Optimal. Leaf size=326 \[ \frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{3/2} f}+\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2} \]
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Rubi [A]
time = 0.34, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3640, 3677,
3610, 3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{3/2} f}+\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (a+i a \tan (e+f x))^2 \sqrt {d \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 3610
Rule 3615
Rule 3640
Rule 3677
Rubi steps
\begin {align*} \int \frac {1}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2} \, dx &=\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {\int \frac {\frac {9 a d}{2}-\frac {5}{2} i a d \tan (e+f x)}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))} \, dx}{4 a^2 d}\\ &=\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {\int \frac {\frac {25 a^2 d^2}{2}-\frac {21}{2} i a^2 d^2 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{8 a^4 d^2}\\ &=-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {\int \frac {-\frac {21}{2} i a^2 d^3-\frac {25}{2} a^2 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{8 a^4 d^4}\\ &=-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {\text {Subst}\left (\int \frac {-\frac {21}{2} i a^2 d^4-\frac {25}{2} a^2 d^3 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 a^4 d^4 f}\\ &=-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 d f}+\frac {\left (\frac {25}{16}-\frac {21 i}{16}\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 d f}\\ &=-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\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^2 d^{3/2} f}+-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\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^2 d^{3/2} f}+-\frac {\left (\frac {25}{32}+\frac {21 i}{32}\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^2 d f}+-\frac {\left (\frac {25}{32}+\frac {21 i}{32}\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^2 d f}\\ &=-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{3/2} f}+\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\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^2 d^{3/2} f}+\frac {\left (\frac {25}{16}+\frac {21 i}{16}\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^2 d^{3/2} f}\\ &=\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{3/2} f}+\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 1.16, size = 231, normalized size = 0.71 \begin {gather*} \frac {\sec ^3(e+f x) \left (23 \cos (e+f x)+41 \cos (3 (e+f x))+43 i \sin (e+f x)-(25-21 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))}-(21+25 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))+(21-25 i) \text {ArcSin}(\cos (e+f x)-\sin (e+f x)) \sqrt {\sin (2 (e+f x))} (-i \cos (2 (e+f x))+\sin (2 (e+f x)))+43 i \sin (3 (e+f x))\right )}{32 a^2 d f \sqrt {d \tan (e+f x)} (-i+\tan (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 131, normalized size = 0.40
method | result | size |
derivativedivides | \(\frac {2 d^{3} \left (-\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}-\frac {\frac {-\frac {9 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}+\frac {11 i d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {23 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d^{4}}-\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d^{4} \sqrt {i d}}\right )}{f \,a^{2}}\) | \(131\) |
default | \(\frac {2 d^{3} \left (-\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}-\frac {\frac {-\frac {9 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2}+\frac {11 i d \sqrt {d \tan \left (f x +e \right )}}{2}}{\left (i d \tan \left (f x +e \right )+d \right )^{2}}+\frac {23 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}}{8 d^{4}}-\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 d^{4} \sqrt {i d}}\right )}{f \,a^{2}}\) | \(131\) |
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. 730 vs. \(2 (253) = 506\).
time = 0.40, size = 730, normalized size = 2.24 \begin {gather*} \frac {4 \, {\left (a^{2} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {\frac {i}{16 \, a^{4} d^{3} f^{2}}} \log \left (-2 \, {\left (4 \, {\left (i \, a^{2} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, a^{2} d^{2} 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}} \sqrt {\frac {i}{16 \, a^{4} d^{3} f^{2}}} + i \, d e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) - 4 \, {\left (a^{2} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {\frac {i}{16 \, a^{4} d^{3} f^{2}}} \log \left (-2 \, {\left (4 \, {\left (-i \, a^{2} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{2} d^{2} 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}} \sqrt {\frac {i}{16 \, a^{4} d^{3} f^{2}}} + i \, d e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) + 4 \, {\left (a^{2} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {529 i}{64 \, a^{4} d^{3} f^{2}}} \log \left (\frac {{\left (8 \, {\left (a^{2} d f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d 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}} \sqrt {-\frac {529 i}{64 \, a^{4} d^{3} f^{2}}} + 23\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{2} d f}\right ) - 4 \, {\left (a^{2} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )}\right )} \sqrt {-\frac {529 i}{64 \, a^{4} d^{3} f^{2}}} \log \left (-\frac {{\left (8 \, {\left (a^{2} d f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d 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}} \sqrt {-\frac {529 i}{64 \, a^{4} d^{3} f^{2}}} - 23\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{2} d 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}} {\left (-42 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 33 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 10 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )}}{16 \, {\left (a^{2} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d^{2} f e^{\left (4 i \, f x + 4 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 {\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )} - 2 i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )} - \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 221, normalized size = 0.68 \begin {gather*} -\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {8 \, \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^{2} \sqrt {d} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} - \frac {23 i \, \sqrt {2} \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^{2} \sqrt {d} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {16}{\sqrt {d \tan \left (f x + e\right )} a^{2} f} + \frac {9 \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) - 11 i \, \sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2} a^{2} f}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.96, size = 185, normalized size = 0.57 \begin {gather*} 2\,\mathrm {atanh}\left (8\,a^2\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {1{}\mathrm {i}}{64\,a^4\,d^3\,f^2}}\right )\,\sqrt {\frac {1{}\mathrm {i}}{64\,a^4\,d^3\,f^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^2\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {529{}\mathrm {i}}{256\,a^4\,d^3\,f^2}}}{23}\right )\,\sqrt {-\frac {529{}\mathrm {i}}{256\,a^4\,d^3\,f^2}}-\frac {\frac {2\,d}{a^2\,f}-\frac {25\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2}{8\,a^2\,f}+\frac {d\,\mathrm {tan}\left (e+f\,x\right )\,43{}\mathrm {i}}{8\,a^2\,f}}{d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}-{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}+d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,2{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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