Optimal. Leaf size=417 \[ \frac{2 d \left (a^2 b^2 d \left (11 c^2+17 d^2\right )+3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{3 f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2} (c+i d)^{3/2}} \]
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Rubi [A] time = 1.97629, antiderivative size = 417, normalized size of antiderivative = 1., 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 = {3569, 3649, 3616, 3615, 93, 208} \[ \frac{2 d \left (a^2 b^2 d \left (11 c^2+17 d^2\right )+3 a^4 d^3-6 a b^3 c \left (c^2+d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{3 f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 \sqrt{c+d \tan (e+f x)}}-\frac{4 b^2 \left (-5 a^2 d+3 a b c-2 b^2 d\right )}{3 f \left (a^2+b^2\right )^2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2} (c+i d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 208
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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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*}
Mathematica [A] time = 6.44415, size = 601, normalized size = 1.44 \[ -\frac{2 b^2}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}-\frac{2 \left (-\frac{2 \left (\frac{1}{2} b^2 \left (4 b^2 d-3 a (b c-a d)\right )-a \left (\frac{3}{2} b^2 (b c-a d)-2 a b^2 d\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{2 \left (-\frac{2 \left (\frac{1}{4} d^2 \left (a^2 b^2 \left (3 c^2+17 d^2\right )-6 a^3 b c d+3 a^4 d^2-6 a b^3 c d-b^4 \left (3 c^2-8 d^2\right )\right )-c \left (b^2 c d \left (-5 a^2 d+3 a b c-2 b^2 d\right )-\frac{3}{2} a b d (b c-a d)^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{f \left (c^2+d^2\right ) (a d-b c) \sqrt{c+d \tan (e+f x)}}-\frac{3 (b c-a d)^3 \left (\frac{(a-i b)^2 (d+i c) \tan ^{-1}\left (\frac{\sqrt{-c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+i b} \sqrt{-c-i d}}+\frac{(a+i b)^2 (-d+i c) \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{-a+i b} \sqrt{c-i d}}\right )}{4 f \left (c^2+d^2\right ) (a d-b c)}\right )}{\left (a^2+b^2\right ) (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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