Optimal. Leaf size=344 \[ \frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {10 b x^{3/2} \text {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {15 i b x \text {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {15 b \sqrt {x} \text {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac {15 i b \text {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.30, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3832, 3813,
2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {15 i b \text {Li}_6\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 d^6 \left (a^2+b^2\right )}-\frac {15 b \sqrt {x} \text {Li}_5\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^5 \left (a^2+b^2\right )}+\frac {15 i b x \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^4 \left (a^2+b^2\right )}+\frac {10 b x^{3/2} \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^3 \left (a^2+b^2\right )}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d^2 \left (a^2+b^2\right )}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{d \left (a^2+b^2\right )}+\frac {x^3}{3 (a+i b)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3813
Rule 3832
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \tan \left (c+d \sqrt {x}\right )} \, dx &=2 \text {Subst}\left (\int \frac {x^5}{a+b \tan (c+d x)} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^3}{3 (a+i b)}+(4 i b) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^5}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {(10 b) \text {Subst}\left (\int x^4 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {(20 i b) \text {Subst}\left (\int x^3 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d^2}\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {10 b x^{3/2} \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}-\frac {(30 b) \text {Subst}\left (\int x^2 \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d^3}\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {10 b x^{3/2} \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {15 i b x \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {(30 i b) \text {Subst}\left (\int x \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d^4}\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {10 b x^{3/2} \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {15 i b x \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {15 b \sqrt {x} \text {Li}_5\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}+\frac {(15 b) \text {Subst}\left (\int \text {Li}_5\left (-\frac {\left (a^2+b^2\right ) e^{2 i (c+d x)}}{(a+i b)^2}\right ) \, dx,x,\sqrt {x}\right )}{\left (a^2+b^2\right ) d^5}\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {10 b x^{3/2} \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {15 i b x \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {15 b \sqrt {x} \text {Li}_5\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac {(15 i b) \text {Subst}\left (\int \frac {\text {Li}_5\left (-\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{2 \left (a^2+b^2\right ) d^6}\\ &=\frac {x^3}{3 (a+i b)}+\frac {2 b x^{5/2} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {5 i b x^2 \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {10 b x^{3/2} \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {15 i b x \text {Li}_4\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {15 b \sqrt {x} \text {Li}_5\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac {15 i b \text {Li}_6\left (-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt {x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.89, size = 303, normalized size = 0.88 \begin {gather*} \frac {2 a d^6 x^3-2 i b d^6 x^3+12 b d^5 x^{5/2} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )-30 i b d^4 x^2 \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )+60 b d^3 x^{3/2} \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )+90 i b d^2 x \text {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )-90 b d \sqrt {x} \text {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )-45 i b \text {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt {x}\right )}}{a+i b}\right )}{6 \left (a^2+b^2\right ) d^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{a +b \tan \left (c +d \sqrt {x}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 813 vs. \(2 (289) = 578\).
time = 0.68, size = 813, normalized size = 2.36 \begin {gather*} -\frac {15 \, {\left (\frac {2 \, {\left (d \sqrt {x} + c\right )} a}{a^{2} + b^{2}} + \frac {2 \, b \log \left (b \tan \left (d \sqrt {x} + c\right ) + a\right )}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (d \sqrt {x} + c\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} c^{5} - \frac {5 \, {\left (d \sqrt {x} + c\right )}^{6} {\left (a - i \, b\right )} - 30 \, {\left (d \sqrt {x} + c\right )}^{5} {\left (a - i \, b\right )} c + 75 \, {\left (d \sqrt {x} + c\right )}^{4} {\left (a - i \, b\right )} c^{2} - 100 \, {\left (d \sqrt {x} + c\right )}^{3} {\left (a - i \, b\right )} c^{3} + 75 \, {\left (d \sqrt {x} + c\right )}^{2} {\left (a - i \, b\right )} c^{4} - 2 \, {\left (48 i \, {\left (d \sqrt {x} + c\right )}^{5} b - 150 i \, {\left (d \sqrt {x} + c\right )}^{4} b c + 200 i \, {\left (d \sqrt {x} + c\right )}^{3} b c^{2} - 150 i \, {\left (d \sqrt {x} + c\right )}^{2} b c^{3} + 75 i \, {\left (d \sqrt {x} + c\right )} b c^{4}\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}\right ) - 15 \, {\left (16 i \, {\left (d \sqrt {x} + c\right )}^{4} b - 40 i \, {\left (d \sqrt {x} + c\right )}^{3} b c + 40 i \, {\left (d \sqrt {x} + c\right )}^{2} b c^{2} - 20 i \, {\left (d \sqrt {x} + c\right )} b c^{3} + 5 i \, b c^{4}\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}\right ) + {\left (48 \, {\left (d \sqrt {x} + c\right )}^{5} b - 150 \, {\left (d \sqrt {x} + c\right )}^{4} b c + 200 \, {\left (d \sqrt {x} + c\right )}^{3} b c^{2} - 150 \, {\left (d \sqrt {x} + c\right )}^{2} b c^{3} + 75 \, {\left (d \sqrt {x} + c\right )} b c^{4}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 4 \, a b \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, d \sqrt {x} + 2 \, c\right )}{a^{2} + b^{2}}\right ) - 360 i \, b {\rm Li}_{6}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}) - 90 \, {\left (8 \, {\left (d \sqrt {x} + c\right )} b - 5 \, b c\right )} {\rm Li}_{5}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}) - 60 \, {\left (-12 i \, {\left (d \sqrt {x} + c\right )}^{2} b + 15 i \, {\left (d \sqrt {x} + c\right )} b c - 5 i \, b c^{2}\right )} {\rm Li}_{4}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b}) + 30 \, {\left (16 \, {\left (d \sqrt {x} + c\right )}^{3} b - 30 \, {\left (d \sqrt {x} + c\right )}^{2} b c + 20 \, {\left (d \sqrt {x} + c\right )} b c^{2} - 5 \, b c^{3}\right )} {\rm Li}_{3}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, d \sqrt {x} + 2 i \, c\right )}}{-i \, a + b})}{a^{2} + b^{2}}}{15 \, d^{6}} \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 {x^{2}}{a + b \tan {\left (c + d \sqrt {x} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________