Optimal. Leaf size=518 \[ \frac {5 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (c^2 x^2+1\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {3 i b f^4 x \left (c^2 x^2+1\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
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Rubi [A] time = 0.42, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5712, 669, 671, 641, 215, 5819, 627, 43, 5675} \[ \frac {5 i f^4 (1-i c x) \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^4 (1-i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (c^2 x^2+1\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (c^2 x^2+1\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {3 i b f^4 x \left (c^2 x^2+1\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 215
Rule 627
Rule 641
Rule 669
Rule 671
Rule 5675
Rule 5712
Rule 5819
Rubi steps
\begin {align*} \int \frac {(f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{(d+i c d x)^{3/2}} \, dx &=\frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(f-i c f x)^4 \left (a+b \sinh ^{-1}(c x)\right )}{\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 {2 i f^4 (1-i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (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 \left (\frac {15 i f^4}{2 c}+\frac {5 i f^4 (1-i c x)}{2 c}+\frac {2 i f^4 (1-i c x)^3}{c \left (1+c^2 x^2\right )}-\frac {15 f^4 \sinh ^{-1}(c x)}{2 c \sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\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 )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(1-i c x)^3}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {\left (15 b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {\sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\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 )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {(1-i c x)^2}{1+i c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {15 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\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 )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (2 i b f^4 \left (1+c^2 x^2\right )^{3/2}\right ) \int \left (-3+i c x+\frac {4}{1+i c x}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ &=-\frac {3 i b f^4 x \left (1+c^2 x^2\right )^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b c f^4 x^2 \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 b f^4 (1-i c x)^2 \left (1+c^2 x^2\right )^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 b f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\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 )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {15 i f^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {5 i f^4 (1-i c x) \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {15 f^4 \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (a+b \sinh ^{-1}(c x)\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b f^4 \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 3.97, size = 779, normalized size = 1.50 \[ \frac {\frac {4 a f^2 \left (c^2 x^2+7 i c x+24\right ) \sqrt {d+i c d x} \sqrt {f-i c f x}}{d^2 (c x-i)}-\frac {60 a f^{5/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{d^{3/2}}+\frac {b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 \sinh ^{-1}(c x) \left (\sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \left (-8 \sqrt {c^2 x^2+1}+i \sinh \left (2 \sinh ^{-1}(c x)\right )+8\right )+\left (\sinh \left (2 \sinh ^{-1}(c x)\right )+8 i \left (\sqrt {c^2 x^2+1}+1\right )\right ) \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-\left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right ) \left (\cosh \left (2 \sinh ^{-1}(c x)\right )+8 \left (\log \left (c^2 x^2+1\right )+2 i c x+4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )-10 \sinh ^{-1}(c x)^2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{d^2 \sqrt {c^2 x^2+1} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}-\frac {4 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right ) \left (\log \left (c^2 x^2+1\right )+4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\sinh ^{-1}(c x)^2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\sinh ^{-1}(c x) \left (-4 \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-4 i \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{d^2 \sqrt {c^2 x^2+1} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}+\frac {16 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\sinh ^{-1}(c x) \left (-\left (\sqrt {c^2 x^2+1}-2\right ) \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \left (\sqrt {c^2 x^2+1}+2\right ) \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+\left (\sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-i \cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right ) \left (-i \log \left (c^2 x^2+1\right )+c x-4 \tan ^{-1}\left (\coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )-\left (\sinh ^{-1}(c x)^2 \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )}{d^2 \sqrt {c^2 x^2+1} \left (\cosh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+i \sinh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}}{8 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b c^{2} f^{2} x^{2} + 2 i \, b c f^{2} x - b f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a c^{2} f^{2} x^{2} + 2 i \, a c f^{2} x - a f^{2}\right )} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f}}{c^{2} d^{2} x^{2} - 2 i \, c d^{2} x - d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )}{\left (i c d x +d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \frac {1}{2} \, {\left (\frac {c^{2} f^{3} x^{3}}{\sqrt {c^{2} d f x^{2} + d f} d} + \frac {8 i \, c f^{3} x^{2}}{\sqrt {c^{2} d f x^{2} + d f} d} + \frac {17 \, f^{3} x}{\sqrt {c^{2} d f x^{2} + d f} d} - \frac {15 \, f^{3} \operatorname {arsinh}\left (c x\right )}{\sqrt {d f} c d} + \frac {24 i \, f^{3}}{\sqrt {c^{2} d f x^{2} + d f} c d}\right )} a + b \int \frac {{\left (-i \, c f x + f\right )}^{\frac {5}{2}} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (i \, c d x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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