Optimal. Leaf size=68 \[ \frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \]
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Rubi [A] time = 0.234762, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {4264, 3859, 2807, 2805} \[ \frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3859
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\sqrt{b+a \cos (c+d x)} \int \frac{\sec (c+d x)}{\sqrt{b+a \cos (c+d x)}} \, dx}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}\\ &=\frac{\sqrt{\frac{b+a \cos (c+d x)}{a+b}} \int \frac{\sec (c+d x)}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}\\ &=\frac{2 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 28.4472, size = 14986, normalized size = 220.38 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.234, size = 208, normalized size = 3.1 \begin{align*} -2\,{\frac{\sqrt{\cos \left ( dx+c \right ) }}{d \left ( b+a\cos \left ( dx+c \right ) \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}}\sqrt{{\frac{b+a\cos \left ( dx+c \right ) }{ \left ( a+b \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) }}} \left ( 2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }\sqrt{{\frac{a-b}{a+b}}}},{\frac{a+b}{a-b}},{i{\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \right ) -{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }\sqrt{{\frac{a-b}{a+b}}}},\sqrt{-{\frac{a+b}{a-b}}} \right ) \right ) \sqrt{{\frac{b+a\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \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{1}{\sqrt{b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \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 \frac{1}{\sqrt{b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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