Optimal. Leaf size=287 \[ -\frac {5 c^{7/2} (-5 B+2 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {5 c^3 (-5 B+2 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 c^2 (-5 B+2 i A) \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 c (-5 B+2 i A) (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 287, 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 = {3588, 78, 47, 50, 63, 217, 203} \[ -\frac {5 c^{7/2} (-5 B+2 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {5 c^3 (-5 B+2 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 c^2 (-5 B+2 i A) \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 c (-5 B+2 i A) (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(-B+i A) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 203
Rule 217
Rule 3588
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(A+B x) (c-i c x)^{5/2}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {((2 A+5 i B) c) \operatorname {Subst}\left (\int \frac {(c-i c x)^{5/2}}{(a+i a x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {\left (5 (2 A+5 i B) c^2\right ) \operatorname {Subst}\left (\int \frac {(c-i c x)^{3/2}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac {5 (2 i A-5 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {\left (5 (2 A+5 i B) c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-i c x}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{2 a f}\\ &=-\frac {5 (2 i A-5 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 (2 i A-5 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {\left (5 (2 A+5 i B) c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{2 a f}\\ &=-\frac {5 (2 i A-5 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 (2 i A-5 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {\left (5 (2 i A-5 B) c^4\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 )}{a^2 f}\\ &=-\frac {5 (2 i A-5 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 (2 i A-5 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {\left (5 (2 i A-5 B) c^4\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 )}{a^2 f}\\ &=-\frac {5 (2 i A-5 B) c^{7/2} \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 )}{a^{3/2} f}-\frac {5 (2 i A-5 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 a^2 f}-\frac {5 (2 i A-5 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{6 a^2 f}-\frac {2 (2 i A-5 B) c (c-i c \tan (e+f x))^{5/2}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i A-B) (c-i c \tan (e+f x))^{7/2}}{3 f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 13.45, size = 255, normalized size = 0.89 \[ \frac {\sqrt {\sec (e+f x)} (A+B \tan (e+f x)) \left (\frac {5 c^4 (5 B-2 i A) e^{i (e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right )}{\sqrt {\frac {c}{1+e^{2 i (e+f x)}}}}+\frac {1}{12} c^3 \sec ^{\frac {3}{2}}(e+f x) \sqrt {c-i c \tan (e+f x)} (33 (5 B-2 i A) \cos (e+f x)+(71 B-26 i A) \cos (3 (e+f x))+2 \sin (e+f x) ((34 A+79 i B) \cos (2 (e+f x))+34 A+82 i B))\right )}{f (a+i a \tan (e+f x))^{3/2} (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 586, normalized size = 2.04 \[ -\frac {3 \, {\left (a^{2} f e^{\left (5 i \, f x + 5 i \, e\right )} + a^{2} f e^{\left (3 i \, f x + 3 i \, e\right )}\right )} \sqrt {\frac {{\left (100 \, A^{2} + 500 i \, A B - 625 \, B^{2}\right )} c^{7}}{a^{3} f^{2}}} \log \left (\frac {2 \, {\left ({\left ({\left (-40 i \, A + 100 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-40 i \, A + 100 \, B\right )} c^{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}} + 2 \, {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - a^{2} f\right )} \sqrt {\frac {{\left (100 \, A^{2} + 500 i \, A B - 625 \, B^{2}\right )} c^{7}}{a^{3} f^{2}}}\right )}}{{\left (-10 i \, A + 25 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-10 i \, A + 25 \, B\right )} c^{3}}\right ) - 3 \, {\left (a^{2} f e^{\left (5 i \, f x + 5 i \, e\right )} + a^{2} f e^{\left (3 i \, f x + 3 i \, e\right )}\right )} \sqrt {\frac {{\left (100 \, A^{2} + 500 i \, A B - 625 \, B^{2}\right )} c^{7}}{a^{3} f^{2}}} \log \left (\frac {2 \, {\left ({\left ({\left (-40 i \, A + 100 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-40 i \, A + 100 \, B\right )} c^{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}} - 2 \, {\left (a^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - a^{2} f\right )} \sqrt {\frac {{\left (100 \, A^{2} + 500 i \, A B - 625 \, B^{2}\right )} c^{7}}{a^{3} f^{2}}}\right )}}{{\left (-10 i \, A + 25 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-10 i \, A + 25 \, B\right )} c^{3}}\right ) - 2 \, {\left ({\left (-60 i \, A + 150 \, B\right )} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-100 i \, A + 250 \, B\right )} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-32 i \, A + 80 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (8 i \, A - 8 \, B\right )} c^{3}\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}}}{12 \, {\left (a^{2} f e^{\left (5 i \, f x + 5 i \, e\right )} + a^{2} f e^{\left (3 i \, f x + 3 i \, e\right )}\right )}} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 733, normalized size = 2.55 \[ \frac {\sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{3} \left (-75 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 +6 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 )+185 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 )+225 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 -30 A \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 +3 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 )-30 i A \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 -225 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 -21 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 )-114 i A \sqrt {c a}\, \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )+90 i A \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 +90 A \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 +74 A \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-118 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+75 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 +279 B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )-46 A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\right )}{6 f \,a^{2} \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \left (-\tan \left (f x +e \right )+i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.88, size = 1379, normalized size = 4.80 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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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