Optimal. Leaf size=436 \[ \frac{a^{3/2} \left (a^2 A b-3 a^3 B-7 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} d \left (a^2+b^2\right )^2}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (-3 a^2 B+a A b-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 d \left (a^2+b^2\right )}+\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.16191, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {3605, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{a^{3/2} \left (a^2 A b-3 a^3 B-7 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} d \left (a^2+b^2\right )^2}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (-3 a^2 B+a A b-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 d \left (a^2+b^2\right )}+\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3605
Rule 3647
Rule 3653
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{\int \frac{\sqrt{\tan (c+d x)} \left (-\frac{3}{2} a (A b-a B)+b (A b-a B) \tan (c+d x)-\frac{1}{2} \left (a A b-3 a^2 B-2 b^2 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{2 \int \frac{\frac{1}{4} a \left (a A b-3 a^2 B-2 b^2 B\right )-\frac{1}{2} b^2 (a A+b B) \tan (c+d x)+\frac{1}{4} \left (a^2 A b+2 A b^3-3 a^3 B-4 a b^2 B\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{2 \int \frac{-\frac{1}{2} b^2 \left (2 a A b-a^2 B+b^2 B\right )-\frac{1}{2} b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )^2}+\frac{\left (a^2 \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )\right ) \int \frac{1+\tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{4 \operatorname{Subst}\left (\int \frac{-\frac{1}{2} b^2 \left (2 a A b-a^2 B+b^2 B\right )-\frac{1}{2} b^2 \left (a^2 A-A b^2+2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}+\frac{\left (a^2 \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 b^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{\left (a^2 \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^2 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}\\ &=\frac{a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}\\ &=\frac{a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}\\ &=\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{a^{3/2} \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a A b-3 a^2 B-2 b^2 B\right ) \sqrt{\tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac{a (A b-a B) \tan ^{\frac{3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 2.2061, size = 275, normalized size = 0.63 \[ \frac{2 \left (\frac{\left (a^2 A b-3 a^3 B-4 a b^2 B+2 A b^3\right ) \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right )}-\frac{(a+b \tan (c+d x)) \left (a^{3/2} \left (-a^2 A b+3 a^3 B+7 a b^2 B-5 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )+\sqrt [4]{-1} b^{5/2} (a+i b)^2 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )+(-1)^{3/4} b^{5/2} (b+i a)^2 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )\right )}{2 \sqrt{b} \left (a^2+b^2\right )^2}+(3 a B-A b) \sqrt{\tan (c+d x)}+b B \tan ^{\frac{3}{2}}(c+d x)\right )}{b^2 d (a+b \tan (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.063, size = 1160, normalized size = 2.7 \begin{align*} \text{result 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(-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]
________________________________________________________________________________________
Giac [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]