Optimal. Leaf size=133 \[ \frac{2 b^2 B \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a+b)}+\frac{2 b B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{2 b B \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.559779, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {21, 2802, 3055, 3059, 2639, 3002, 2641, 2805} \[ \frac{2 b^2 B \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a+b)}+\frac{2 b B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{2 b B \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2802
Rule 3055
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx &=B \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx\\ &=\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(2 B) \int \frac{-\frac{3 b}{2}+\frac{1}{2} a \cos (c+d x)+\frac{1}{2} b \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{3 a}\\ &=\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b B \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{(4 B) \int \frac{\frac{1}{4} \left (a^2+3 b^2\right )+a b \cos (c+d x)+\frac{3}{4} b^2 \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^2}\\ &=\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b B \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(4 B) \int \frac{-\frac{1}{4} b \left (a^2+3 b^2\right )-\frac{1}{4} a b^2 \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^2 b}+\frac{(b B) \int \sqrt{\cos (c+d x)} \, dx}{a^2}\\ &=\frac{2 b B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b B \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{B \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a}+\frac{\left (b^2 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^2}\\ &=\frac{2 b B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 B F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{2 b^2 B \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 (a+b) d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b B \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.06872, size = 215, normalized size = 1.62 \[ \frac{B \left (\frac{2 \left (2 a^2+9 b^2\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{6 \sin (c+d x) \left (\left (2 a^2-b^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a \sqrt{\sin ^2(c+d x)}}+\frac{4 \sin (c+d x) (a-3 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+8 a \left (2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\frac{2 a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )\right )}{6 a^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 8.419, size = 452, normalized size = 3.4 \begin{align*} -2\,{\frac{\sqrt{- \left ( -2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}B}{\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d} \left ( -2\,{\frac{{b}^{3}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}}{{a}^{2} \left ( -2\,ab+2\,{b}^{2} \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,-2\,{\frac{b}{a-b}},\sqrt{2} \right ) }-{\frac{b \left ( -\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +2\,\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }{{a}^{2} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }}+{\frac{1}{a} \left ( -1/6\,{\frac{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\cos \left ( 1/2\,dx+c/2 \right ) }{ \left ( \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1/2 \right ) ^{2}}}+1/3\,{\frac{\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) }{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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{B b \cos \left (d x + c\right ) + B a}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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