Optimal. Leaf size=136 \[ \frac {i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3604, 47, 37}
\begin {gather*} \frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 3604
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{7/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{35 a f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.17, size = 101, normalized size = 0.74 \begin {gather*} \frac {i c \sec ^2(e+f x) (21+25 \cos (2 (e+f x))+10 i \sin (2 (e+f x))) (i+\tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{210 a^3 f (-i+\tan (e+f x))^3 \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 85, normalized size = 0.62
method | result | size |
derivativedivides | \(-\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (10 i \tan \left (f x +e \right )-2 \left (\tan ^{2}\left (f x +e \right )\right )+23\right )}{105 f \,a^{4} \left (-\tan \left (f x +e \right )+i\right )^{5}}\) | \(85\) |
default | \(-\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (10 i \tan \left (f x +e \right )-2 \left (\tan ^{2}\left (f x +e \right )\right )+23\right )}{105 f \,a^{4} \left (-\tan \left (f x +e \right )+i\right )^{5}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.63, size = 146, normalized size = 1.07 \begin {gather*} \frac {{\left (15 i \, c \cos \left (7 \, f x + 7 \, e\right ) + 42 i \, c \cos \left (\frac {5}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 35 i \, c \cos \left (\frac {3}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 15 \, c \sin \left (7 \, f x + 7 \, e\right ) + 42 \, c \sin \left (\frac {5}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 35 \, c \sin \left (\frac {3}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right )\right )} \sqrt {c}}{420 \, a^{\frac {7}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.43, size = 97, normalized size = 0.71 \begin {gather*} \frac {{\left (35 i \, c e^{\left (6 i \, f x + 6 i \, e\right )} + 77 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 57 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 15 i \, c\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}} e^{\left (-7 i \, f x - 7 i \, e\right )}}{420 \, a^{4} 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*} \int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {7}{2}}}\, dx \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 = 6.51, size = 182, normalized size = 1.34 \begin {gather*} \frac {c\,\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}}\,\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}}\,\left (35\,\sin \left (2\,e+2\,f\,x\right )+77\,\sin \left (4\,e+4\,f\,x\right )+57\,\sin \left (6\,e+6\,f\,x\right )+15\,\sin \left (8\,e+8\,f\,x\right )+\cos \left (2\,e+2\,f\,x\right )\,35{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,77{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,57{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,15{}\mathrm {i}\right )}{840\,a^4\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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