Optimal. Leaf size=364 \[ -\frac {2 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (c^2 x^2+1\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b f^4 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^4 \left (c^2 x^2+1\right )^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^4 \left (c^2 x^2+1\right )^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
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Rubi [A] time = 0.38, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5712, 669, 653, 215, 5819, 627, 43, 31, 5675} \[ -\frac {2 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (c^2 x^2+1\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 i b f^4 \left (c^2 x^2+1\right )^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^4 \left (c^2 x^2+1\right )^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^4 \left (c^2 x^2+1\right )^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 215
Rule 627
Rule 653
Rule 669
Rule 5675
Rule 5712
Rule 5819
Rubi steps
\begin {align*} \int \frac {(f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{5/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{5/2} \int \frac {(f-i c f x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b c \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (\frac {2 i f^4 (1-i c x)^3}{3 c \left (1+c^2 x^2\right )^2}-\frac {2 i f^4 (1-i c x)}{c \left (1+c^2 x^2\right )}+\frac {f^4 \sinh ^{-1}(c x)}{c \sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {(1-i c x)^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1-i c x}{1+c^2 x^2} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {\sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1-i c x}{(1+i c x)^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac {1}{1+i c x} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b f^4 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac {2}{(-i+c x)^2}+\frac {i}{-i+c x}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac {4 i b f^4 \left (1+c^2 x^2\right )^{5/2}}{3 c (i-c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {b f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {2 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {f^4 \left (1+c^2 x^2\right )^{5/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {8 b f^4 \left (1+c^2 x^2\right )^{5/2} \log (i-c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 5.84, size = 706, normalized size = 1.94 \[ \frac {\frac {12 a f^{3/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{d^{5/2}}-\frac {16 a f (2 c x-i) \sqrt {d+i c d x} \sqrt {f-i c f x}}{d^3 (c x-i)^2}-\frac {b f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right ) \left (2 \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (14 \log \left (c^2 x^2+1\right )+\sqrt {c^2 x^2+1} \left (7 \log \left (c^2 x^2+1\right )+\sinh ^{-1}(c x) \left (3 \sinh ^{-1}(c x)-14 i\right )+28 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )+6 \sinh ^{-1}(c x)^2-4 i \sinh ^{-1}(c x)+56 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+4\right )+\cosh \left (\frac {3}{2} \sinh ^{-1}(c x)\right ) \left (7 i \log \left (c^2 x^2+1\right )+\left (-14+3 i \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)-28 \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (84 \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-i \left (21 \log \left (c^2 x^2+1\right )+9 \sinh ^{-1}(c x)^2-6 i \sinh ^{-1}(c x)+8\right )\right )\right )}{d^3 (c x+i) \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )^4}+\frac {2 i b f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right ) \left (2 \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (\frac {1}{2} i \left (\left (\sqrt {c^2 x^2+1}+2\right ) \log \left (c^2 x^2+1\right )+4\right )+\left (\sqrt {c^2 x^2+1}+2\right ) \sinh ^{-1}(c x)+2 \left (\sqrt {c^2 x^2+1}+2\right ) \tan ^{-1}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )-i \cosh \left (\frac {3}{2} \sinh ^{-1}(c x)\right ) \left (-\frac {1}{2} i \log \left (c^2 x^2+1\right )+\sinh ^{-1}(c x)-2 \tan ^{-1}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (\frac {3}{2} \log \left (c^2 x^2+1\right )+3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+4\right )\right )}{d^3 (c x+i) \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )^4}}{12 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b c f x + i \, b f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a c f x + i \, a f\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{c^{3} d^{3} x^{3} - 3 i \, c^{2} d^{3} x^{2} - 3 \, c d^{3} x + i \, d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )}{\left (i c d x +d \right )^{\frac {5}{2}}}\, dx \]
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 \[ -\frac {1}{3} \, a {\left (-\frac {3 i \, {\left (c^{2} d f x^{2} + d f\right )}^{\frac {3}{2}}}{-3 i \, c^{4} d^{4} x^{3} - 9 \, c^{3} d^{4} x^{2} + 9 i \, c^{2} d^{4} x + 3 \, c d^{4}} + \frac {2 i \, \sqrt {c^{2} d f x^{2} + d f} f}{c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}} + \frac {21 i \, \sqrt {c^{2} d f x^{2} + d f} f}{3 i \, c^{2} d^{3} x + 3 \, c d^{3}} - \frac {3 \, f^{2} \operatorname {arsinh}\left (c x\right )}{c d^{3} \sqrt {\frac {f}{d}}}\right )} + b \int \frac {{\left (-i \, c f x + f\right )}^{\frac {3}{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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