Optimal. Leaf size=251 \[ -\frac {2 a^{7/2} B \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{7/2} f}+\frac {2 a^3 B \sqrt {a+i a \tan (e+f x)}}{c^3 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3588, 78, 47, 63, 217, 203} \[ -\frac {2 a^{7/2} B \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{7/2} f}+\frac {2 a^3 B \sqrt {a+i a \tan (e+f x)}}{c^3 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 203
Rule 217
Rule 3588
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))^{7/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^{5/2} (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))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {(i a B) \operatorname {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(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}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {\left (i a^2 B\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{c f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {\left (i a^3 B\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{c^2 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a^3 B \sqrt {a+i a \tan (e+f x)}}{c^3 f \sqrt {c-i c \tan (e+f x)}}-\frac {\left (i a^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{c^3 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a^3 B \sqrt {a+i a \tan (e+f x)}}{c^3 f \sqrt {c-i c \tan (e+f x)}}-\frac {\left (2 a^3 B\right ) \operatorname {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^3 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a^3 B \sqrt {a+i a \tan (e+f x)}}{c^3 f \sqrt {c-i c \tan (e+f x)}}-\frac {\left (2 a^3 B\right ) \operatorname {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^3 f}\\ &=-\frac {2 a^{7/2} 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^{7/2} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {2 a B (a+i a \tan (e+f x))^{5/2}}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac {2 a^2 B (a+i a \tan (e+f x))^{3/2}}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a^3 B \sqrt {a+i a \tan (e+f x)}}{c^3 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 17.75, size = 570, normalized size = 2.27 \[ \frac {\cos ^4(e+f x) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left ((9 B-5 i A) \cos (6 f x) \left (\frac {\cos (3 e)}{70 c^4}+\frac {i \sin (3 e)}{70 c^4}\right )+(A-i B) \cos (8 f x) \left (\frac {\sin (5 e)}{14 c^4}-\frac {i \cos (5 e)}{14 c^4}\right )+(5 A+9 i B) \sin (6 f x) \left (\frac {\cos (3 e)}{70 c^4}+\frac {i \sin (3 e)}{70 c^4}\right )+(A-i B) \sin (8 f x) \left (\frac {\cos (5 e)}{14 c^4}+\frac {i \sin (5 e)}{14 c^4}\right )+\cos (4 f x) \left (-\frac {2 B \cos (e)}{15 c^4}-\frac {2 i B \sin (e)}{15 c^4}\right )+\cos (2 f x) \left (\frac {2 B \cos (e)}{3 c^4}-\frac {2 i B \sin (e)}{3 c^4}\right )+\sin (2 f x) \left (\frac {2 B \sin (e)}{3 c^4}+\frac {2 i B \cos (e)}{3 c^4}\right )+\sin (4 f x) \left (\frac {2 B \sin (e)}{15 c^4}-\frac {2 i B \cos (e)}{15 c^4}\right )-\frac {i B \sin (3 e)}{c^4}+\frac {B \cos (3 e)}{c^4}\right )}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))}-\frac {2 B \sqrt {e^{i f x}} e^{-i (4 e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{c^3 f \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \sec ^{\frac {9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.86, size = 433, normalized size = 1.73 \[ \frac {105 \, c^{4} f \sqrt {-\frac {B^{2} a^{7}}{c^{7} f^{2}}} \log \left (\frac {4 \, {\left (2 \, {\left (B a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + B a^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + {\left (c^{4} f e^{\left (2 i \, f x + 2 i \, e\right )} - c^{4} f\right )} \sqrt {-\frac {B^{2} a^{7}}{c^{7} f^{2}}}\right )}}{B a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + B a^{3}}\right ) - 105 \, c^{4} f \sqrt {-\frac {B^{2} a^{7}}{c^{7} f^{2}}} \log \left (\frac {4 \, {\left (2 \, {\left (B a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + B a^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (c^{4} f e^{\left (2 i \, f x + 2 i \, e\right )} - c^{4} f\right )} \sqrt {-\frac {B^{2} a^{7}}{c^{7} f^{2}}}\right )}}{B a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + B a^{3}}\right ) + {\left ({\left (-30 i \, A - 30 \, B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (-30 i \, A + 54 \, B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )} - 56 \, B a^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 280 \, B a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 420 \, B a^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{210 \, c^{4} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 638, normalized size = 2.54 \[ \frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{3} \left (-105 i B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \left (\tan ^{5}\left (f x +e \right )\right ) a c +1050 i B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \left (\tan ^{3}\left (f x +e \right )\right ) a c +337 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (\tan ^{4}\left (f x +e \right )\right )+525 B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \left (\tan ^{4}\left (f x +e \right )\right ) a c +30 i A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (\tan ^{3}\left (f x +e \right )\right )-15 A \left (\tan ^{4}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-525 i B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \tan \left (f x +e \right ) a c -1176 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (\tan ^{2}\left (f x +e \right )\right )-1050 B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) \left (\tan ^{2}\left (f x +e \right )\right ) a c -950 B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (\tan ^{3}\left (f x +e \right )\right )+30 i A \sqrt {c a}\, \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )+167 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+105 B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) a c +730 B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )+15 A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\right )}{105 f \,c^{4} \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{5} \sqrt {c a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 245, normalized size = 0.98 \[ -\frac {{\left (210 \, B a^{3} \arctan \left (\cos \left (f x + e\right ), \sin \left (f x + e\right ) + 1\right ) + 210 \, B a^{3} \arctan \left (\cos \left (f x + e\right ), -\sin \left (f x + e\right ) + 1\right ) + {\left (30 i \, A + 30 \, B\right )} a^{3} \cos \left (7 \, f x + 7 \, e\right ) - 84 \, B a^{3} \cos \left (5 \, f x + 5 \, e\right ) + 140 \, B a^{3} \cos \left (3 \, f x + 3 \, e\right ) - 420 \, B a^{3} \cos \left (f x + e\right ) + 105 i \, B a^{3} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right ) - 105 i \, B a^{3} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right ) - 30 \, {\left (A - i \, B\right )} a^{3} \sin \left (7 \, f x + 7 \, e\right ) - 84 i \, B a^{3} \sin \left (5 \, f x + 5 \, e\right ) + 140 i \, B a^{3} \sin \left (3 \, f x + 3 \, e\right ) - 420 i \, B a^{3} \sin \left (f x + e\right )\right )} \sqrt {a}}{210 \, c^{\frac {7}{2}} f} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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