Optimal. Leaf size=385 \[ \frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}+\frac{\left (-26 a^2 b^2+3 a^4+3 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b} d \left (a^2+b^2\right )^3}+\frac{a \sqrt{\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.805196, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {3567, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}+\frac{\left (-26 a^2 b^2+3 a^4+3 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b} d \left (a^2+b^2\right )^3}+\frac{a \sqrt{\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3}-\frac{(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3567
Rule 3649
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{3}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{\int \frac{\frac{a}{2}-2 b \tan (c+d x)-\frac{3}{2} a \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx}{2 \left (a^2+b^2\right )}\\ &=\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\int \frac{\frac{1}{4} a \left (5 a^2-3 b^2\right )-4 a^2 b \tan (c+d x)-\frac{1}{4} a \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a \left (a^2+b^2\right )^2}\\ &=\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\int \frac{2 a^2 \left (a^2-3 b^2\right )-2 a b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{2 a \left (a^2+b^2\right )^3}+\frac{\left (3 a^4-26 a^2 b^2+3 b^4\right ) \int \frac{1+\tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 \left (a^2+b^2\right )^3}\\ &=\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\operatorname{Subst}\left (\int \frac{2 a^2 \left (a^2-3 b^2\right )-2 a b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a \left (a^2+b^2\right )^3 d}+\frac{\left (3 a^4-26 a^2 b^2+3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}\\ &=\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}-\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}+\frac{\left (3 a^4-26 a^2 b^2+3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right )^3 d}\\ &=\frac{\left (3 a^4-26 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b} \left (a^2+b^2\right )^3 d}+\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}+\frac{\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}\\ &=\frac{\left (3 a^4-26 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b} \left (a^2+b^2\right )^3 d}+\frac{(a-b) \left (a^2+4 a b+b^2\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 )^3 d}-\frac{(a-b) \left (a^2+4 a b+b^2\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 )^3 d}+\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}+\frac{\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}\\ &=\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d}-\frac{(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d}+\frac{\left (3 a^4-26 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b} \left (a^2+b^2\right )^3 d}+\frac{(a-b) \left (a^2+4 a b+b^2\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 )^3 d}-\frac{(a-b) \left (a^2+4 a b+b^2\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 )^3 d}+\frac{a \sqrt{\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\left (3 a^2-5 b^2\right ) \sqrt{\tan (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 3.47426, size = 326, normalized size = 0.85 \[ \frac{\frac{(a+b \tan (c+d x)) \left (a^{5/2} b^{5/2} \left (a^2+b^2\right )^2 \sqrt{\tan (c+d x)}+\frac{1}{2} \left (a^{5/2} b^{5/2} \left (3 a^2-5 b^2\right ) \left (a^2+b^2\right ) \sqrt{\tan (c+d x)}+(a+b \tan (c+d x)) \left (4 \sqrt [4]{-1} a^{5/2} b^{5/2} (a+i b)^3 \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )+a^2 b^2 \left (-26 a^2 b^2+3 a^4+3 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )+4 \sqrt [4]{-1} a^{5/2} b^{5/2} (a-i b)^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )\right )\right )\right )}{a^{3/2} \left (a^2+b^2\right )^2}-b^{7/2} \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))+b^{9/2} \tan ^{\frac{5}{2}}(c+d x)}{2 a b^{5/2} d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 895, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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(-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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