Optimal. Leaf size=199 \[ \frac{35 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{64 \sqrt{2} a^2 c^{3/2} f}-\frac{35 i}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.211571, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3522, 3487, 51, 63, 206} \[ \frac{35 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{64 \sqrt{2} a^2 c^{3/2} f}-\frac{35 i}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac{\int \cos ^4(e+f x) \sqrt{c-i c \tan (e+f x)} \, dx}{a^2 c^2}\\ &=\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^3 (c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{a^2 f}\\ &=\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{\left (7 i c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^2 (c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{8 a^2 f}\\ &=\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{(35 i c) \operatorname{Subst}\left (\int \frac{1}{(c-x) (c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{32 a^2 f}\\ &=-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}+\frac{(35 i) \operatorname{Subst}\left (\int \frac{1}{(c-x) (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{64 a^2 f}\\ &=-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{35 i}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}+\frac{(35 i) \operatorname{Subst}\left (\int \frac{1}{(c-x) \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{128 a^2 c f}\\ &=-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{35 i}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}+\frac{(35 i) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{64 a^2 c f}\\ &=\frac{35 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{64 \sqrt{2} a^2 c^{3/2} f}-\frac{35 i}{96 a^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{i}{4 a^2 f (1+i \tan (e+f x))^2 (c-i c \tan (e+f x))^{3/2}}+\frac{7 i}{16 a^2 f (1+i \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}}-\frac{35 i}{64 a^2 c f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.99822, size = 145, normalized size = 0.73 \[ -\frac{i e^{-4 i (e+f x)} \left (-45 e^{2 i (e+f x)}+41 e^{4 i (e+f x)}+88 e^{6 i (e+f x)}+8 e^{8 i (e+f x)}-105 e^{4 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (e+f x)}}\right )-6\right ) \sqrt{c-i c \tan (e+f x)}}{384 a^2 c^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 140, normalized size = 0.7 \begin{align*}{\frac{-2\,i{c}^{3}}{f{a}^{2}} \left ({\frac{1}{16\,{c}^{4}} \left ({\frac{1}{ \left ( -c-ic\tan \left ( fx+e \right ) \right ) ^{2}} \left ({\frac{11}{8} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{13\,c}{4}\sqrt{c-ic\tan \left ( fx+e \right ) }} \right ) }-{\frac{35\,\sqrt{2}}{16}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \right ) }+{\frac{3}{16\,{c}^{4}}{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}}+{\frac{1}{24\,{c}^{3}} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53078, size = 938, normalized size = 4.71 \begin{align*} \frac{{\left (105 i \, \sqrt{\frac{1}{2}} a^{2} c^{2} f \sqrt{\frac{1}{a^{4} c^{3} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (1120 i \, a^{2} c f e^{\left (2 i \, f x + 2 i \, e\right )} + 1120 i \, a^{2} c f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{1}{a^{4} c^{3} f^{2}}} + 1120 i\right )} e^{\left (-i \, f x - i \, e\right )}}{1024 \, a^{2} c f}\right ) - 105 i \, \sqrt{\frac{1}{2}} a^{2} c^{2} f \sqrt{\frac{1}{a^{4} c^{3} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (-1120 i \, a^{2} c f e^{\left (2 i \, f x + 2 i \, e\right )} - 1120 i \, a^{2} c f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{1}{a^{4} c^{3} f^{2}}} + 1120 i\right )} e^{\left (-i \, f x - i \, e\right )}}{1024 \, a^{2} c f}\right ) + \sqrt{2} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-8 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 88 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 41 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 45 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{384 \, a^{2} c^{2} f} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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