Optimal. Leaf size=90 \[ -\frac {8 i a^3}{f \sqrt {c-i c \tan (e+f x)}}-\frac {8 i a^3 \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f} \]
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Rubi [A]
time = 0.11, 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 = {3603, 3568, 45}
\begin {gather*} \frac {2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}-\frac {8 i a^3 \sqrt {c-i c \tan (e+f x)}}{c f}-\frac {8 i a^3}{f \sqrt {c-i c \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx &=\left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx\\ &=\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {(c-x)^2}{(c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=\frac {\left (i a^3\right ) \text {Subst}\left (\int \left (\frac {4 c^2}{(c+x)^{3/2}}-\frac {4 c}{\sqrt {c+x}}+\sqrt {c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f}\\ &=-\frac {8 i a^3}{f \sqrt {c-i c \tan (e+f x)}}-\frac {8 i a^3 \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}\\ \end {align*}
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Mathematica [A]
time = 1.29, size = 94, normalized size = 1.04 \begin {gather*} \frac {2 a^3 \sec (e+f x) (12+11 \cos (2 (e+f x))-5 i \sin (2 (e+f x))) (-i \cos (e+4 f x)+\sin (e+4 f x)) \sqrt {c-i c \tan (e+f x)}}{3 c f (\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 66, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-4 c \sqrt {c -i c \tan \left (f x +e \right )}-\frac {4 c^{2}}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}\) | \(66\) |
default | \(\frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-4 c \sqrt {c -i c \tan \left (f x +e \right )}-\frac {4 c^{2}}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 73, normalized size = 0.81 \begin {gather*} -\frac {2 i \, {\left (\frac {12 \, a^{3} c}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} - \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{3} - 12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a^{3} c}{c}\right )}}{3 \, c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 78, normalized size = 0.87 \begin {gather*} -\frac {4 \, \sqrt {2} {\left (3 i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \, {\left (c f e^{\left (2 i \, f x + 2 i \, e\right )} + c f\right )}} \end {gather*}
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 \begin {gather*} - i a^{3} \left (\int \frac {i}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.28, size = 113, normalized size = 1.26 \begin {gather*} -\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 )\,23{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}-7\,\sin \left (2\,e+2\,f\,x\right )-3\,\sin \left (4\,e+4\,f\,x\right )+20{}\mathrm {i}\right )}{3\,c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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