Optimal. Leaf size=283 \[ \frac {2 a^{7/2} (i A+6 B) \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{5/2} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^2 (i A+6 B) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {a^3 (i A+6 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^3 f} \]
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Rubi [A]
time = 0.23, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3669, 79, 49,
52, 65, 223, 209} \begin {gather*} \frac {2 a^{7/2} (6 B+i A) \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{5/2} f}-\frac {a^3 (6 B+i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^3 f}-\frac {2 a^2 (6 B+i A) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}+\frac {2 a (6 B+i A) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 79
Rule 209
Rule 223
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{5/2} (A+B x)}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac {(a (A-6 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}+\frac {\left (a^2 (A-6 i B)\right ) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 c f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^2 (i A+6 B) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {\left (a^3 (A-6 i B)\right ) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{c^2 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^2 (i A+6 B) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {a^3 (i A+6 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^3 f}-\frac {\left (a^4 (A-6 i B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{c^2 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^2 (i A+6 B) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {a^3 (i A+6 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^3 f}+\frac {\left (2 a^3 (i A+6 B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{c^2 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^2 (i A+6 B) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {a^3 (i A+6 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^3 f}+\frac {\left (2 a^3 (i A+6 B)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{c^2 f}\\ &=\frac {2 a^{7/2} (i A+6 B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{5/2} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac {2 a (i A+6 B) (a+i a \tan (e+f x))^{5/2}}{15 c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 a^2 (i A+6 B) (a+i a \tan (e+f x))^{3/2}}{3 c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {a^3 (i A+6 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^3 f}\\ \end {align*}
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Mathematica [A]
time = 11.76, size = 528, normalized size = 1.87 \begin {gather*} \frac {2 (i A+6 B) e^{-i (4 e+f x)} \sqrt {e^{i f x}} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \text {ArcTan}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{c^2 \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} f \sec ^{\frac {9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}+\frac {\cos ^4(e+f x) \left ((A-6 i B) \cos (2 f x) \left (-\frac {2 i \cos (e)}{3 c^3}-\frac {2 \sin (e)}{3 c^3}\right )+(i A+6 B) \cos (4 f x) \left (\frac {2 \cos (e)}{15 c^3}+\frac {2 i \sin (e)}{15 c^3}\right )+(A-6 i B) \left (-\frac {i \cos (3 e)}{c^3}-\frac {\sin (3 e)}{c^3}\right )+(A-i B) \cos (6 f x) \left (-\frac {i \cos (3 e)}{5 c^3}+\frac {\sin (3 e)}{5 c^3}\right )+(A-6 i B) \left (\frac {2 \cos (e)}{3 c^3}-\frac {2 i \sin (e)}{3 c^3}\right ) \sin (2 f x)+(A-6 i B) \left (-\frac {2 \cos (e)}{15 c^3}-\frac {2 i \sin (e)}{15 c^3}\right ) \sin (4 f x)+(A-i B) \left (\frac {\cos (3 e)}{5 c^3}+\frac {i \sin (3 e)}{5 c^3}\right ) \sin (6 f x)\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 832 vs. \(2 (234 ) = 468\).
time = 0.43, size = 833, normalized size = 2.94 Too large to display
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. 990 vs. \(2 (231) = 462\).
time = 0.66, size = 990, normalized size = 3.50 \begin {gather*} \frac {15 \, {\left (30 \, {\left ({\left (A - 6 i \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 6 \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 6 i \, B\right )} a^{3}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 30 \, {\left ({\left (A - 6 i \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 6 \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 6 i \, B\right )} a^{3}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 12 \, {\left ({\left (A - i \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - i \, B\right )} a^{3}\right )} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 20 \, {\left ({\left (A - 3 i \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 3 \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 3 i \, B\right )} a^{3}\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 60 \, {\left ({\left (A - 5 i \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 5 \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 6 i \, B\right )} a^{3}\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 15 \, {\left ({\left (-i \, A - 6 \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) + {\left (A - 6 i \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (-i \, A - 6 \, B\right )} a^{3}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 15 \, {\left ({\left (i \, A + 6 \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - 6 i \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 6 \, B\right )} a^{3}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 12 \, {\left ({\left (i \, A + B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - i \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + B\right )} a^{3}\right )} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 20 \, {\left ({\left (-i \, A - 3 \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) + {\left (A - 3 i \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (-i \, A - 3 \, B\right )} a^{3}\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 60 \, {\left ({\left (i \, A + 5 \, B\right )} a^{3} \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - 5 i \, B\right )} a^{3} \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 6 \, B\right )} a^{3}\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a} \sqrt {c}}{-450 \, {\left (i \, c^{3} \cos \left (2 \, f x + 2 \, e\right ) - c^{3} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{3}\right )} 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. 533 vs. \(2 (231) = 462\).
time = 1.62, size = 533, normalized size = 1.88 \begin {gather*} \frac {15 \, c^{3} \sqrt {\frac {{\left (A^{2} - 12 i \, A B - 36 \, B^{2}\right )} a^{7}}{c^{5} f^{2}}} f \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-i \, A - 6 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, A - 6 \, B\right )} a^{3} e^{\left (i \, f x + 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}} + {\left (c^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} - c^{3} f\right )} \sqrt {\frac {{\left (A^{2} - 12 i \, A B - 36 \, B^{2}\right )} a^{7}}{c^{5} f^{2}}}\right )}}{{\left (i \, A + 6 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 6 \, B\right )} a^{3}}\right ) - 15 \, c^{3} \sqrt {\frac {{\left (A^{2} - 12 i \, A B - 36 \, B^{2}\right )} a^{7}}{c^{5} f^{2}}} f \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-i \, A - 6 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, A - 6 \, B\right )} a^{3} e^{\left (i \, f x + 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}} - {\left (c^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} - c^{3} f\right )} \sqrt {\frac {{\left (A^{2} - 12 i \, A B - 36 \, B^{2}\right )} a^{7}}{c^{5} f^{2}}}\right )}}{{\left (i \, A + 6 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 6 \, B\right )} a^{3}}\right ) - 4 \, {\left (3 \, {\left (i \, A + B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 2 \, {\left (-i \, A - 6 \, B\right )} a^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 10 \, {\left (i \, A + 6 \, B\right )} a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (i \, A + 6 \, B\right )} a^{3} e^{\left (i \, f x + 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}}}{30 \, c^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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