Optimal. Leaf size=199 \[ \frac {3 (i A-9 B) c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a^2 f}-\frac {3 (i A-9 B) c^2 \sqrt {c-i c \tan (e+f x)}}{8 a^2 f}-\frac {(i A-9 B) c (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3669, 79, 43,
52, 65, 214} \begin {gather*} \frac {3 c^{5/2} (-9 B+i A) \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a^2 f}-\frac {3 c^2 (-9 B+i A) \sqrt {c-i c \tan (e+f x)}}{8 a^2 f}-\frac {c (-9 B+i A) (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(-B+i A) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rule 3669
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^2} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{3/2}}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2}-\frac {((A+9 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{(a+i a x)^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {(i A-9 B) c (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (3 (A+9 i B) c^2\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{a+i a x} \, dx,x,\tan (e+f x)\right )}{16 a f}\\ &=-\frac {3 (i A-9 B) c^2 \sqrt {c-i c \tan (e+f x)}}{8 a^2 f}-\frac {(i A-9 B) c (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (3 (A+9 i B) c^3\right ) \text {Subst}\left (\int \frac {1}{(a+i a x) \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 a f}\\ &=-\frac {3 (i A-9 B) c^2 \sqrt {c-i c \tan (e+f x)}}{8 a^2 f}-\frac {(i A-9 B) c (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2}+\frac {\left (3 (i A-9 B) c^2\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{4 a f}\\ &=\frac {3 (i A-9 B) c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a^2 f}-\frac {3 (i A-9 B) c^2 \sqrt {c-i c \tan (e+f x)}}{8 a^2 f}-\frac {(i A-9 B) c (c-i c \tan (e+f x))^{3/2}}{8 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c-i c \tan (e+f x))^{5/2}}{4 a^2 f (1+i \tan (e+f x))^2}\\ \end {align*}
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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.35, size = 140, normalized size = 0.70
method | result | size |
derivativedivides | \(-\frac {2 i c^{2} \left (i B \sqrt {c -i c \tan \left (f x +e \right )}-c \left (\frac {4 \left (\frac {13 i B}{32}+\frac {5 A}{32}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+4 \left (-\frac {11}{16} i B c -\frac {3}{16} A c \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{2}}+\frac {3 \left (\frac {9 i B}{4}+\frac {A}{4}\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )\right )}{f \,a^{2}}\) | \(140\) |
default | \(-\frac {2 i c^{2} \left (i B \sqrt {c -i c \tan \left (f x +e \right )}-c \left (\frac {4 \left (\frac {13 i B}{32}+\frac {5 A}{32}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}+4 \left (-\frac {11}{16} i B c -\frac {3}{16} A c \right ) \sqrt {c -i c \tan \left (f x +e \right )}}{\left (c +i c \tan \left (f x +e \right )\right )^{2}}+\frac {3 \left (\frac {9 i B}{4}+\frac {A}{4}\right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )\right )}{f \,a^{2}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 200, normalized size = 1.01 \begin {gather*} -\frac {i \, {\left (\frac {3 \, \sqrt {2} {\left (A + 9 i \, B\right )} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a^{2}} + \frac {32 i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} B c^{3}}{a^{2}} - \frac {4 \, {\left ({\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (5 \, A + 13 i \, B\right )} c^{4} - 2 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (3 \, A + 11 i \, B\right )} c^{5}\right )}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2} a^{2} - 4 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )} a^{2} c + 4 \, a^{2} c^{2}}\right )}}{16 \, c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 393 vs. \(2 (160) = 320\).
time = 3.74, size = 393, normalized size = 1.97 \begin {gather*} -\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {{\left (A^{2} + 18 i \, A B - 81 \, B^{2}\right )} c^{5}}{a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-\frac {3 \, {\left ({\left (-i \, A + 9 \, B\right )} c^{3} + \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {-\frac {{\left (A^{2} + 18 i \, A B - 81 \, B^{2}\right )} c^{5}}{a^{4} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a^{2} f}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {{\left (A^{2} + 18 i \, A B - 81 \, B^{2}\right )} c^{5}}{a^{4} f^{2}}} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (-\frac {3 \, {\left ({\left (-i \, A + 9 \, B\right )} c^{3} - \sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} f\right )} \sqrt {-\frac {{\left (A^{2} + 18 i \, A B - 81 \, B^{2}\right )} c^{5}}{a^{4} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a^{2} f}\right ) + \sqrt {2} {\left (3 \, {\left (i \, A - 9 \, B\right )} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - {\left (-i \, A + 9 \, B\right )} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (-i \, A + B\right )} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{8 \, a^{2} f} \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*} - \frac {\int \frac {A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \left (- \frac {A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \frac {B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\, dx + \int \left (- \frac {B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \left (- \frac {2 i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx + \int \left (- \frac {2 i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan ^{2}{\left (e + f x \right )} - 2 i \tan {\left (e + f x \right )} - 1}\right )\, dx}{a^{2}} \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 = 9.36, size = 294, normalized size = 1.48 \begin {gather*} \frac {\frac {11\,B\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2}-\frac {13\,B\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{4}}{4\,a^2\,c^2\,f+a^2\,f\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,a^2\,c\,f\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}-\frac {\frac {A\,c^4\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,3{}\mathrm {i}}{2\,a^2\,f}-\frac {A\,c^3\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,5{}\mathrm {i}}{4\,a^2\,f}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2-4\,c\,\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )+4\,c^2}+\frac {2\,B\,c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{a^2\,f}+\frac {\sqrt {2}\,A\,{\left (-c\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,3{}\mathrm {i}}{8\,a^2\,f}-\frac {27\,\sqrt {2}\,B\,c^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {c}}\right )}{8\,a^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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