Optimal. Leaf size=94 \[ -\frac{\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^3}-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}-\frac{(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.137888, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3190, 414, 527, 522, 206, 205} \[ -\frac{\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^3}-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}-\frac{(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3190
Rule 414
Rule 527
Rule 522
Rule 206
Rule 205
Rubi steps
\begin{align*} \int \frac{\csc ^5(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}-\frac{\operatorname{Subst}\left (\int \frac{3 a+4 b+3 b x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\cos (x)\right )}{4 (a+b)}\\ &=-\frac{(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}-\frac{\operatorname{Subst}\left (\int \frac{3 a^2+7 a b+8 b^2+b (3 a+7 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\cos (x)\right )}{8 (a+b)^2}\\ &=-\frac{(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\cos (x)\right )}{(a+b)^3}-\frac{\left (3 a^2+10 a b+15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (x)\right )}{8 (a+b)^3}\\ &=-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^3}-\frac{\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac{(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}\\ \end{align*}
Mathematica [B] time = 1.23215, size = 204, normalized size = 2.17 \[ \frac{\sqrt{a} \left (-2 \left (3 a^2+10 a b+7 b^2\right ) \csc ^2\left (\frac{x}{2}\right )+2 \left (3 a^2+10 a b+7 b^2\right ) \sec ^2\left (\frac{x}{2}\right )-8 \left (3 a^2+10 a b+15 b^2\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )-(a+b)^2 \csc ^4\left (\frac{x}{2}\right )+(a+b)^2 \sec ^4\left (\frac{x}{2}\right )\right )-64 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b}-\sqrt{a+b} \tan \left (\frac{x}{2}\right )}{\sqrt{a}}\right )-64 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan \left (\frac{x}{2}\right )+\sqrt{b}}{\sqrt{a}}\right )}{64 \sqrt{a} (a+b)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.049, size = 205, normalized size = 2.2 \begin{align*}{\frac{1}{ \left ( 16\,a+16\,b \right ) \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}+{\frac{3\,a}{16\, \left ( a+b \right ) ^{2} \left ( 1+\cos \left ( x \right ) \right ) }}+{\frac{7\,b}{16\, \left ( a+b \right ) ^{2} \left ( 1+\cos \left ( x \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\cos \left ( x \right ) \right ){a}^{2}}{16\, \left ( a+b \right ) ^{3}}}-{\frac{5\,\ln \left ( 1+\cos \left ( x \right ) \right ) ab}{8\, \left ( a+b \right ) ^{3}}}-{\frac{15\,\ln \left ( 1+\cos \left ( x \right ) \right ){b}^{2}}{16\, \left ( a+b \right ) ^{3}}}-{\frac{1}{ \left ( 16\,a+16\,b \right ) \left ( \cos \left ( x \right ) -1 \right ) ^{2}}}+{\frac{3\,a}{16\, \left ( a+b \right ) ^{2} \left ( \cos \left ( x \right ) -1 \right ) }}+{\frac{7\,b}{16\, \left ( a+b \right ) ^{2} \left ( \cos \left ( x \right ) -1 \right ) }}+{\frac{3\,\ln \left ( \cos \left ( x \right ) -1 \right ){a}^{2}}{16\, \left ( a+b \right ) ^{3}}}+{\frac{5\,\ln \left ( \cos \left ( x \right ) -1 \right ) ab}{8\, \left ( a+b \right ) ^{3}}}+{\frac{15\,\ln \left ( \cos \left ( x \right ) -1 \right ){b}^{2}}{16\, \left ( a+b \right ) ^{3}}}-{\frac{{b}^{3}}{ \left ( a+b \right ) ^{3}}\arctan \left ({b\cos \left ( x \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.49897, size = 1490, normalized size = 15.85 \begin{align*} \text{result too large to display} \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 [B] time = 1.15854, size = 240, normalized size = 2.55 \begin{align*} -\frac{b^{3} \arctan \left (\frac{b \cos \left (x\right )}{\sqrt{a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt{a b}} - \frac{{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (x\right ) + 1\right )}{16 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (-\cos \left (x\right ) + 1\right )}{16 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{3 \, a \cos \left (x\right )^{3} + 7 \, b \cos \left (x\right )^{3} - 5 \, a \cos \left (x\right ) - 9 \, b \cos \left (x\right )}{8 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (\cos \left (x\right )^{2} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]