Optimal. Leaf size=314 \[ -\frac{d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{b^{5/2} \left (-7 a^2 d+4 a b c-3 b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{5/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2 (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2 (c+i d)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.47136, antiderivative size = 314, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 3569
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
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*}
Mathematica [A] time = 6.22077, size = 628, normalized size = 2. \[ -\frac{b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{-\frac{2 \left (\frac{1}{2} d^2 \left (2 a^2 d-2 a b c+3 b^2 d\right )-c \left (b d (b c-a d)-\frac{3}{2} b^2 c d\right )\right )}{f \left (c^2+d^2\right ) (a d-b c) \sqrt{c+d \tan (e+f x)}}-\frac{2 \left (\frac{2 \sqrt{b c-a d} \left (\frac{1}{4} a^2 b d \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )+\frac{1}{4} b^2 \left (4 a^2 b d \left (c^2+d^2\right )-2 a^3 c d^2-2 a b^2 c \left (c^2+2 d^2\right )+3 b^3 d \left (c^2+d^2\right )\right )-\frac{1}{2} a b (b c-a d)^2 (a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\sqrt{b} f \left (a^2+b^2\right ) (a d-b c)}+\frac{\frac{i \sqrt{c-i d} \left (-\frac{1}{2} \left (a^2 c-2 a b d-b^2 c\right ) (b c-a d)^2-\frac{1}{2} i \left (a^2 d+2 a b c-b^2 d\right ) (b c-a d)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (-c+i d)}-\frac{i \sqrt{c+i d} \left (-\frac{1}{2} (b c-a d)^2 \left (a^2 c-2 a b d-b^2 c\right )+\frac{1}{2} i (b c-a d)^2 \left (a^2 d+2 a b c-b^2 d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (-c-i d)}}{a^2+b^2}\right )}{\left (c^2+d^2\right ) (a d-b c)}}{\left (a^2+b^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.103, size = 12889, normalized size = 41.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \tan \left (f x + e\right ) + a\right )}^{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.
[In]
[Out]