Optimal. Leaf size=101 \[ -\frac {4 a^2 (i A+B)}{f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^2 (i A+3 B) \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{3/2}}{3 c^2 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3669, 78}
\begin {gather*} -\frac {2 a^2 (3 B+i A) \sqrt {c-i c \tan (e+f x)}}{c f}-\frac {4 a^2 (B+i A)}{f \sqrt {c-i c \tan (e+f x)}}+\frac {2 a^2 B (c-i c \tan (e+f x))^{3/2}}{3 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x) (A+B x)}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (\frac {2 a (A-i B)}{(c-i c x)^{3/2}}-\frac {a (A-3 i B)}{c \sqrt {c-i c x}}-\frac {i a B \sqrt {c-i c x}}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {4 a^2 (i A+B)}{f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^2 (i A+3 B) \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}\\ \end {align*}
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Mathematica [A]
time = 1.44, size = 138, normalized size = 1.37 \begin {gather*} \frac {a^2 (9 A-15 i B+(9 A-13 i B) \cos (2 (e+f x))+(-3 i A-7 B) \sin (2 (e+f x))) (-i \cos (e+3 f x)+\sin (e+3 f x)) (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{3 c f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 93, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {2 i a^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-3 i B c \sqrt {c -i c \tan \left (f x +e \right )}+A c \sqrt {c -i c \tan \left (f x +e \right )}+\frac {2 c^{2} \left (-i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}\) | \(93\) |
default | \(-\frac {2 i a^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-3 i B c \sqrt {c -i c \tan \left (f x +e \right )}+A c \sqrt {c -i c \tan \left (f x +e \right )}+\frac {2 c^{2} \left (-i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 84, normalized size = 0.83 \begin {gather*} -\frac {2 i \, {\left (\frac {6 \, {\left (A - i \, B\right )} a^{2} c}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} + \frac {i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} B a^{2} + 3 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (A - 3 i \, B\right )} a^{2} 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 = 4.13, size = 97, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (3 \, {\left (i \, A + B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (3 i \, A + 5 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (3 i \, A + 5 \, B\right )} a^{2}\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*} - a^{2} \left (\int \left (- \frac {A}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan ^{2}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {B \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {B \tan ^{3}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {2 i A \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \left (- \frac {2 i B \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 = 10.77, size = 176, normalized size = 1.74 \begin {gather*} -\frac {2\,\sqrt {2}\,a^2\,\sqrt {\frac {c}{\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}}}\,\left (A\,6{}\mathrm {i}+10\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,9{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}+15\,B\,\cos \left (2\,e+2\,f\,x\right )+3\,B\,\cos \left (4\,e+4\,f\,x\right )-9\,A\,\sin \left (2\,e+2\,f\,x\right )-3\,A\,\sin \left (4\,e+4\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,15{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}\right )}{3\,c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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