Optimal. Leaf size=368 \[ \frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^3 d^{3/2} f}+\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {7}{6 d f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}+\frac {5}{12 a d f (a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f (a+i a \tan (e+f x))^3 \sqrt {d \tan (e+f x)}} \]
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Rubi [A] time = 0.69, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^3 d^{3/2} f}+\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {7}{6 d f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {d \tan (e+f x)}}+\frac {5}{12 a d f (a+i a \tan (e+f x))^2 \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f (a+i a \tan (e+f x))^3 \sqrt {d \tan (e+f x)}} \]
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))^3} \, dx &=\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {\int \frac {\frac {13 a d}{2}-\frac {7}{2} i a d \tan (e+f x)}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^2} \, dx}{6 a^2 d}\\ &=\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {\int \frac {31 a^2 d^2-25 i a^2 d^2 \tan (e+f x)}{(d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))} \, dx}{24 a^4 d^2}\\ &=\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\int \frac {90 a^3 d^3-84 i a^3 d^3 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{48 a^6 d^3}\\ &=-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\int \frac {-84 i a^3 d^4-90 a^3 d^4 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{48 a^6 d^5}\\ &=-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {-84 i a^3 d^5-90 a^3 d^4 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{24 a^6 d^5 f}\\ &=-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}+-\frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \operatorname {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^3 d f}+\frac {\left (\frac {15}{8}-\frac {7 i}{4}\right ) \operatorname {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^3 d f}\\ &=-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}+-\frac {\left (\frac {15}{16}-\frac {7 i}{8}\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^3 d^{3/2} f}+-\frac {\left (\frac {15}{16}-\frac {7 i}{8}\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^3 d^{3/2} f}+-\frac {\left (\frac {15}{16}+\frac {7 i}{8}\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^3 d f}+-\frac {\left (\frac {15}{16}+\frac {7 i}{8}\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^3 d f}\\ &=-\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^3 d^{3/2} f}+\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}+-\frac {\left (\frac {15}{8}+\frac {7 i}{4}\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^3 d^{3/2} f}+\frac {\left (\frac {15}{8}+\frac {7 i}{4}\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^3 d^{3/2} f}\\ &=\frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {\left (\frac {15}{8}+\frac {7 i}{4}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^3 d^{3/2} f}+\frac {\left (\frac {15}{16}-\frac {7 i}{8}\right ) \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a^3 d^{3/2} f}-\frac {15}{4 a^3 d f \sqrt {d \tan (e+f x)}}+\frac {1}{6 d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^3}+\frac {5}{12 a d f \sqrt {d \tan (e+f x)} (a+i a \tan (e+f x))^2}+\frac {7}{6 d f \sqrt {d \tan (e+f x)} \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 2.09, size = 234, normalized size = 0.64 \[ \frac {e^{-6 i (e+f x)} \left (9 e^{2 i (e+f x)}+49 e^{4 i (e+f x)}-105 e^{6 i (e+f x)}-146 e^{8 i (e+f x)}-87 e^{6 i (e+f x)} \sqrt {-1+e^{4 i (e+f x)}} \tan ^{-1}\left (\sqrt {-1+e^{4 i (e+f x)}}\right )+6 e^{6 i (e+f x)} \sqrt {-1+e^{2 i (e+f x)}} \sqrt {1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right )+1\right )}{48 a^3 d f \left (1+e^{2 i (e+f x)}\right ) \sqrt {d \tan (e+f x)}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.59, size = 705, normalized size = 1.92 \[ -\frac {12 \, {\left (a^{3} d^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} - a^{3} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \sqrt {\frac {i}{64 \, a^{6} d^{3} f^{2}}} \log \left ({\left ({\left (16 i \, a^{3} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3} 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}{64 \, a^{6} 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 ) - 12 \, {\left (a^{3} d^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} - a^{3} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \sqrt {\frac {i}{64 \, a^{6} d^{3} f^{2}}} \log \left ({\left ({\left (-16 i \, a^{3} d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 16 i \, a^{3} 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}{64 \, a^{6} 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 ) - 12 \, {\left (a^{3} d^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} - a^{3} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \sqrt {-\frac {841 i}{64 \, a^{6} d^{3} f^{2}}} \log \left (\frac {{\left (8 \, {\left (a^{3} d f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} 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 {841 i}{64 \, a^{6} d^{3} f^{2}}} + 29\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{3} d f}\right ) + 12 \, {\left (a^{3} d^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} - a^{3} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \sqrt {-\frac {841 i}{64 \, a^{6} d^{3} f^{2}}} \log \left (-\frac {{\left (8 \, {\left (a^{3} d f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} 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 {841 i}{64 \, a^{6} d^{3} f^{2}}} - 29\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a^{3} 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 (-146 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 105 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 49 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 9 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )}}{48 \, {\left (a^{3} d^{2} f e^{\left (8 i \, f x + 8 i \, e\right )} - a^{3} d^{2} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.39, size = 252, normalized size = 0.68 \[ -\frac {\frac {87 \, \sqrt {2} \arctan \left (\frac {16 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-8 i \, \sqrt {2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{3} \sqrt {d} f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {3 i \, \sqrt {2} \arctan \left (\frac {16 i \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-8 i \, \sqrt {2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a^{3} \sqrt {d} f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {48}{\sqrt {d \tan \left (f x + e\right )} a^{3} f} + \frac {2 \, {\left (21 i \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )^{2} + 49 \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right ) - 30 i \, \sqrt {d \tan \left (f x + e\right )} d^{2}\right )}}{{\left (-i \, d \tan \left (f x + e\right ) - d\right )}^{3} a^{3} f}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 197, normalized size = 0.54 \[ -\frac {7 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{4 f \,a^{3} d \left (d \tan \left (f x +e \right )-i d \right )^{3}}+\frac {49 i \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{12 f \,a^{3} \left (d \tan \left (f x +e \right )-i d \right )^{3}}+\frac {5 d \sqrt {d \tan \left (f x +e \right )}}{2 f \,a^{3} \left (d \tan \left (f x +e \right )-i d \right )^{3}}-\frac {29 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{8 f \,a^{3} d \sqrt {-i d}}-\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{8 f \,a^{3} d \sqrt {i d}}-\frac {2}{a^{3} d f \sqrt {d \tan \left (f x +e \right )}} \]
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 = 6.12, size = 227, normalized size = 0.62 \[ \frac {\frac {15\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{4\,a^3\,f}-\frac {17\,d^2\,\mathrm {tan}\left (e+f\,x\right )}{2\,a^3\,f}+\frac {d^2\,2{}\mathrm {i}}{a^3\,f}-\frac {d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\,121{}\mathrm {i}}{12\,a^3\,f}}{3\,d^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}-{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}+d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,3{}\mathrm {i}-d^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}+2\,\mathrm {atanh}\left (16\,a^3\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {1{}\mathrm {i}}{256\,a^6\,d^3\,f^2}}\right )\,\sqrt {\frac {1{}\mathrm {i}}{256\,a^6\,d^3\,f^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^3\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {841{}\mathrm {i}}{256\,a^6\,d^3\,f^2}}}{29}\right )\,\sqrt {-\frac {841{}\mathrm {i}}{256\,a^6\,d^3\,f^2}} \]
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 \[ \frac {i \int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (e + f x \right )} - 3 i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )} - 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )} + i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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