Optimal. Leaf size=436 \[ \frac {f^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i b^2 f^2 \left (c^2 x^2+1\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 f^2 x \left (c^2 x^2+1\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 f^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rubi [A] time = 0.63, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {5712, 5831, 3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac {f^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {f^2 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i b^2 f^2 \left (c^2 x^2+1\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 f^2 x \left (c^2 x^2+1\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 f^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3317
Rule 5712
Rule 5831
Rubi steps
\begin {align*} \int \frac {(f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+i c d x}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {(f-i c f x)^2 \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} \operatorname {Subst}\left (\int (a+b x)^2 (c f-i c f \sinh (x))^2 \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {\sqrt {1+c^2 x^2} \operatorname {Subst}\left (\int \left (c^2 f^2 (a+b x)^2-2 i c^2 f^2 (a+b x)^2 \sinh (x)-c^2 f^2 (a+b x)^2 \sinh ^2(x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {f^2 \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 f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=\frac {b c f^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \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 {f^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^2 \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 (f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (4 i b f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (b^2 f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=-\frac {b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \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 {f^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (4 i b^2 f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (b^2 f^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\sinh ^{-1}(c x)\right )}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ &=-\frac {4 i b^2 f^2 \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 f^2 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 f^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {4 i b f^2 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c f^2 x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i f^2 \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 {f^2 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}\\ \end {align*}
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Mathematica [A] time = 2.26, size = 532, normalized size = 1.22 \[ \frac {12 a^2 \sqrt {d} f^{3/2} \sqrt {c^2 x^2+1} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-16 i a^2 f \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}-4 a^2 c f x \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+2 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^2 \left (6 a-8 i b \sqrt {c^2 x^2+1}-b \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+2 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x) \left (b \cosh \left (2 \sinh ^{-1}(c x)\right )+4 i \left (4 b c x+a (-4+i c x) \sqrt {c^2 x^2+1}\right )\right )+32 i a b c f x \sqrt {d+i c d x} \sqrt {f-i c f x}+2 a b f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-32 i b^2 f \sqrt {c^2 x^2+1} \sqrt {d+i c d x} \sqrt {f-i c f x}+4 b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh ^{-1}(c x)^3-b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )}{8 c d \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} c f x + i \, b^{2} f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (a b c f x + i \, a b f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a^{2} c f x + i \, a^{2} f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{c d x - i \, d}, x\right ) \]
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 \[ \int \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {i \, c d x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}{\sqrt {i c d x +d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \]
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 \[ \int \frac {\left (- i f \left (c x + i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {i d \left (c x - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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