Optimal. Leaf size=82 \[ -\frac{\sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b d^2 \sqrt{d \tan (a+b x)}}-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0873308, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2609, 2614, 2573, 2641} \[ -\frac{\sqrt{\sin (2 a+2 b x)} \sec (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b d^2 \sqrt{d \tan (a+b x)}}-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2609
Rule 2614
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec (a+b x)}{(d \tan (a+b x))^{5/2}} \, dx &=-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}}-\frac{\int \frac{\sec (a+b x)}{\sqrt{d \tan (a+b x)}} \, dx}{3 d^2}\\ &=-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}}-\frac{\sqrt{\sin (a+b x)} \int \frac{1}{\sqrt{\cos (a+b x)} \sqrt{\sin (a+b x)}} \, dx}{3 d^2 \sqrt{\cos (a+b x)} \sqrt{d \tan (a+b x)}}\\ &=-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}}-\frac{\left (\sec (a+b x) \sqrt{\sin (2 a+2 b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx}{3 d^2 \sqrt{d \tan (a+b x)}}\\ &=-\frac{2 \sec (a+b x)}{3 b d (d \tan (a+b x))^{3/2}}-\frac{F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt{\sin (2 a+2 b x)}}{3 b d^2 \sqrt{d \tan (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.694892, size = 113, normalized size = 1.38 \[ \frac{2 \cos (2 (a+b x)) \csc (a+b x) \sqrt{\sec ^2(a+b x)} \left (\sqrt{\sec ^2(a+b x)}-\sqrt [4]{-1} \tan ^{\frac{3}{2}}(a+b x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (a+b x)}\right )\right |-1\right )\right )}{3 b d^2 \left (\tan ^2(a+b x)-1\right ) \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 302, normalized size = 3.7 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( bx+a \right ) -1 \right ) ^{2} \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}{3\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{3}} \left ( \sin \left ( bx+a \right ) \cos \left ( bx+a \right ){\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{\cos \left ( bx+a \right ) -1}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{\cos \left ( bx+a \right ) -1+\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}+\sin \left ( bx+a \right ) \sqrt{{\frac{\cos \left ( bx+a \right ) -1}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{\cos \left ( bx+a \right ) -1+\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +\cos \left ( bx+a \right ) \sqrt{2} \right ) \left ({\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \tan \left (b x + a\right )} \sec \left (b x + a\right )}{d^{3} \tan \left (b x + a\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (a + b x \right )}}{\left (d \tan{\left (a + b x \right )}\right )^{\frac{5}{2}}}\, dx \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 \frac{\sec \left (b x + a\right )}{\left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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