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 = 0.691828, antiderivative size = 368, normalized size of antiderivative = 1., 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 3559
Rule 3596
Rule 3529
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
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*}
Mathematica [A] time = 1.9872, 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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Maple [A] time = 0.059, size = 197, normalized size = 0.5 \begin{align*} -{\frac{7}{4\,f{a}^{3}d \left ( -id+d\tan \left ( fx+e \right ) \right ) ^{3}} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{\frac{49\,i}{12}}}{f{a}^{3} \left ( -id+d\tan \left ( fx+e \right ) \right ) ^{3}} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,d}{2\,f{a}^{3} \left ( -id+d\tan \left ( fx+e \right ) \right ) ^{3}}\sqrt{d\tan \left ( fx+e \right ) }}-{\frac{29}{8\,f{a}^{3}d}\arctan \left ({\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{-id}}}} \right ){\frac{1}{\sqrt{-id}}}}-2\,{\frac{1}{f{a}^{3}d\sqrt{d\tan \left ( fx+e \right ) }}}-{\frac{1}{8\,f{a}^{3}d}\arctan \left ({\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{id}}}} \right ){\frac{1}{\sqrt{id}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.68087, size = 1912, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19203, size = 348, normalized size = 0.95 \begin{align*} -\frac{1}{24} \, d^{4}{\left (\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} d^{\frac{11}{2}} 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} d^{\frac{11}{2}} f{\left (-\frac{i \, d}{\sqrt{d^{2}}} + 1\right )}} + \frac{48}{\sqrt{d \tan \left (f x + e\right )} a^{3} d^{5} 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} d^{5} f}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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