Optimal. Leaf size=230 \[ -\frac{3 \left (4 a^2-10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 d (a-b)^{5/2}}+\frac{3 \left (4 a^2+10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )}{32 d (a+b)^{5/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 d \left (a^2-b^2\right )}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^2-7 b^2\right )-6 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )^2} \]
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Rubi [A] time = 0.389479, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2668, 741, 823, 827, 1166, 206} \[ -\frac{3 \left (4 a^2-10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 d (a-b)^{5/2}}+\frac{3 \left (4 a^2+10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )}{32 d (a+b)^{5/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 d \left (a^2-b^2\right )}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^2-7 b^2\right )-6 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{16 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 741
Rule 823
Rule 827
Rule 1166
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+x} \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 \left (a^2-b^2\right ) d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (6 a^2-7 b^2\right )+\frac{5 a x}{2}}{\sqrt{a+x} \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d}\\ &=-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 \left (a^2-b^2\right ) d}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^2-7 b^2\right )-6 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d}-\frac{b \operatorname{Subst}\left (\int \frac{-\frac{3}{4} \left (4 a^4-9 a^2 b^2+7 b^4\right )-\frac{3}{2} a \left (a^2-2 b^2\right ) x}{\sqrt{a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 \left (a^2-b^2\right ) d}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^2-7 b^2\right )-6 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d}-\frac{b \operatorname{Subst}\left (\int \frac{\frac{3}{2} a^2 \left (a^2-2 b^2\right )-\frac{3}{4} \left (4 a^4-9 a^2 b^2+7 b^4\right )-\frac{3}{2} a \left (a^2-2 b^2\right ) x^2}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt{a+b \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^2 d}\\ &=-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 \left (a^2-b^2\right ) d}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^2-7 b^2\right )-6 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d}-\frac{\left (3 \left (4 a^2-10 a b+7 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-b-x^2} \, dx,x,\sqrt{a+b \sin (c+d x)}\right )}{32 (a-b)^2 d}+\frac{\left (3 \left (4 a^2+10 a b+7 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-x^2} \, dx,x,\sqrt{a+b \sin (c+d x)}\right )}{32 (a+b)^2 d}\\ &=-\frac{3 \left (4 a^2-10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 (a-b)^{5/2} d}+\frac{3 \left (4 a^2+10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )}{32 (a+b)^{5/2} d}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 \left (a^2-b^2\right ) d}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^2-7 b^2\right )-6 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [A] time = 1.86918, size = 244, normalized size = 1.06 \[ \frac{\sqrt{a-b} \left (3 (a-b)^2 \left (4 a^2+10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right )+\sqrt{a+b} \sec ^4(c+d x) \sqrt{a+b \sin (c+d x)} \left (a \left (11 a^2-14 b^2\right ) \sin (c+d x)+3 \left (a^3-2 a b^2\right ) \sin (3 (c+d x))+\left (7 b^3-a^2 b\right ) \cos (2 (c+d x))-9 a^2 b+15 b^3\right )\right )-3 (a+b)^{5/2} \left (4 a^2-10 a b+7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right )}{32 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.816, size = 618, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [B] time = 1.13278, size = 566, normalized size = 2.46 \begin{align*} \frac{b^{5}{\left (\frac{3 \,{\left (4 \, a^{2} - 10 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac{\sqrt{b \sin \left (d x + c\right ) + a}}{\sqrt{-a + b}}\right )}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} \sqrt{-a + b}} - \frac{3 \,{\left (4 \, a^{2} + 10 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac{\sqrt{b \sin \left (d x + c\right ) + a}}{\sqrt{-a - b}}\right )}{{\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} \sqrt{-a - b}} - \frac{2 \,{\left (6 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3} - 18 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{4} + 18 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{5} - 6 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{6} - 12 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a b^{2} + 35 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} b^{2} - 44 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} b^{2} + 21 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{4} b^{2} + 7 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} b^{4} + 2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a b^{4} - 4 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{2} b^{4} - 11 \, \sqrt{b \sin \left (d x + c\right ) + a} b^{6}\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )}{\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{2} - 2 \,{\left (b \sin \left (d x + c\right ) + a\right )} a + a^{2} - b^{2}\right )}^{2}}\right )}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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