Optimal. Leaf size=101 \[ \frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3624, 3559,
3561, 212} \begin {gather*} \frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 3559
Rule 3561
Rule 3624
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}-\int (a+i a \tan (c+d x))^{3/2} \, dx\\ &=-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}-(2 a) \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}+\frac {\left (4 i a^2\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=\frac {2 i \sqrt {2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {2 i a \sqrt {a+i a \tan (c+d x)}}{d}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.52, size = 162, normalized size = 1.60 \begin {gather*} \frac {a e^{-\frac {1}{2} i (2 c+3 d x)} \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (-i \cos \left (\frac {d x}{2}\right )+\sin \left (\frac {d x}{2}\right )\right ) \left (-20 \sinh ^{-1}\left (e^{i (c+d x)}\right )+\sqrt {1+e^{2 i (c+d x)}} \sec ^3(c+d x) (5+7 \cos (2 (c+d x))+2 i \sin (2 (c+d x)))\right )}{5 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 76, normalized size = 0.75
method | result | size |
derivativedivides | \(-\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-a^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d a}\) | \(76\) |
default | \(-\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-a^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d a}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 102, normalized size = 1.01 \begin {gather*} -\frac {i \, {\left (5 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} + 10 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4}\right )}}{5 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 315 vs. \(2 (74) = 148\).
time = 0.38, size = 315, normalized size = 3.12 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \log \left (\frac {4 \, {\left (a^{2} e^{\left (i \, d x + i \, c\right )} + {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) - 5 \, \sqrt {2} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \log \left (\frac {4 \, {\left (a^{2} e^{\left (i \, d x + i \, c\right )} + {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) + 2 \, \sqrt {2} {\left (9 i \, a e^{\left (5 i \, d x + 5 i \, c\right )} + 10 i \, a e^{\left (3 i \, d x + 3 i \, c\right )} + 5 i \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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 \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.30, size = 84, normalized size = 0.83 \begin {gather*} -\frac {\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,2{}\mathrm {i}}{5}-\sqrt {2}\,{\left (-a\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,2{}\mathrm {i}}{a\,d}-\frac {a\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________