Optimal. Leaf size=174 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}-\frac {10 i \sqrt {a+i a \tan (c+d x)}}{a^2 d}+\frac {7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3639, 3676,
3673, 3608, 3561, 212} \begin {gather*} -\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}-\frac {10 i \sqrt {a+i a \tan (c+d x)}}{a^2 d}-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3561
Rule 3608
Rule 3639
Rule 3673
Rule 3676
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}-\frac {\int \frac {\tan ^2(c+d x) \left (-3 a+\frac {9}{2} i a \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2}\\ &=-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \tan (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-15 i a^2-\frac {63}{4} a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}+\frac {7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}+\frac {\int \sqrt {a+i a \tan (c+d x)} \left (\frac {63 a^2}{4}-15 i a^2 \tan (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}-\frac {10 i \sqrt {a+i a \tan (c+d x)}}{a^2 d}+\frac {7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}+\frac {\int \sqrt {a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}-\frac {10 i \sqrt {a+i a \tan (c+d x)}}{a^2 d}+\frac {7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}-\frac {i \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2 a d}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tan ^3(c+d x)}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {5 i \tan ^2(c+d x)}{2 a d \sqrt {a+i a \tan (c+d x)}}-\frac {10 i \sqrt {a+i a \tan (c+d x)}}{a^2 d}+\frac {7 i (a+i a \tan (c+d x))^{3/2}}{2 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 1.35, size = 165, normalized size = 0.95 \begin {gather*} -\frac {i e^{-4 i (c+d x)} \left (\sqrt {1+e^{2 i (c+d x)}} \left (-1+18 e^{2 i (c+d x)}+87 e^{4 i (c+d x)}+52 e^{6 i (c+d x)}\right )+3 e^{3 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^2 \sinh ^{-1}\left (e^{i (c+d x)}\right )\right ) \sec ^2(c+d x)}{24 a d \sqrt {1+e^{2 i (c+d x)}} \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 113, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 a \sqrt {a +i a \tan \left (d x +c \right )}-\frac {7 a^{2}}{4 \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {a^{3}}{6 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {a^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8}\right )}{d \,a^{3}}\) | \(113\) |
default | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 a \sqrt {a +i a \tan \left (d x +c \right )}-\frac {7 a^{2}}{4 \sqrt {a +i a \tan \left (d x +c \right )}}+\frac {a^{3}}{6 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {a^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8}\right )}{d \,a^{3}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 139, normalized size = 0.80 \begin {gather*} \frac {i \, {\left (3 \, \sqrt {2} a^{\frac {7}{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 ) + 16 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} - 96 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{3} - \frac {4 \, {\left (21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} - 2 \, a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\right )}}{24 \, a^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 336 vs. \(2 (131) = 262\).
time = 0.43, size = 336, normalized size = 1.93 \begin {gather*} -\frac {3 \, \sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + i \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - i \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) - \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-52 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 87 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}}{12 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 0.28, size = 129, normalized size = 0.74 \begin {gather*} \frac {\frac {1{}\mathrm {i}}{3\,d}-\frac {\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{2\,a\,d}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}-\frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,4{}\mathrm {i}}{a^2\,d}+\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,2{}\mathrm {i}}{3\,a^3\,d}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-a}}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{3/2}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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