Optimal. Leaf size=326 \[ \frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{3/2} f}+\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}} \]
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Rubi [A] time = 0.52, 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 = {3559, 3596, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{16}+\frac {21 i}{16}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{3/2} f}+\frac {\left (\frac {25}{32}-\frac {21 i}{32}\right ) \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^2 d^{3/2} f}-\frac {25}{8 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {7}{8 a^2 d f (1+i \tan (e+f x)) \sqrt {d \tan (e+f x)}}+\frac {1}{4 d f (a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3529
Rule 3534
Rule 3559
Rule 3596
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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 1.19, size = 231, normalized size = 0.71 \[ \frac {\sec ^3(e+f x) \left (43 i \sin (e+f x)+43 i \sin (3 (e+f x))+23 \cos (e+f x)+41 \cos (3 (e+f x))+(21-25 i) \sqrt {\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))+(-21-25 i) \sin ^{\frac {3}{2}}(2 (e+f x)) \log \left (\sin (e+f x)+\sqrt {\sin (2 (e+f x))}+\cos (e+f x)\right )-(25-21 i) \sqrt {\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left (\sin (e+f x)+\sqrt {\sin (2 (e+f x))}+\cos (e+f x)\right )\right )}{32 a^2 d f (\tan (e+f x)-i)^2 \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 694, normalized size = 2.13 \[ -\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 ({\left ({\left (8 i \, a^{2} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + 8 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}}} - 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 ({\left ({\left (-8 i \, a^{2} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 8 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}}} - 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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.84, size = 221, normalized size = 0.68 \[ -\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {16 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{8 i \, \sqrt {2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{2} \sqrt {d} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} - \frac {23 i \, \sqrt {2} \arctan \left (-\frac {16 i \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{8 i \, \sqrt {2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{2} \sqrt {d} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {16}{\sqrt {d \tan \left (f x + e\right )} a^{2} f} + \frac {9 \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) - 11 i \, \sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2} a^{2} f}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 163, normalized size = 0.50 \[ -\frac {9 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{8 f \,a^{2} d \left (d \tan \left (f x +e \right )-i d \right )^{2}}+\frac {11 i \sqrt {d \tan \left (f x +e \right )}}{8 f \,a^{2} \left (d \tan \left (f x +e \right )-i d \right )^{2}}-\frac {23 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{8 f \,a^{2} d \sqrt {-i d}}-\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 f \,a^{2} d \sqrt {i d}}-\frac {2}{a^{2} d f \sqrt {d \tan \left (f x +e \right )}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ 2\,\mathrm {atanh}\left (8\,a^2\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {1{}\mathrm {i}}{64\,a^4\,d^3\,f^2}}\right )\,\sqrt {\frac {1{}\mathrm {i}}{64\,a^4\,d^3\,f^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^2\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {529{}\mathrm {i}}{256\,a^4\,d^3\,f^2}}}{23}\right )\,\sqrt {-\frac {529{}\mathrm {i}}{256\,a^4\,d^3\,f^2}}-\frac {\frac {2\,d}{a^2\,f}-\frac {25\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2}{8\,a^2\,f}+\frac {d\,\mathrm {tan}\left (e+f\,x\right )\,43{}\mathrm {i}}{8\,a^2\,f}}{d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}-{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}+d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,2{}\mathrm {i}} \]
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 \[ - \frac {\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )} - 2 i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )} - \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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