Optimal. Leaf size=427 \[ \frac {\log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {2}}+\frac {2 (-a+i)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(a+i)^{3/4}}+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )-\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )-\frac {2 (-a+i)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(a+i)^{3/4}} \]
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Rubi [A] time = 0.25, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5095, 105, 63, 240, 211, 1165, 628, 1162, 617, 204, 93, 298, 205, 208} \[ \frac {\log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt {-i a-i b x+1}}{\sqrt {i a+i b x+1}}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {2}}+\frac {2 (-a+i)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(a+i)^{3/4}}+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )-\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )-\frac {2 (-a+i)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{(a+i)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 204
Rule 205
Rule 208
Rule 211
Rule 240
Rule 298
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{\frac {3}{2} i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac {(1+i a+i b x)^{3/4}}{x (1-i a-i b x)^{3/4}} \, dx\\ &=-\left ((-1-i a) \int \frac {1}{x (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}} \, dx\right )+(i b) \int \frac {1}{(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}} \, dx\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a-i b x}\right )\right )+(4 (1+i a)) \operatorname {Subst}\left (\int \frac {x^2}{-1-i a-(-1+i a) x^4} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )\right )-\frac {(2 (i-a)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-a}-\sqrt {i+a} x^2} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{\sqrt {i+a}}+\frac {(2 (i-a)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {i-a}+\sqrt {i+a} x^2} \, dx,x,\frac {\sqrt [4]{1+i a+i b x}}{\sqrt [4]{1-i a-i b x}}\right )}{\sqrt {i+a}}\\ &=\frac {2 (i-a)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}-\frac {2 (i-a)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}-2 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )\\ &=\frac {2 (i-a)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}-\frac {2 (i-a)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2}}-\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )\\ &=\frac {2 (i-a)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}-\frac {2 (i-a)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}+\frac {\log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2}}-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )+\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )\\ &=\frac {2 (i-a)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}+\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )-\frac {2 (i-a)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i+a)^{3/4}}+\frac {\log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 122, normalized size = 0.29 \[ 2 \sqrt [4]{-i (a+b x+i)} \left (\frac {2 (a-i) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )}{(a+i) \sqrt [4]{i a+i b x+1}}-2^{3/4} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {1}{2} i (a+b x+i)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.51, size = 690, normalized size = 1.62 \[ \frac {1}{2} \, \sqrt {4 i} \log \left (\frac {1}{2} i \, \sqrt {4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - \frac {1}{2} \, \sqrt {4 i} \log \left (-\frac {1}{2} i \, \sqrt {4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - \frac {1}{2} \, \sqrt {-4 i} \log \left (\frac {1}{2} i \, \sqrt {-4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \frac {1}{2} \, \sqrt {-4 i} \log \left (-\frac {1}{2} i \, \sqrt {-4 i} + \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a^{2} + 2 i \, a - 1\right )} \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {3}{4}} + {\left (a^{2} - 2 i \, a - 1\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a^{2} - 2 i \, a - 1}\right ) + \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a^{2} + 2 i \, a - 1\right )} \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {3}{4}} - {\left (a^{2} - 2 i \, a - 1\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a^{2} - 2 i \, a - 1}\right ) + i \, \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a^{2} - 2 \, a - i\right )} \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {3}{4}} + {\left (a^{2} - 2 i \, a - 1\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a^{2} - 2 i \, a - 1}\right ) - i \, \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a^{2} + 2 \, a + i\right )} \left (-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}\right )^{\frac {3}{4}} + {\left (a^{2} - 2 i \, a - 1\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a^{2} - 2 i \, a - 1}\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 = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )^{\frac {3}{2}}}{x}\, dx \]
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 \[ \int \frac {\left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}}{x}\,{d x} \]
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 \[ \int \frac {{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}}{x} \,d x \]
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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