Optimal. Leaf size=364 \[ -\frac{7 (-4 B+i A) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+6 i) A-(29+i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{(-B+i A) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.771327, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3595, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{7 (-4 B+i A) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+6 i) A-(29+i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{(-B+i A) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3595
Rule 3528
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac{\int \frac{\tan ^{\frac{5}{2}}(c+d x) \left (\frac{7}{2} a (i A-B)+\frac{1}{2} a (A+13 i B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\tan ^{\frac{3}{2}}(c+d x) \left (-5 a^2 (2 A+5 i B)+a^2 (4 i A-31 B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\int \sqrt{\tan (c+d x)} \left (-21 a^3 (i A-4 B)-15 a^3 (A+6 i B) \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\int \frac{15 a^3 (A+6 i B)-21 a^3 (i A-4 B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{48 a^6}\\ &=\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{15 a^3 (A+6 i B)-21 a^3 (i A-4 B) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{24 a^6 d}\\ &=\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{((5+7 i) A-(28-30 i) B) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{16 a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{16 a^3 d}\\ &=\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{((5+7 i) A-(28-30 i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{32 \sqrt{2} a^3 d}+\frac{((5+7 i) A-(28-30 i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{32 \sqrt{2} a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{32 a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{32 a^3 d}\\ &=\frac{((5+7 i) A-(28-30 i) B) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt{2} a^3 d}-\frac{((5+7 i) A-(28-30 i) B) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{((5-7 i) A+(28+30 i) B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{((5-7 i) A+(28+30 i) B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{16 \sqrt{2} a^3 d}\\ &=\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{16 \sqrt{2} a^3 d}-\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{((5+7 i) A-(28-30 i) B) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt{2} a^3 d}-\frac{((5+7 i) A-(28-30 i) B) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{5 (A+6 i B) \sqrt{\tan (c+d x)}}{8 a^3 d}+\frac{(i A-B) \tan ^{\frac{7}{2}}(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac{(2 A+5 i B) \tan ^{\frac{5}{2}}(c+d x)}{12 a d (a+i a \tan (c+d x))^2}-\frac{7 (i A-4 B) \tan ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 3.41184, size = 286, normalized size = 0.79 \[ \frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left (\frac{2}{3} \tan (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((9 A+33 i B) \cos (c+d x)+21 (A+7 i B) \cos (3 (c+d x))+2 i \sin (c+d x) ((19 A+145 i B) \cos (2 (c+d x))+19 A+97 i B))-i (\cos (3 c)+i \sin (3 c)) \sqrt{\sin (2 (c+d x))} \sec (c+d x) \left (((7+5 i) A-(30-28 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(1-i) ((6+i) A+(1+29 i) B) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt{\tan (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 369, normalized size = 1. \begin{align*}{\frac{2\,iB}{{a}^{3}d}\sqrt{\tan \left ( dx+c \right ) }}-{\frac{{\frac{9\,i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B}{2\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{19\,A}{12\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{{\frac{49\,i}{12}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{7\,B}{4\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}\sqrt{\tan \left ( dx+c \right ) }}+{\frac{{\frac{5\,i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}\sqrt{\tan \left ( dx+c \right ) }}-{\frac{29\,B}{4\,{a}^{3}d \left ( \sqrt{2}-i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}-i\sqrt{2}}} \right ) }+{\frac{{\frac{3\,i}{2}}A}{{a}^{3}d \left ( \sqrt{2}-i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}-i\sqrt{2}}} \right ) }+{\frac{{\frac{i}{4}}A}{{a}^{3}d \left ( \sqrt{2}+i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}+i\sqrt{2}}} \right ) }+{\frac{B}{4\,{a}^{3}d \left ( \sqrt{2}+i\sqrt{2} \right ) }\arctan \left ( 2\,{\frac{\sqrt{\tan \left ( dx+c \right ) }}{\sqrt{2}+i\sqrt{2}}} \right ) } \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 = 1.79629, size = 1860, normalized size = 5.11 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26151, size = 223, normalized size = 0.61 \begin{align*} \frac{\left (i - 1\right ) \, \sqrt{2}{\left (6 \, A + 29 i \, B\right )} \arctan \left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} + \frac{\left (i + 1\right ) \, \sqrt{2}{\left (A - i \, B\right )} \arctan \left (-\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\tan \left (d x + c\right )}\right )}{16 \, a^{3} d} + \frac{2 i \, B \sqrt{\tan \left (d x + c\right )}}{a^{3} d} + \frac{-27 i \, A \tan \left (d x + c\right )^{\frac{5}{2}} + 60 \, B \tan \left (d x + c\right )^{\frac{5}{2}} - 38 \, A \tan \left (d x + c\right )^{\frac{3}{2}} - 98 i \, B \tan \left (d x + c\right )^{\frac{3}{2}} + 15 i \, A \sqrt{\tan \left (d x + c\right )} - 42 \, B \sqrt{\tan \left (d x + c\right )}}{24 \, a^{3} d{\left (\tan \left (d x + c\right ) - i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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