Optimal. Leaf size=243 \[ -\frac{2 b \left (5 a^2+3 b^2\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (5 a^2+3 b^2\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34118, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {3016, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 b \left (5 a^2+3 b^2\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (5 a^2+3 b^2\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3016
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx &=-\int \frac{-a+b \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx\\ &=-\frac{4 a b \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \int \frac{\frac{3}{2} \left (a^2+b^2\right )-a b \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac{4 a b \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 b \left (5 a^2+3 b^2\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 \int \frac{-\frac{1}{4} a \left (3 a^2+5 b^2\right )-\frac{1}{4} b \left (5 a^2+3 b^2\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=-\frac{4 a b \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 b \left (5 a^2+3 b^2\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(2 a) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{3 \left (a^2-b^2\right )}+\frac{\left (5 a^2+3 b^2\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=-\frac{4 a b \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 b \left (5 a^2+3 b^2\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\left (\left (5 a^2+3 b^2\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{3 \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (2 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{3 \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (5 a^2+3 b^2\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 \left (a^2-b^2\right )^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3 \left (a^2-b^2\right ) d \sqrt{a+b \cos (c+d x)}}-\frac{4 a b \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 b \left (5 a^2+3 b^2\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.979563, size = 158, normalized size = 0.65 \[ \frac{2 \left (\frac{\left (\frac{a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (\left (5 a^2+3 b^2\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+2 a (b-a) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )}{(a-b)^2}-\frac{b \sin (c+d x) \left (b \left (5 a^2+3 b^2\right ) \cos (c+d x)+a \left (7 a^2+b^2\right )\right )}{\left (a^2-b^2\right )^2}\right )}{3 d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 1.384, size = 792, normalized size = 3.3 \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{b^{2} \cos \left (d x + c\right )^{2} - a^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \cos \left (d x + c\right ) + a}{\left (b \cos \left (d x + c\right ) - a\right )}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \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{b^{2} \cos \left (d x + c\right )^{2} - a^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]