Optimal. Leaf size=557 \[ \frac{b^2 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}-\frac{\sqrt{b} g^{5/2} \left (b^2-a^2\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{a^3 f}-\frac{b^2 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}+\frac{\sqrt{b} g^{5/2} \left (b^2-a^2\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{a^3 f}+\frac{g^3 \left (a^2-b^2\right ) \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \left (b-\sqrt{b^2-a^2}\right ) \sqrt{g \cos (e+f x)}}+\frac{g^3 \left (a^2-b^2\right ) \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \left (\sqrt{b^2-a^2}+b\right ) \sqrt{g \cos (e+f x)}}+\frac{b g^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{a^2 f \sqrt{\cos (e+f x)}}+\frac{b g \csc (e+f x) (g \cos (e+f x))^{3/2}}{a^2 f}-\frac{3 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}+\frac{3 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}-\frac{g \csc ^2(e+f x) (g \cos (e+f x))^{3/2}}{2 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.34346, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {2898, 2565, 321, 329, 298, 203, 206, 2567, 2640, 2639, 288, 2695, 2867, 2701, 2807, 2805, 205, 208} \[ \frac{b^2 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}-\frac{\sqrt{b} g^{5/2} \left (b^2-a^2\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{a^3 f}-\frac{b^2 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}+\frac{\sqrt{b} g^{5/2} \left (b^2-a^2\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt [4]{b^2-a^2}}\right )}{a^3 f}+\frac{g^3 \left (a^2-b^2\right ) \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \left (b-\sqrt{b^2-a^2}\right ) \sqrt{g \cos (e+f x)}}+\frac{g^3 \left (a^2-b^2\right ) \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \left (\sqrt{b^2-a^2}+b\right ) \sqrt{g \cos (e+f x)}}+\frac{b g^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{a^2 f \sqrt{\cos (e+f x)}}+\frac{b g \csc (e+f x) (g \cos (e+f x))^{3/2}}{a^2 f}-\frac{3 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}+\frac{3 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}-\frac{g \csc ^2(e+f x) (g \cos (e+f x))^{3/2}}{2 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2898
Rule 2565
Rule 321
Rule 329
Rule 298
Rule 203
Rule 206
Rule 2567
Rule 2640
Rule 2639
Rule 288
Rule 2695
Rule 2867
Rule 2701
Rule 2807
Rule 2805
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{5/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx &=\int \left (\frac{b^2 (g \cos (e+f x))^{5/2} \csc (e+f x)}{a^3}-\frac{b (g \cos (e+f x))^{5/2} \csc ^2(e+f x)}{a^2}+\frac{(g \cos (e+f x))^{5/2} \csc ^3(e+f x)}{a}-\frac{b^3 (g \cos (e+f x))^{5/2}}{a^3 (a+b \sin (e+f x))}\right ) \, dx\\ &=\frac{\int (g \cos (e+f x))^{5/2} \csc ^3(e+f x) \, dx}{a}-\frac{b \int (g \cos (e+f x))^{5/2} \csc ^2(e+f x) \, dx}{a^2}+\frac{b^2 \int (g \cos (e+f x))^{5/2} \csc (e+f x) \, dx}{a^3}-\frac{b^3 \int \frac{(g \cos (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx}{a^3}\\ &=-\frac{2 b^2 g (g \cos (e+f x))^{3/2}}{3 a^3 f}+\frac{b g (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^2 f}-\frac{\operatorname{Subst}\left (\int \frac{x^{5/2}}{\left (1-\frac{x^2}{g^2}\right )^2} \, dx,x,g \cos (e+f x)\right )}{a f g}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x^{5/2}}{1-\frac{x^2}{g^2}} \, dx,x,g \cos (e+f x)\right )}{a^3 f g}+\frac{\left (3 b g^2\right ) \int \sqrt{g \cos (e+f x)} \, dx}{2 a^2}-\frac{\left (b^2 g^2\right ) \int \frac{\sqrt{g \cos (e+f x)} (b+a \sin (e+f x))}{a+b \sin (e+f x)} \, dx}{a^3}\\ &=\frac{b g (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^2 f}-\frac{g (g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{2 a f}+\frac{(3 g) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{g^2}} \, dx,x,g \cos (e+f x)\right )}{4 a f}-\frac{\left (b^2 g\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{g^2}} \, dx,x,g \cos (e+f x)\right )}{a^3 f}-\frac{\left (b g^2\right ) \int \sqrt{g \cos (e+f x)} \, dx}{a^2}+\frac{\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac{\sqrt{g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{a^3}+\frac{\left (3 b g^2 \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{2 a^2 \sqrt{\cos (e+f x)}}\\ &=\frac{b g (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^2 f}-\frac{g (g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{2 a f}+\frac{3 b g^2 \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \sqrt{\cos (e+f x)}}+\frac{(3 g) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{g^2}} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{2 a f}-\frac{\left (2 b^2 g\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{g^2}} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{a^3 f}-\frac{\left (\left (a^2-b^2\right ) g^3\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \left (\sqrt{-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 a^2}+\frac{\left (\left (a^2-b^2\right ) g^3\right ) \int \frac{1}{\sqrt{g \cos (e+f x)} \left (\sqrt{-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 a^2}+\frac{\left (b^2 \left (a^2-b^2\right ) g^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{a^3 f}-\frac{\left (b g^2 \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{a^2 \sqrt{\cos (e+f x)}}\\ &=\frac{b g (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^2 f}-\frac{g (g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{2 a f}+\frac{b g^2 \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \sqrt{\cos (e+f x)}}+\frac{\left (3 g^3\right ) \operatorname{Subst}\left (\int \frac{1}{g-x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{4 a f}-\frac{\left (3 g^3\right ) \operatorname{Subst}\left (\int \frac{1}{g+x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{4 a f}-\frac{\left (b^2 g^3\right ) \operatorname{Subst}\left (\int \frac{1}{g-x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{a^3 f}+\frac{\left (b^2 g^3\right ) \operatorname{Subst}\left (\int \frac{1}{g+x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{a^3 f}+\frac{\left (2 b^2 \left (a^2-b^2\right ) g^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{a^3 f}-\frac{\left (\left (a^2-b^2\right ) g^3 \sqrt{\cos (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \left (\sqrt{-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 a^2 \sqrt{g \cos (e+f x)}}+\frac{\left (\left (a^2-b^2\right ) g^3 \sqrt{\cos (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)} \left (\sqrt{-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 a^2 \sqrt{g \cos (e+f x)}}\\ &=-\frac{3 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}+\frac{b^2 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}+\frac{3 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}-\frac{b^2 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}+\frac{b g (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^2 f}-\frac{g (g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{2 a f}+\frac{b g^2 \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \sqrt{\cos (e+f x)}}+\frac{\left (a^2-b^2\right ) g^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 \left (b-\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}+\frac{\left (a^2-b^2\right ) g^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 \left (b+\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}-\frac{\left (b \left (a^2-b^2\right ) g^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} g-b x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{a^3 f}+\frac{\left (b \left (a^2-b^2\right ) g^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} g+b x^2} \, dx,x,\sqrt{g \cos (e+f x)}\right )}{a^3 f}\\ &=-\frac{3 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}+\frac{b^2 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}-\frac{\sqrt{b} \left (-a^2+b^2\right )^{3/4} g^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt{g}}\right )}{a^3 f}+\frac{3 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{4 a f}-\frac{b^2 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g}}\right )}{a^3 f}+\frac{\sqrt{b} \left (-a^2+b^2\right )^{3/4} g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt{g}}\right )}{a^3 f}+\frac{b g (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^2 f}-\frac{g (g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{2 a f}+\frac{b g^2 \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 f \sqrt{\cos (e+f x)}}+\frac{\left (a^2-b^2\right ) g^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 \left (b-\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}+\frac{\left (a^2-b^2\right ) g^3 \sqrt{\cos (e+f x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (e+f x)\right |2\right )}{a^2 \left (b+\sqrt{-a^2+b^2}\right ) f \sqrt{g \cos (e+f x)}}\\ \end{align*}
Mathematica [C] time = 29.1487, size = 1590, normalized size = 2.85 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 2.873, size = 318, normalized size = 0.6 \begin{align*}{\frac{3}{8\,af}{g}^{{\frac{5}{2}}}\ln \left ({ \left ( 4\,g\cos \left ( 1/2\,fx+e/2 \right ) +2\,\sqrt{g}\sqrt{-2\, \left ( \sin \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}g+g}-2\,g \right ) \left ( -1+\cos \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \right ) }-{\frac{{g}^{2}}{16\,af}\sqrt{-2\, \left ( \sin \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}g+g} \left ( \cos \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3}{8\,af}{g}^{{\frac{5}{2}}}\ln \left ({ \left ( -4\,g\cos \left ( 1/2\,fx+e/2 \right ) +2\,\sqrt{g}\sqrt{-2\, \left ( \sin \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}g+g}-2\,g \right ) \left ( \cos \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}} \right ) }+{\frac{3\,{g}^{3}}{4\,af}\ln \left ({ \left ( -2\,g+2\,\sqrt{-g}\sqrt{2\, \left ( \cos \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}g-g} \right ) \left ( \cos \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \right ){\frac{1}{\sqrt{-g}}}}+{\frac{{g}^{2}}{16\,af}\sqrt{-2\, \left ( \sin \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}g+g} \left ( -1+\cos \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}}+{\frac{{g}^{2}}{8\,af}\sqrt{2\, \left ( \cos \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}g-g} \left ( \cos \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-2}} \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 (g \cos \left (f x + e\right )\right )^{\frac{5}{2}} \csc \left (f x + e\right )^{3}}{b \sin \left (f x + e\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 (g \cos \left (f x + e\right )\right )^{\frac{5}{2}} \csc \left (f x + e\right )^{3}}{b \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]