Optimal. Leaf size=223 \[ \frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac{\left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \left (a^2-b^2\right )}-\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \left (a^2-b^2\right )}-\frac{a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a-b) (a+b)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.411831, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3238, 3843, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac{\left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \left (a^2-b^2\right )}-\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b d \left (a^2-b^2\right )}-\frac{a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3238
Rule 3843
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx &=\int \frac{\sqrt{\sec (c+d x)}}{(b+a \sec (c+d x))^2} \, dx\\ &=\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{\int \frac{-\frac{a}{2}-b \sec (c+d x)+\frac{1}{2} a \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{a^2-b^2}\\ &=\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d (b+a \sec (c+d x))}+\frac{\left (a \left (3-\frac{a^2}{b^2}\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{2 \left (a^2-b^2\right )}+\frac{\int \frac{-\frac{a b}{2}-\left (-\frac{a^2}{2}+b^2\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d (b+a \sec (c+d x))}-\frac{a \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{2 b \left (a^2-b^2\right )}+\frac{\left (a^2-2 b^2\right ) \int \sqrt{\sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}+\frac{\left (a \left (3-\frac{a^2}{b^2}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{2 \left (a^2-b^2\right )}\\ &=-\frac{a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{(a-b) b^2 (a+b)^2 d}+\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d (b+a \sec (c+d x))}-\frac{\left (a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 b \left (a^2-b^2\right )}+\frac{\left (\left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=-\frac{a \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{b \left (a^2-b^2\right ) d}+\frac{\left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{b^2 \left (a^2-b^2\right ) d}-\frac{a \left (a^2-3 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{(a-b) b^2 (a+b)^2 d}+\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d (b+a \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 5.14587, size = 251, normalized size = 1.13 \[ \frac{\cos (2 (c+d x)) \csc (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (\left (a^2-3 b^2\right ) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-b (b-a) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+a b \left (a \tan ^2(c+d x)-\sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )\right )}{b^2 d (a-b) (a+b) \left (\sec ^2(c+d x)-2\right ) (a \sec (c+d x)+b)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 6.832, size = 794, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]