Optimal. Leaf size=417 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} (c-i d)^{3/2} f}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} (c+i d)^{3/2} f}-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]
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Rubi [A]
time = 1.32, antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3650, 3730,
3697, 3696, 95, 214} \begin {gather*} -\frac {4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{3 f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3+a^2 b^2 d \left (11 c^2+17 d^2\right )-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 \sqrt {c+d \tan (e+f x)}}-\frac {i \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 (a-i b)^{5/2} (c-i d)^{3/2}}+\frac {i \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 (a+i b)^{5/2} (c+i d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 214
Rule 3650
Rule 3696
Rule 3697
Rule 3730
Rubi steps
\begin {align*} \int \frac {1}{(a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \int \frac {\frac {1}{2} \left (4 b^2 d-3 a (b c-a d)\right )+\frac {3}{2} b (b c-a d) \tan (e+f x)+2 b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} \left (-6 a^3 b c d-6 a b^3 c d+3 a^4 d^2-b^4 \left (3 c^2-8 d^2\right )+a^2 b^2 \left (3 c^2+17 d^2\right )\right )-\frac {3}{2} a b (b c-a d)^2 \tan (e+f x)-b^2 d \left (3 a b c-5 a^2 d-2 b^2 d\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}} \, dx}{3 \left (a^2+b^2\right )^2 (b c-a d)^2}\\ &=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {8 \int \frac {\frac {3}{8} (b c-a d)^3 \left (a^2 c-b^2 c-2 a b d\right )-\frac {3}{8} (b c-a d)^3 \left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right )}\\ &=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2 (c-i d)}+\frac {\int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2 (c+i d)}\\ &=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^2 (c-i d) f}+\frac {\text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^2 (c+i d) f}\\ &=-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{i a+b-(i c+d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^2 (c-i d) f}+\frac {\text {Subst}\left (\int \frac {1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^2 (c+i d) f}\\ &=-\frac {i \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 )}{(a-i b)^{5/2} (c-i d)^{3/2} f}+\frac {i \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 )}{(a+i b)^{5/2} (c+i d)^{3/2} f}-\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {4 b^2 \left (3 a b c-5 a^2 d-2 b^2 d\right )}{3 \left (a^2+b^2\right )^2 (b c-a d)^2 f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d \left (3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )+a^2 b^2 d \left (11 c^2+17 d^2\right )\right ) \sqrt {a+b \tan (e+f x)}}{3 \left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 6.48, size = 601, normalized size = 1.44 \begin {gather*} -\frac {2 b^2}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-\frac {2 \left (\frac {1}{2} b^2 \left (4 b^2 d-3 a (b c-a d)\right )-a \left (-2 a b^2 d+\frac {3}{2} b^2 (b c-a d)\right )\right )}{\left (a^2+b^2\right ) (b c-a d) f \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (-\frac {3 (b c-a d)^3 \left (\frac {(a+i b)^2 (i c-d) \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 )}{\sqrt {-a+i b} \sqrt {-c+i d}}+\frac {(a-i b)^2 (i c+d) \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 )}{\sqrt {a+i b} \sqrt {c+i d}}\right )}{4 (-b c+a d) \left (c^2+d^2\right ) f}-\frac {2 \left (-c \left (-\frac {3}{2} a b d (b c-a d)^2+b^2 c d \left (3 a b c-5 a^2 d-2 b^2 d\right )\right )+\frac {1}{4} d^2 \left (-6 a^3 b c d-6 a b^3 c d+3 a^4 d^2-b^4 \left (3 c^2-8 d^2\right )+a^2 b^2 \left (3 c^2+17 d^2\right )\right )\right ) \sqrt {a+b \tan (e+f x)}}{(-b c+a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}\right )}{\left (a^2+b^2\right ) (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
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 {1}{\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]
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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\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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