Optimal. Leaf size=225 \[ -\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{4 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b}-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b} \]
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Rubi [A] time = 0.162636, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2607, 288, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{4 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{8 \sqrt{2} b}-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 288
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(d x)^{3/2}}{\left (1+x^2\right )^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{d x} \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{4 b}\\ &=-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}+\frac{d \operatorname{Subst}\left (\int \frac{1}{1+\frac{x^4}{d^2}} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{2 b}\\ &=-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{d-x^2}{1+\frac{x^4}{d^2}} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{4 b}+\frac{\operatorname{Subst}\left (\int \frac{d+x^2}{1+\frac{x^4}{d^2}} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{4 b}\\ &=-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}-\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b}-\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 b}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{8 b}\\ &=-\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b}-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}+\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b}-\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b}\\ &=-\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b}+\frac{d^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{4 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{8 \sqrt{2} b}-\frac{d \cos ^2(a+b x) \sqrt{d \tan (a+b x)}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.258561, size = 110, normalized size = 0.49 \[ -\frac{d \csc (a+b x) \sqrt{d \tan (a+b x)} \left (\sin (a+b x)+\sin (3 (a+b x))+\sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-\sqrt{\sin (2 (a+b x))} \log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.143, size = 660, normalized size = 2.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{3}{2}} \cos \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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