Optimal. Leaf size=273 \[ -\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]
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Rubi [A]
time = 2.19, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3646, 3736,
6857, 65, 223, 212, 95, 214} \begin {gather*} \frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}+\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3646
Rule 3736
Rule 6857
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} \left (b^3 c^2+a^3 c d-3 a b^2 c d+3 a^2 b d^2\right )+\frac {1}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)+\frac {1}{2} b^3 \left (c^2+d^2\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (b^3 c^2+a^3 c d-3 a b^2 c d+3 a^2 b d^2\right )+\frac {1}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) x+\frac {1}{2} b^3 \left (c^2+d^2\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \text {Subst}\left (\int \left (\frac {b^3 \left (c^2+d^2\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )+d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {b^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{d f}+\frac {\text {Subst}\left (\int \frac {d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )+d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{d f}+\frac {\text {Subst}\left (\int \left (\frac {-d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right )+i d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right )+i d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {(i a+b)^3 \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c-i d) f}-\frac {(i a-b)^3 \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c+i d) f}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{d f}\\ &=\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {(i a+b)^3 \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(c-i d) f}-\frac {(i a-b)^3 \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(c+i d) f}\\ &=-\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1503\) vs. \(2(273)=546\).
time = 6.19, size = 1503, normalized size = 5.51 \begin {gather*} -\frac {i (-a-i b) \left (-\left ((-a-i b) \left (-\frac {2 \sqrt {a+i b} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-c-i d) \sqrt {c+i d}}+\frac {2 \sqrt {a+b \tan (e+f x)}}{(-c-i d) \sqrt {c+d \tan (e+f x)}}\right )\right )-\frac {2 (b c-a d) \left (\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}\right )^{3/2} \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )^2 \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right ) \left (-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right ) \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}-\frac {\sqrt {b} \sqrt {d} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{b d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{2 f}-\frac {i (-a+i b) \left (-\left ((-a+i b) \left (-\frac {2 \sqrt {-a+i b} \tanh ^{-1}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d) \sqrt {-c+i d}}+\frac {2 \sqrt {a+b \tan (e+f x)}}{(c-i d) \sqrt {c+d \tan (e+f x)}}\right )\right )+\frac {2 (b c-a d) \left (\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}\right )^{3/2} \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )^2 \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right ) \left (-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right ) \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}-\frac {\sqrt {b} \sqrt {d} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{b d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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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 )}^{5/2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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