Optimal. Leaf size=127 \[ \frac {\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {8 \sqrt {a+i a \tan (c+d x)}}{5 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 (a+i a \tan (c+d x))^{3/2}}{15 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3641, 3673,
3608, 3561, 212} \begin {gather*} \frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 (a+i a \tan (c+d x))^{3/2}}{15 a d}-\frac {8 \sqrt {a+i a \tan (c+d x)}}{5 d}+\frac {\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 3561
Rule 3608
Rule 3641
Rule 3673
Rubi steps
\begin {align*} \int \tan ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx &=\frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 \int \tan (c+d x) \left (2 a+\frac {1}{2} i a \tan (c+d x)\right ) \sqrt {a+i a \tan (c+d x)} \, dx}{5 a}\\ &=\frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 (a+i a \tan (c+d x))^{3/2}}{15 a d}-\frac {2 \int \sqrt {a+i a \tan (c+d x)} \left (-\frac {i a}{2}+2 a \tan (c+d x)\right ) \, dx}{5 a}\\ &=-\frac {8 \sqrt {a+i a \tan (c+d x)}}{5 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 (a+i a \tan (c+d x))^{3/2}}{15 a d}+i \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=-\frac {8 \sqrt {a+i a \tan (c+d x)}}{5 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 (a+i a \tan (c+d x))^{3/2}}{15 a d}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}\\ &=\frac {\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {8 \sqrt {a+i a \tan (c+d x)}}{5 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{5 d}-\frac {2 (a+i a \tan (c+d x))^{3/2}}{15 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.80, size = 95, normalized size = 0.75 \begin {gather*} \frac {\sec ^2(c+d x) \left (-10+\frac {30 \sinh ^{-1}\left (e^{i (c+d x)}\right ) \cos ^3(c+d x)}{\sqrt {1+e^{2 i (c+d x)}}}-16 \cos (2 (c+d x))-i \sin (2 (c+d x))\right ) \sqrt {a+i a \tan (c+d x)}}{15 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 92, normalized size = 0.72
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-\frac {a^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2}\right )}{d \,a^{2}}\) | \(92\) |
default | \(-\frac {2 \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{2} \sqrt {a +i a \tan \left (d x +c \right )}-\frac {a^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2}\right )}{d \,a^{2}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 120, normalized size = 0.94 \begin {gather*} -\frac {15 \, \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 ) + 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 20 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 60 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4}}{30 \, a^{4} 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. 285 vs. \(2 (98) = 196\).
time = 0.46, size = 285, normalized size = 2.24 \begin {gather*} \frac {15 \, \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}{d^{2}}} \log \left (4 \, {\left ({\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {a}{d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 15 \, \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}{d^{2}}} \log \left (-4 \, {\left ({\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {a}{d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - 4 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (17 \, e^{\left (5 i \, d x + 5 i \, c\right )} + 20 \, e^{\left (3 i \, d x + 3 i \, c\right )} + 15 \, e^{\left (i \, d x + i \, c\right )}\right )}}{30 \, {\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 \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \tan ^{3}{\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 = 4.15, size = 100, normalized size = 0.79 \begin {gather*} -\frac {2\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{d}+\frac {2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{3\,a\,d}-\frac {2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{5\,a^2\,d}-\frac {\sqrt {2}\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,1{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________