Optimal. Leaf size=102 \[ -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {(i A-6 B) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3669, 79, 37}
\begin {gather*} -\frac {(-6 B+i A) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{3/2} (A+B x)}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {(a (A+6 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {(i A-6 B) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 6.41, size = 121, normalized size = 1.19 \begin {gather*} \frac {a^2 \cos (e+f x) ((-6 i A+B) \cos (e+f x)-(A+6 i B) \sin (e+f x)) (\cos (6 e+8 f x)+i \sin (6 e+8 f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{35 c^4 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 115, normalized size = 1.13
method | result | size |
risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (5 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+5 B \,{\mathrm e}^{6 i \left (f x +e \right )}+7 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-7 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{70 c^{3} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(106\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (i A \left (\tan ^{2}\left (f x +e \right )\right )+5 i B \tan \left (f x +e \right )-6 B \left (\tan ^{2}\left (f x +e \right )\right )+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(115\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (i A \left (\tan ^{2}\left (f x +e \right )\right )+5 i B \tan \left (f x +e \right )-6 B \left (\tan ^{2}\left (f x +e \right )\right )+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (i+\tan \left (f x +e \right )\right )^{5}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 174 vs. \(2 (82) = 164\).
time = 0.62, size = 174, normalized size = 1.71 \begin {gather*} -\frac {70 \, {\left (5 \, {\left (A - i \, B\right )} a^{2} \cos \left (9 \, f x + 9 \, e\right ) + 2 \, {\left (6 \, A + i \, B\right )} a^{2} \cos \left (7 \, f x + 7 \, e\right ) + 7 \, {\left (A + i \, B\right )} a^{2} \cos \left (5 \, f x + 5 \, e\right ) - 5 \, {\left (-i \, A - B\right )} a^{2} \sin \left (9 \, f x + 9 \, e\right ) - 2 \, {\left (-6 i \, A + B\right )} a^{2} \sin \left (7 \, f x + 7 \, e\right ) - 7 \, {\left (-i \, A + B\right )} a^{2} \sin \left (5 \, f x + 5 \, e\right )\right )} \sqrt {a} \sqrt {c}}{-4900 \, {\left (i \, c^{4} \cos \left (2 \, f x + 2 \, e\right ) - c^{4} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{4}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.73, size = 109, normalized size = 1.07 \begin {gather*} -\frac {{\left (5 \, {\left (i \, A + B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 2 \, {\left (6 i \, A - B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 7 \, {\left (i \, A - B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{70 \, c^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 10.28, size = 192, normalized size = 1.88 \begin {gather*} -\frac {a^2\,\sqrt {\frac {a\,\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 (A\,\cos \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}-7\,B\,\cos \left (4\,e+4\,f\,x\right )+5\,B\,\cos \left (6\,e+6\,f\,x\right )-7\,A\,\sin \left (4\,e+4\,f\,x\right )-5\,A\,\sin \left (6\,e+6\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}\right )}{70\,c^3\,f\,\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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