Optimal. Leaf size=326 \[ -\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{\pi } \sqrt{i b}}\right )}{3 \sqrt{i b} b^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right )}{3 \sqrt{i b} b^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{x}{3 b^2 \sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}-\frac{\sqrt{d^2 x^4+2 i d x^2}}{3 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}} \]
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Rubi [A] time = 0.0722763, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4828, 4819} \[ -\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{\pi } \sqrt{i b}}\right )}{3 \sqrt{i b} b^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right )}{3 \sqrt{i b} b^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{x}{3 b^2 \sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}-\frac{\sqrt{d^2 x^4+2 i d x^2}}{3 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4828
Rule 4819
Rubi steps
\begin{align*} \int \frac{1}{\left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{5/2}} \, dx &=-\frac{\sqrt{2 i d x^2+d^2 x^4}}{3 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}-\frac{x}{3 b^2 \sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}+\frac{\int \frac{1}{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}} \, dx}{3 b^2}\\ &=-\frac{\sqrt{2 i d x^2+d^2 x^4}}{3 b d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}-\frac{x}{3 b^2 \sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}-\frac{\sqrt{\pi } x S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right ) \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right )}{3 \sqrt{i b} b^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{\sqrt{\pi } x C\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right ) \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right )}{3 \sqrt{i b} b^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.77374, size = 308, normalized size = 0.94 \[ -\frac{\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{\pi } \sqrt{i b}}\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}+\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}+\frac{b \sqrt{d x^2 \left (d x^2+2 i\right )}}{d x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^{3/2}}+\frac{x}{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( i+d{x}^{2} \right ) \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsinh}\left (d x^{2} + i\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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