Optimal. Leaf size=257 \[ -\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{32 \sqrt{2} b d^{3/2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{32 \sqrt{2} b d^{3/2}}+\frac{3 \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{64 \sqrt{2} b d^{3/2}}-\frac{3 \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{64 \sqrt{2} b d^{3/2}}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3} \]
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Rubi [A] time = 0.181913, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {2591, 288, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{32 \sqrt{2} b d^{3/2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{32 \sqrt{2} b d^{3/2}}+\frac{3 \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{64 \sqrt{2} b d^{3/2}}-\frac{3 \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{64 \sqrt{2} b d^{3/2}}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 288
Rule 290
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{x^{5/2}}{\left (d^2+x^2\right )^3} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (d^2+x^2\right )^2} \, dx,x,d \tan (a+b x)\right )}{8 b}\\ &=\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{32 b d}\\ &=\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{16 b d}\\ &=\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}-\frac{3 \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{32 b d}+\frac{3 \operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{32 b d}\\ &=\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \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 )}{64 \sqrt{2} b d^{3/2}}+\frac{3 \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 )}{64 \sqrt{2} b d^{3/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{64 b d}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{64 b d}\\ &=\frac{3 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{64 \sqrt{2} b d^{3/2}}-\frac{3 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{64 \sqrt{2} b d^{3/2}}+\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{32 \sqrt{2} b d^{3/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{32 \sqrt{2} b d^{3/2}}\\ &=-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{32 \sqrt{2} b d^{3/2}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{32 \sqrt{2} b d^{3/2}}+\frac{3 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{64 \sqrt{2} b d^{3/2}}-\frac{3 \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{64 \sqrt{2} b d^{3/2}}+\frac{3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac{\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3}\\ \end{align*}
Mathematica [A] time = 0.326889, size = 123, normalized size = 0.48 \[ \frac{\csc (a+b x) \sqrt{d \tan (a+b x)} \left (\cos (a+b x)-2 \cos (3 (a+b x))+\cos (5 (a+b x))-3 \sqrt{\sin (2 (a+b x))} \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-3 \sqrt{\sin (2 (a+b x))} \log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{64 b d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.14, size = 558, normalized size = 2.2 \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 [A] time = 1.40565, size = 348, normalized size = 1.35 \begin{align*} \frac{1}{128} \, d{\left (\frac{6 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{b d^{4}} + \frac{6 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{b d^{4}} - \frac{3 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt{2} \sqrt{d \tan \left (b x + a\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{b d^{4}} + \frac{3 \, \sqrt{2}{\left | d \right |}^{\frac{3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt{2} \sqrt{d \tan \left (b x + a\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{b d^{4}} + \frac{8 \,{\left (3 \, \sqrt{d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )^{3} - \sqrt{d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b d^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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