Optimal. Leaf size=314 \[ \frac{\left (\frac{7}{4}-\frac{5 i}{4}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} a d^{5/2} f}-\frac{\left (\frac{7}{4}-\frac{5 i}{4}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} a d^{5/2} f}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{\left (\frac{7}{8}+\frac{5 i}{8}\right ) \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} a d^{5/2} f}-\frac{\left (\frac{7}{8}+\frac{5 i}{8}\right ) \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} a d^{5/2} f}+\frac{1}{2 d f (a+i a \tan (e+f x)) (d \tan (e+f x))^{3/2}}-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.391818, antiderivative size = 314, normalized size of antiderivative = 1., 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 = {3552, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (\frac{7}{4}-\frac{5 i}{4}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} a d^{5/2} f}-\frac{\left (\frac{7}{4}-\frac{5 i}{4}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} a d^{5/2} f}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{\left (\frac{7}{8}+\frac{5 i}{8}\right ) \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} a d^{5/2} f}-\frac{\left (\frac{7}{8}+\frac{5 i}{8}\right ) \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} a d^{5/2} f}+\frac{1}{2 d f (a+i a \tan (e+f x)) (d \tan (e+f x))^{3/2}}-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3552
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))^{5/2} (a+i a \tan (e+f x))} \, dx &=\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}-\frac{\int \frac{-\frac{7 a d}{2}+\frac{5}{2} i a d \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx}{2 a^2 d}\\ &=-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}-\frac{\int \frac{\frac{5}{2} i a d^2+\frac{7}{2} a d^2 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{2 a^2 d^3}\\ &=-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}-\frac{\int \frac{\frac{7 a d^3}{2}-\frac{5}{2} i a d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{2 a^2 d^5}\\ &=-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}-\frac{\operatorname{Subst}\left (\int \frac{\frac{7 a d^4}{2}-\frac{5}{2} i a d^3 x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{a^2 d^5 f}\\ &=-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}-\frac{\left (\frac{7}{4}-\frac{5 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 d^2 f}-\frac{\left (\frac{7}{4}+\frac{5 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 d^2 f}\\ &=-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}--\frac{\left (\frac{7}{8}+\frac{5 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 d^{5/2} f}--\frac{\left (\frac{7}{8}+\frac{5 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 d^{5/2} f}-\frac{\left (\frac{7}{8}-\frac{5 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 d^2 f}-\frac{\left (\frac{7}{8}-\frac{5 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 d^2 f}\\ &=\frac{\left (\frac{7}{8}+\frac{5 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 d^{5/2} f}-\frac{\left (\frac{7}{8}+\frac{5 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 d^{5/2} f}-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}--\frac{\left (\frac{7}{4}-\frac{5 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 d^{5/2} f}-\frac{\left (\frac{7}{4}-\frac{5 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 d^{5/2} f}\\ &=\frac{\left (\frac{7}{4}-\frac{5 i}{4}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} a d^{5/2} f}-\frac{\left (\frac{7}{4}-\frac{5 i}{4}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} a d^{5/2} f}+\frac{\left (\frac{7}{8}+\frac{5 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 d^{5/2} f}-\frac{\left (\frac{7}{8}+\frac{5 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 d^{5/2} f}-\frac{7}{6 a d f (d \tan (e+f x))^{3/2}}+\frac{5 i}{2 a d^2 f \sqrt{d \tan (e+f x)}}+\frac{1}{2 d f (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 2.22423, size = 273, normalized size = 0.87 \[ \frac{\sec ^3(e+f x) \left (16 \sin (e+f x)+16 \sin (3 (e+f x))+54 i \cos (e+f x)-22 i \cos (3 (e+f x))-(15+21 i) \left (\sin (2 (e+f x))+2 i \sin ^2(e+f x)\right ) \sqrt{\sin (2 (e+f x))} \sin ^{-1}(\cos (e+f x)-\sin (e+f x))+(-15+21 i) \sin ^{\frac{3}{2}}(2 (e+f x)) \log \left (\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right )+(21+15 i) \sqrt{\sin (2 (e+f x))} \cos (2 (e+f x)) \log \left (\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right )-(21+15 i) \sqrt{\sin (2 (e+f x))} \log \left (\sin (e+f x)+\sqrt{\sin (2 (e+f x))}+\cos (e+f x)\right )\right )}{48 a d f (\tan (e+f x)-i) (d \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 154, normalized size = 0.5 \begin{align*}{\frac{{\frac{i}{2}}}{fa{d}^{2} \left ( -id+d\tan \left ( fx+e \right ) \right ) }\sqrt{d\tan \left ( fx+e \right ) }}+{\frac{3\,i}{fa{d}^{2}}\arctan \left ({\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{-id}}}} \right ){\frac{1}{\sqrt{-id}}}}-{\frac{2}{3\,adf} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,i}{fa{d}^{2}}{\frac{1}{\sqrt{d\tan \left ( fx+e \right ) }}}}-{\frac{{\frac{i}{2}}}{fa{d}^{2}}\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 = 2.72132, size = 1968, normalized size = 6.27 \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 = 1.19749, size = 304, normalized size = 0.97 \begin{align*} -\frac{1}{6} \, d^{2}{\left (\frac{3 i \, \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 d^{\frac{9}{2}} f{\left (\frac{i \, d}{\sqrt{d^{2}}} + 1\right )}} - \frac{18 i \, \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 d^{\frac{9}{2}} f{\left (-\frac{i \, d}{\sqrt{d^{2}}} + 1\right )}} + \frac{3 \, \sqrt{d \tan \left (f x + e\right )}}{{\left (i \, d \tan \left (f x + e\right ) + d\right )} a d^{4} f} - \frac{4 \,{\left (3 i \, d \tan \left (f x + e\right ) - d\right )}}{\sqrt{d \tan \left (f x + e\right )} a d^{5} f \tan \left (f x + e\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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