Optimal. Leaf size=301 \[ -\frac{2 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2}{f \left (a^2+b^2\right ) (b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2} (c+i d)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.26918, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{2 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{2 b^2}{f \left (a^2+b^2\right ) (b c-a d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2} (c-i d)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2} (c+i d)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
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))^{3/2} (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 b^2}{\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 \int \frac{\frac{1}{2} \left (2 b^2 d-a (b c-a d)\right )+\frac{1}{2} b (b c-a d) \tan (e+f x)+b^2 d \tan ^2(e+f x)}{\sqrt{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{2 b^2}{\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 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{\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{4 \int \frac{-\frac{1}{4} (b c-a d)^2 (a c-b d)+\frac{1}{4} (b c-a d)^2 (b c+a d) \tan (e+f x)}{\sqrt{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{2 b^2}{\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 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{\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{\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) (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) (c+i d)}\\ &=-\frac{2 b^2}{\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 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{\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{\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) (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) (c+i d) f}\\ &=-\frac{2 b^2}{\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 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{\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{\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) (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) (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)^{3/2} (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)^{3/2} (c+i d)^{3/2} f}-\frac{2 b^2}{\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 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{\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)}}\\ \end{align*}
Mathematica [A] time = 4.6199, size = 349, normalized size = 1.16 \[ -\frac{\frac{2 d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \sqrt{a+b \tan (e+f x)}}{\left (c^2+d^2\right ) (b c-a d) \sqrt{c+d \tan (e+f x)}}+\frac{2 b^2}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}+\frac{(b c-a d)^2 \left (\frac{(b+i a) (c-i d) \tan ^{-1}\left (\frac{\sqrt{-c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+i b} \sqrt{-c-i d}}+\frac{i (a+i b) (c+i d) \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{-a+i b} \sqrt{c-i d}}\right )}{\left (c^2+d^2\right ) (a d-b c)}}{f \left (a^2+b^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{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.
[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 )^{\frac{3}{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(-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.
[In]
[Out]