Optimal. Leaf size=211 \[ -\frac {\sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{(1-i a) x}-\frac {3 i b \tan ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{\sqrt [4]{-a+i} (a+i)^{7/4}}+\frac {3 i b \tanh ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{\sqrt [4]{-a+i} (a+i)^{7/4}} \]
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Rubi [A] time = 0.11, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5095, 94, 93, 298, 205, 208} \[ -\frac {\sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{(1-i a) x}-\frac {3 i b \tan ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{\sqrt [4]{-a+i} (a+i)^{7/4}}+\frac {3 i b \tanh ^{-1}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{\sqrt [4]{-a+i} (a+i)^{7/4}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 205
Rule 208
Rule 298
Rule 5095
Rubi steps
\begin {align*} \int \frac {e^{\frac {3}{2} i \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1+i a+i b x)^{3/4}}{x^2 (1-i a-i b x)^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {(3 b) \int \frac {1}{x (1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}} \, dx}{2 (i+a)}\\ &=-\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {(6 b) \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 )}{i+a}\\ &=-\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}+\frac {(3 i b) \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 )}{(i+a)^{3/2}}-\frac {(3 i b) \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 )}{(i+a)^{3/2}}\\ &=-\frac {\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{(1-i a) x}-\frac {3 i b \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 )}{\sqrt [4]{i-a} (i+a)^{7/4}}+\frac {3 i b \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 )}{\sqrt [4]{i-a} (i+a)^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 106, normalized size = 0.50 \[ \frac {\sqrt [4]{-i (a+b x+i)} \left (6 i b x \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )+a^2+a b x+i b x+1\right )}{(a+i)^2 x \sqrt [4]{i a+i b x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.48, size = 711, normalized size = 3.37 \[ \frac {3 \, \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {1}{4}} {\left (-i \, a + 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (8 \, a^{6} + 32 i \, a^{5} - 40 \, a^{4} - 40 \, a^{2} - 32 i \, a + 8\right )} \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 3 \, \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {1}{4}} {\left (i \, a - 1\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - {\left (8 \, a^{6} + 32 i \, a^{5} - 40 \, a^{4} - 40 \, a^{2} - 32 i \, a + 8\right )} \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) - 3 \, \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {1}{4}} {\left (a + i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (8 i \, a^{6} - 32 \, a^{5} - 40 i \, a^{4} - 40 i \, a^{2} + 32 \, a + 8 i\right )} \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 3 \, \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {1}{4}} {\left (a + i\right )} x \log \left (\frac {b^{3} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + {\left (-8 i \, a^{6} + 32 \, a^{5} + 40 i \, a^{4} + 40 i \, a^{2} - 32 \, a - 8 i\right )} \left (-\frac {b^{4}}{16 \, a^{8} + 96 i \, a^{7} - 224 \, a^{6} - 224 i \, a^{5} - 224 i \, a^{3} + 224 \, a^{2} + 96 i \, a - 16}\right )^{\frac {3}{4}}}{b^{3}}\right ) - i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{{\left (a + i\right )} x} \]
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 \[ \int \frac {\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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 \[ \int \frac {\left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}}{x^2} \,d x \]
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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