Optimal. Leaf size=141 \[ \frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac {42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac {42 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {49, 42, 235,
233, 202} \begin {gather*} -\frac {42 \sqrt [4]{x^2+1} E\left (\left .\frac {\text {ArcTan}(x)}{2}\right |2\right )}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}+\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac {42 x}{5 \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 42
Rule 49
Rule 202
Rule 233
Rule 235
Rubi steps
\begin {align*} \int \frac {(a-i a x)^{7/4}}{(a+i a x)^{9/4}} \, dx &=\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac {7}{5} \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{5/4}} \, dx\\ &=\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac {21}{5} \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx\\ &=\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac {\left (21 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}+\frac {\left (21 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac {42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac {\left (21 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=\frac {4 i (a-i a x)^{7/4}}{5 a (a+i a x)^{5/4}}+\frac {42 x}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {28 i (a-i a x)^{3/4}}{5 a \sqrt [4]{a+i a x}}-\frac {42 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 70, normalized size = 0.50 \begin {gather*} \frac {i \sqrt [4]{1+i x} (a-i a x)^{11/4} \, _2F_1\left (\frac {9}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2}-\frac {i x}{2}\right )}{11 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.18, size = 101, normalized size = 0.72
method | result | size |
risch | \(-\frac {8 \left (4 x^{2}+i x +3\right )}{5 \left (x -i\right ) \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {21 x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________