Optimal. Leaf size=90 \[ \frac {2 i a^3 \sqrt {c-i c \tan (e+f x)}}{c^2 f}+\frac {8 i a^3}{c f \sqrt {c-i c \tan (e+f x)}}-\frac {8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3522, 3487, 43} \[ \frac {2 i a^3 \sqrt {c-i c \tan (e+f x)}}{c^2 f}+\frac {8 i a^3}{c f \sqrt {c-i c \tan (e+f x)}}-\frac {8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^{9/2}} \, dx\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {(c-x)^2}{(c+x)^{5/2}} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \left (\frac {4 c^2}{(c+x)^{5/2}}-\frac {4 c}{(c+x)^{3/2}}+\frac {1}{\sqrt {c+x}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac {8 i a^3}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {8 i a^3}{c f \sqrt {c-i c \tan (e+f x)}}+\frac {2 i a^3 \sqrt {c-i c \tan (e+f x)}}{c^2 f}\\ \end {align*}
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Mathematica [A] time = 5.36, size = 94, normalized size = 1.04 \[ \frac {2 a^3 \sqrt {c-i c \tan (e+f x)} (9 \sin (2 (e+f x))+7 i \cos (2 (e+f x))+4 i) (\cos (2 e+5 f x)+i \sin (2 e+5 f x))}{3 c^2 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 62, normalized size = 0.69 \[ \frac {\sqrt {2} {\left (-2 i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \, c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 64, normalized size = 0.71 \[ \frac {2 i a^{3} \left (\sqrt {c -i c \tan \left (f x +e \right )}-\frac {4 c^{2}}{3 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {4 c}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 68, normalized size = 0.76 \[ \frac {2 i \, {\left (\frac {3 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a^{3}}{c} + \frac {4 \, {\left (3 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{3} - a^{3} c\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\right )}}{3 \, c f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.15, size = 98, normalized size = 1.09 \[ \frac {2\,a^3\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,4{}\mathrm {i}-\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-4\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )+8{}\mathrm {i}\right )}{3\,c^2\,f} \]
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 \[ - i a^{3} \left (\int \frac {i}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 \tan {\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (e + f x \right )}}{- i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )} + c \sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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