Optimal. Leaf size=103 \[ \frac {x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b \left (c^2 x^2+1\right )^{3/2} \log \left (c^2 x^2+1\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {5712, 5687, 260} \[ \frac {x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b \left (c^2 x^2+1\right )^{3/2} \log \left (c^2 x^2+1\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5687
Rule 5712
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=\frac {x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 118, normalized size = 1.15 \[ \frac {i \sqrt {f-i c f x} \left (2 a c x-b \sqrt {c^2 x^2+1} \log (d (-1+i c x))-b \sqrt {c^2 x^2+1} \log (d+i c d x)+2 b c x \sinh ^{-1}(c x)\right )}{2 c d f^2 (c x+i) \sqrt {d+i c d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ \frac {4 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 4 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a x + {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac {b c^{2} x^{4} + \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} c d f x^{2} \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} + b x^{2}}{b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b}\right ) - {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac {b c^{2} x^{4} - \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} c d f x^{2} \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} + b x^{2}}{b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b}\right ) - 2 \, {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac {b c^{2} x^{3} + \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} c d f x \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} + b x}{b c^{2} x^{2} + b}\right ) + 2 \, {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} \log \left (\frac {b c^{2} x^{3} - \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} c d f x \sqrt {\frac {b^{2}}{c^{2} d^{3} f^{3}}} + b x}{b c^{2} x^{2} + b}\right ) + 4 \, {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )} {\rm integral}\left (-\frac {\sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b c x}{c^{4} d^{2} f^{2} x^{4} + 2 \, c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}}, x\right )}{4 \, {\left (c^{2} d^{2} f^{2} x^{2} + d^{2} f^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arcsinh \left (c x \right )}{\left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 82, normalized size = 0.80 \[ \frac {b x \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d f x^{2} + d f} d f} + \frac {a x}{\sqrt {c^{2} d f x^{2} + d f} d f} - \frac {b \sqrt {\frac {1}{d f}} \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, c d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \left (- i f \left (c x + i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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