Optimal. Leaf size=259 \[ -\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 d \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rubi [A]
time = 0.34, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {5796, 5838,
5783, 5798, 5772, 267} \begin {gather*} -\frac {2 i a b d x \sqrt {c^2 x^2+1}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 d \left (c^2 x^2+1\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rule 5783
Rule 5796
Rule 5798
Rule 5838
Rubi steps
\begin {align*} \int \frac {\sqrt {d+i c d x} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {f-i c f x}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {(d+i c d x) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {\sqrt {1+c^2 x^2} \int \left (\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}+\frac {i c d x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {\left (d \sqrt {1+c^2 x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (i c d \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i b d \sqrt {1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=-\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i b^2 d \sqrt {1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=-\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (2 i b^2 c d \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=-\frac {2 i a b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i b^2 d \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b^2 d x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 315, normalized size = 1.22 \begin {gather*} \frac {3 i \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-2 a b c x+a^2 \sqrt {1+c^2 x^2}+2 b^2 \sqrt {1+c^2 x^2}\right )-6 i b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (b c x-a \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)+3 b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (a+i b \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)^2+b^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^3+3 a^2 \sqrt {d} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{3 c f \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{2} \sqrt {i c d x +d}}{\sqrt {-i c f x +f}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {i d \left (c x - i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {- i f \left (c x + i\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}}{\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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