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3} d c}{2 \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) d c}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \arcsinh \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c \arcsinh \left (c x \right )}{4 \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________