Optimal. Leaf size=384 \[ \frac{e^{5/2} \left (b^2-a^2\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} d}-\frac{e^{5/2} \left (b^2-a^2\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} d}-\frac{a e^3 \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}-\frac{a e^3 \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.869977, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2695, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{e^{5/2} \left (b^2-a^2\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} d}-\frac{e^{5/2} \left (b^2-a^2\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} d}-\frac{a e^3 \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}-\frac{a e^3 \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 d \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2695
Rule 2867
Rule 2640
Rule 2639
Rule 2701
Rule 2807
Rule 2805
Rule 329
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx &=\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac{e^2 \int \frac{\sqrt{e \cos (c+d x)} (b+a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b}\\ &=\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac{\left (a e^2\right ) \int \sqrt{e \cos (c+d x)} \, dx}{b^2}+\frac{\left (\left (-a^2+b^2\right ) e^2\right ) \int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{b^2}\\ &=\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac{\left (a \left (a^2-b^2\right ) e^3\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{2 b^3}-\frac{\left (a \left (a^2-b^2\right ) e^3\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{2 b^3}-\frac{\left (\left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{b d}+\frac{\left (a e^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{b^2 \sqrt{\cos (c+d x)}}\\ &=\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac{2 a e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}-\frac{\left (2 \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{b d}+\frac{\left (a \left (a^2-b^2\right ) e^3 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{2 b^3 \sqrt{e \cos (c+d x)}}-\frac{\left (a \left (a^2-b^2\right ) e^3 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{2 b^3 \sqrt{e \cos (c+d x)}}\\ &=\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac{2 a e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}-\frac{a \left (a^2-b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{a \left (a^2-b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{\left (\left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e-b x^2} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{b^2 d}-\frac{\left (\left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e+b x^2} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{b^2 d}\\ &=\frac{\left (-a^2+b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{b^{5/2} d}-\frac{\left (-a^2+b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{b^{5/2} d}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac{2 a e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}-\frac{a \left (a^2-b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{a \left (a^2-b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^3 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 21.4337, size = 709, normalized size = 1.85 \[ \frac{(e \cos (c+d x))^{5/2} \left (-\frac{a \left (a+b \sqrt{\sin ^2(c+d x)}\right ) \left (8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (-\log \left (-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right )+\log \left (\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )\right )\right )}{4 b^{3/2} \left (b^2-a^2\right ) (a+b \sin (c+d x))}-\frac{6 b \sin (c+d x) \left (a+b \sqrt{\sin ^2(c+d x)}\right ) \left (\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )}{3 \left (a^2-b^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (-\log \left (-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right )+\log \left ((1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right )+2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )\right )}{\sqrt{b} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{\sin ^2(c+d x)} (a+b \sin (c+d x))}+2 \cos ^{\frac{3}{2}}(c+d x)\right )}{3 b d \cos ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 2.72, size = 1131, normalized size = 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{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{b \sin \left (d x + c\right ) + a}\,{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{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{b \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]