Optimal. Leaf size=314 \[ -\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {b^{5/2} \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2 (c-i d)^{3/2}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2 (c+i d)^{3/2}} \]
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Rubi [A] time = 1.47, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3569, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {b^{5/2} \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2 (c-i d)^{3/2}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2 (c+i d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 3537
Rule 3539
Rule 3569
Rule 3634
Rule 3649
Rule 3653
Rubi steps
\begin {align*} \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {\frac {1}{2} \left (-2 a b c+2 a^2 d+3 b^2 d\right )+b (b c-a d) \tan (e+f x)+\frac {3}{2} b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {2 \int \frac {\frac {1}{4} \left (-2 a^3 c d^2+4 a^2 b d \left (c^2+d^2\right )+3 b^3 d \left (c^2+d^2\right )-2 a b^2 c \left (c^2+2 d^2\right )\right )+\frac {1}{2} (b c-a d)^2 (b c+a d) \tan (e+f x)+\frac {1}{4} b d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )}\\ &=-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {\left (b^3 \left (4 a b c-7 a^2 d-3 b^2 d\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)^2}-\frac {2 \int \frac {-\frac {1}{2} (b c-a d)^2 \left (a^2 c-b^2 c-2 a b d\right )+\frac {1}{2} (b c-a d)^2 \left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)^2 \left (c^2+d^2\right )}\\ &=-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \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 {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2 (c+i d)}+\frac {\left (b^3 \left (4 a b c-7 a^2 d-3 b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f}\\ &=-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 (i c-d) f}-\frac {\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 (i c+d) f}+\frac {\left (b^3 \left (4 a b c-7 a^2 d-3 b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2 d (b c-a d)^2 f}\\ &=-\frac {b^{5/2} \left (4 a b c-7 a^2 d-3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{5/2} f}-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {\operatorname {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 )}{(a-i b)^2 (c-i d) d f}-\frac {\operatorname {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 )}{(a+i b)^2 (c+i d) d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 (c-i d)^{3/2} f}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 (c+i d)^{3/2} f}-\frac {b^{5/2} \left (4 a b c-7 a^2 d-3 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{5/2} f}-\frac {d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.25, size = 628, normalized size = 2.00 \[ -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {-\frac {2 \left (\frac {1}{2} d^2 \left (2 a^2 d-2 a b c+3 b^2 d\right )-c \left (b d (b c-a d)-\frac {3}{2} b^2 c d\right )\right )}{f \left (c^2+d^2\right ) (a d-b c) \sqrt {c+d \tan (e+f x)}}-\frac {2 \left (\frac {2 \sqrt {b c-a d} \left (\frac {1}{4} a^2 b d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )+\frac {1}{4} b^2 \left (-2 a^3 c d^2+4 a^2 b d \left (c^2+d^2\right )-2 a b^2 c \left (c^2+2 d^2\right )+3 b^3 d \left (c^2+d^2\right )\right )-\frac {1}{2} a b (b c-a d)^2 (a d+b c)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right ) (a d-b c)}+\frac {\frac {i \sqrt {c-i d} \left (-\frac {1}{2} \left (a^2 c-2 a b d-b^2 c\right ) (b c-a d)^2-\frac {1}{2} i \left (a^2 d+2 a b c-b^2 d\right ) (b c-a d)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (-c+i d)}-\frac {i \sqrt {c+i d} \left (-\frac {1}{2} (b c-a d)^2 \left (a^2 c-2 a b d-b^2 c\right )+\frac {1}{2} i (b c-a d)^2 \left (a^2 d+2 a b c-b^2 d\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-c-i d)}}{a^2+b^2}\right )}{\left (c^2+d^2\right ) (a d-b c)}}{\left (a^2+b^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [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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giac [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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maple [B] time = 0.43, size = 12889, normalized size = 41.05 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \text {Hanged} \]
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 \[ \int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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