Optimal. Leaf size=312 \[ \frac {\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))} \]
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Rubi [A]
time = 0.23, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {3631, 3609,
3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a f}+\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a f}+\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3631
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^{7/2}}{a+i a \tan (e+f x)} \, dx &=-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int (d \tan (e+f x))^{3/2} \left (\frac {5 a d^2}{2}-\frac {7}{2} i a d^2 \tan (e+f x)\right ) \, dx}{2 a^2}\\ &=-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int \sqrt {d \tan (e+f x)} \left (\frac {7}{2} i a d^3+\frac {5}{2} a d^3 \tan (e+f x)\right ) \, dx}{2 a^2}\\ &=\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int \frac {-\frac {5 a d^4}{2}+\frac {7}{2} i a d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^2}\\ &=\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+\frac {\text {Subst}\left (\int \frac {-\frac {5 a d^5}{2}+\frac {7}{2} i a d^4 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f}\\ &=\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+-\frac {\left (\left (\frac {5}{4}+\frac {7 i}{4}\right ) d^4\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}+-\frac {\left (\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^4\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}\\ &=\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+\frac {\left (\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {\left (\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+-\frac {\left (\left (\frac {5}{8}-\frac {7 i}{8}\right ) d^4\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}+-\frac {\left (\left (\frac {5}{8}-\frac {7 i}{8}\right ) d^4\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}\\ &=\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}+-\frac {\left (\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}\\ &=\frac {\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{4}-\frac {7 i}{4}\right ) d^{7/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {\left (\frac {5}{8}+\frac {7 i}{8}\right ) d^{7/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {5 d^3 \sqrt {d \tan (e+f x)}}{2 a f}-\frac {7 i d^2 (d \tan (e+f x))^{3/2}}{6 a f}-\frac {d (d \tan (e+f x))^{5/2}}{2 f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 1.91, size = 275, normalized size = 0.88 \begin {gather*} -\frac {d^4 \sec ^3(e+f x) \left (-16 \cos (e+f x)+16 \cos (3 (e+f x))+54 i \sin (e+f x)+(21-15 i) \log \left (\cos (e+f x)+\sin (e+f x)+\sqrt {\sin (2 (e+f x))}\right ) \sqrt {\sin (2 (e+f x))}+(21-15 i) \cos (2 (e+f x)) \log \left (\cos (e+f x)+\sin (e+f x)+\sqrt {\sin (2 (e+f x))}\right ) \sqrt {\sin (2 (e+f x))}+(42+30 i) \text {ArcSin}(\cos (e+f x)-\sin (e+f x)) \cos (e+f x) (\cos (e+f x)+i \sin (e+f x)) \sqrt {\sin (2 (e+f x))}+(15+21 i) \log \left (\cos (e+f x)+\sin (e+f x)+\sqrt {\sin (2 (e+f x))}\right ) \sin ^{\frac {3}{2}}(2 (e+f x))+22 i \sin (3 (e+f x))\right )}{48 a f \sqrt {d \tan (e+f x)} (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 126, normalized size = 0.40
method | result | size |
derivativedivides | \(\frac {2 d^{2} \left (-\frac {i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+d \sqrt {d \tan \left (f x +e \right )}+\frac {d^{2} \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{i d \tan \left (f x +e \right )+d}+\frac {6 i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{\sqrt {-i d}}\right )}{4}+\frac {i d^{2} \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 \sqrt {i d}}\right )}{f a}\) | \(126\) |
default | \(\frac {2 d^{2} \left (-\frac {i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+d \sqrt {d \tan \left (f x +e \right )}+\frac {d^{2} \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{i d \tan \left (f x +e \right )+d}+\frac {6 i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{\sqrt {-i d}}\right )}{4}+\frac {i d^{2} \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 \sqrt {i d}}\right )}{f a}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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.
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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. 646 vs. \(2 (238) = 476\).
time = 0.38, size = 646, normalized size = 2.07 \begin {gather*} \frac {3 \, \sqrt {\frac {9 i \, d^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac {{\left (-3 i \, d^{4} + \sqrt {\frac {9 i \, d^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a f}\right ) - 3 \, \sqrt {\frac {9 i \, d^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac {{\left (-3 i \, d^{4} - \sqrt {\frac {9 i \, d^{7}}{a^{2} f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a f}\right ) - 3 \, \sqrt {-\frac {i \, d^{7}}{4 \, a^{2} f^{2}}} {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (-\frac {2 \, {\left (i \, d^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, \sqrt {-\frac {i \, d^{7}}{4 \, a^{2} f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d^{3}}\right ) + 3 \, \sqrt {-\frac {i \, d^{7}}{4 \, a^{2} f^{2}}} {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (-\frac {2 \, {\left (i \, d^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 \, \sqrt {-\frac {i \, d^{7}}{4 \, a^{2} f^{2}}} {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d^{3}}\right ) + {\left (19 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 38 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 \, d^{3}\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{12 \, {\left (a f e^{\left (4 i \, f x + 4 i \, e\right )} + a f e^{\left (2 i \, f x + 2 i \, e\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*} - \frac {i \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 242, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, d^{3} {\left (\frac {18 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {8 i \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {8 i \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 i \, \sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) - i \, d\right )} a f} + \frac {4 \, {\left (i \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{3} f^{2} \tan \left (f x + e\right ) - 3 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{3} f^{2}\right )}}{a^{3} d^{3} f^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.21, size = 180, normalized size = 0.58 \begin {gather*} \mathrm {atan}\left (\frac {2\,a\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {d^7\,9{}\mathrm {i}}{4\,a^2\,f^2}}}{3\,d^4}\right )\,\sqrt {\frac {d^7\,9{}\mathrm {i}}{4\,a^2\,f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {4\,a\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {d^7\,1{}\mathrm {i}}{16\,a^2\,f^2}}}{d^4}\right )\,\sqrt {-\frac {d^7\,1{}\mathrm {i}}{16\,a^2\,f^2}}\,2{}\mathrm {i}+\frac {2\,d^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{a\,f}-\frac {d^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,2{}\mathrm {i}}{3\,a\,f}+\frac {d^4\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{2\,a\,f\,\left (-d\,\mathrm {tan}\left (e+f\,x\right )+d\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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