Optimal. Leaf size=90 \[ \frac{(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^2 d}-\frac{(2 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d} \]
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Rubi [A] time = 0.14345, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3676, 414, 522, 206, 208} \[ \frac{(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^2 d}-\frac{(2 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 3676
Rule 414
Rule 522
Rule 206
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \left (a-(a-b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\sec (c+d x) \tan (c+d x)}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{-a+2 b+(-a+b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{2 b d}\\ &=\frac{\sec (c+d x) \tan (c+d x)}{2 b d}-\frac{(2 a-3 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 b^2 d}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{b^2 d}\\ &=-\frac{(2 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^2 d}+\frac{\sec (c+d x) \tan (c+d x)}{2 b d}\\ \end{align*}
Mathematica [B] time = 1.32839, size = 207, normalized size = 2.3 \[ \frac{-\frac{2 (a-b)^{3/2} \log \left (\sqrt{a}-\sqrt{a-b} \sin (c+d x)\right )}{\sqrt{a}}+\frac{2 (a-b)^{3/2} \log \left (\sqrt{a-b} \sin (c+d x)+\sqrt{a}\right )}{\sqrt{a}}+2 (2 a-3 b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 (3 b-2 a) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{b}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{b}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}}{4 b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.087, size = 224, normalized size = 2.5 \begin{align*}{\frac{{a}^{2}}{d{b}^{2}}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}-2\,{\frac{a}{db\sqrt{a \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \sin \left ( dx+c \right ) }{\sqrt{a \left ( a-b \right ) }}} \right ) }+{\frac{1}{d}{\it Artanh} \left ({ \left ( a-b \right ) \sin \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a-b \right ) }}}}-{\frac{1}{4\,db \left ( \sin \left ( dx+c \right ) +1 \right ) }}-{\frac{\ln \left ( \sin \left ( dx+c \right ) +1 \right ) a}{2\,d{b}^{2}}}+{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) +1 \right ) }{4\,db}}-{\frac{1}{4\,db \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) a}{2\,d{b}^{2}}}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{4\,db}} \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 [A] time = 2.03724, size = 724, normalized size = 8.04 \begin{align*} \left [-\frac{2 \,{\left (a - b\right )} \sqrt{\frac{a - b}{a}} \cos \left (d x + c\right )^{2} \log \left (-\frac{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) +{\left (2 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, b \sin \left (d x + c\right )}{4 \, b^{2} d \cos \left (d x + c\right )^{2}}, -\frac{4 \,{\left (a - b\right )} \sqrt{-\frac{a - b}{a}} \arctan \left (\sqrt{-\frac{a - b}{a}} \sin \left (d x + c\right )\right ) \cos \left (d x + c\right )^{2} +{\left (2 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, b \sin \left (d x + c\right )}{4 \, b^{2} d \cos \left (d x + c\right )^{2}}\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 ^{5}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.75967, size = 177, normalized size = 1.97 \begin{align*} -\frac{\frac{{\left (2 \, a - 3 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b^{2}} - \frac{{\left (2 \, a - 3 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b^{2}} - \frac{4 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (-\frac{a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} b^{2}} + \frac{2 \, \sin \left (d x + c\right )}{{\left (\sin \left (d x + c\right )^{2} - 1\right )} b}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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