Optimal. Leaf size=322 \[ \frac {(i a+b)^3 (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f} \]
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Rubi [A]
time = 0.63, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3647, 3711,
3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}+\frac {2 \left (2 a^3 c d+3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (a^3 d+3 a^2 b c-3 a b^2 d-b^3 c\right ) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {(b+i a)^3 (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(-b+i a)^3 (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3609
Rule 3618
Rule 3620
Rule 3647
Rule 3711
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^{5/2} \, dx &=\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \tan (e+f x))^{5/2} \left (\frac {1}{2} \left (9 a^3 d-2 b^2 \left (b c+\frac {7 a d}{2}\right )\right )+\frac {9}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (b c-10 a d) \tan ^2(e+f x)\right ) \, dx}{9 d}\\ &=-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \tan (e+f x))^{5/2} \left (\frac {9}{2} a \left (a^2-3 b^2\right ) d+\frac {9}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {9}{2} d \left (a^3 c-3 a b^2 c-3 a^2 b d+b^3 d\right )+\frac {9}{2} d \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (-\frac {9}{2} d \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right )+\frac {9}{2} d \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \tan (e+f x)\right ) \, dx}{9 d}\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \int \frac {\frac {9}{2} d (a c-b d) \left (a^2 c^2-3 b^2 c^2-8 a b c d-3 a^2 d^2+b^2 d^2\right )+\frac {9}{2} d (b c+a d) \left (3 a^2 c^2-b^2 c^2-8 a b c d-a^2 d^2+3 b^2 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{9 d}\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {1}{2} \left ((a-i b)^3 (c-i d)^3\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (c+i d)^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac {\left ((i a+b)^3 (c-i d)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}+\frac {\left ((i a-b)^3 (c+i d)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}-\frac {\left ((a-i b)^3 (c-i d)^3\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b)^3 (c+i d)^3\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=\frac {(i a+b)^3 (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a^3 c d-6 a b^2 c d+3 a^2 b \left (c^2-d^2\right )-b^3 \left (c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b \left (3 a^2-b^2\right ) (c+d \tan (e+f x))^{5/2}}{5 f}-\frac {4 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{63 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}\\ \end {align*}
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Mathematica [A]
time = 6.21, size = 413, normalized size = 1.28 \begin {gather*} \frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{7/2}}{9 d f}+\frac {2 \left (-\frac {2 b^2 (b c-10 a d) (c+d \tan (e+f x))^{7/2}}{7 d f}+\frac {i \left (\frac {9}{2} a \left (a^2-3 b^2\right ) d-\frac {9}{2} i b \left (3 a^2-b^2\right ) d\right ) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+(c-i d) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c-i d) \left (\frac {2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{-c+i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{2 f}-\frac {i \left (\frac {9}{2} a \left (a^2-3 b^2\right ) d+\frac {9}{2} i b \left (3 a^2-b^2\right ) d\right ) \left (\frac {2}{5} (c+d \tan (e+f x))^{5/2}+(c+i d) \left (\frac {2}{3} (c+d \tan (e+f x))^{3/2}+(c+i d) \left (\frac {2 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{-c-i d}+2 \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{2 f}\right )}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2490\) vs.
\(2(286)=572\).
time = 0.50, size = 2491, normalized size = 7.74
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2491\) |
default | \(\text {Expression too large to display}\) | \(2491\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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 \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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