3.4.77 \(\int \frac {(1+c^2 x^2)^{5/2}}{x^2 (a+b \sinh ^{-1}(c x))} \, dx\) [377]

Optimal. Leaf size=159 \[ \frac {c \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b}+\frac {c \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b}+\frac {15 c \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b}-\frac {c \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{b}-\frac {c \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

c*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/b+1/8*c*Chi(4*(a+b*arcsinh(c*x))/b)*cosh(4*a/b)/b+15/8*c*ln(a+b*arcs
inh(c*x))/b-c*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b-1/8*c*Shi(4*(a+b*arcsinh(c*x))/b)*sinh(4*a/b)/b+Uninte
grable(1/x^2/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

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Rubi [A]
time = 0.59, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])),x]

[Out]

(c*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/b + (c*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh
[c*x]))/b])/(8*b) + (15*c*Log[a + b*ArcSinh[c*x]])/(8*b) - (c*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x
]))/b])/b - (c*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b) + Defer[Int][1/(x^2*Sqrt[1 + c^2*
x^2]*(a + b*ArcSinh[c*x])), x]

Rubi steps

\begin {align*} \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx &=\int \left (\frac {3 c^2}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}+\frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 c^4 x^2}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}+\frac {c^6 x^4}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}\right ) \, dx\\ &=\left (3 c^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+\left (3 c^4\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+c^6 \int \frac {x^4}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac {3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{b}+c \text {Subst}\left (\int \frac {\sinh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+(3 c) \text {Subst}\left (\int \frac {\sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac {3 c \log \left (a+b \sinh ^{-1}(c x)\right )}{b}+c \text {Subst}\left (\int \left (\frac {3}{8 (a+b x)}-\frac {\cosh (2 x)}{2 (a+b x)}+\frac {\cosh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )-(3 c) \text {Subst}\left (\int \left (\frac {1}{2 (a+b x)}-\frac {\cosh (2 x)}{2 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac {15 c \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b}+\frac {1}{8} c \text {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{2} c \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{2} (3 c) \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac {15 c \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b}-\frac {1}{2} \left (c \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{2} \left (3 c \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{8} \left (c \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\frac {1}{2} \left (c \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{2} \left (3 c \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{8} \left (c \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\frac {c \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b}+\frac {c \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b}+\frac {15 c \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b}-\frac {c \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{b}-\frac {c \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b}+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \arcsinh \left (c x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*x^2+1)^(5/2)/x^2/(a+b*arcsinh(c*x)),x)

[Out]

int((c^2*x^2+1)^(5/2)/x^2/(a+b*arcsinh(c*x)),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(5/2)/x^2/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate((c^2*x^2 + 1)^(5/2)/((b*arcsinh(c*x) + a)*x^2), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(5/2)/x^2/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^4 + 2*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1)/(b*x^2*arcsinh(c*x) + a*x^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c**2*x**2+1)**(5/2)/x**2/(a+b*asinh(c*x)),x)

[Out]

Integral((c**2*x**2 + 1)**(5/2)/(x**2*(a + b*asinh(c*x))), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(5/2)/x^2/(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate((c^2*x^2 + 1)^(5/2)/((b*arcsinh(c*x) + a)*x^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c^2\,x^2+1\right )}^{5/2}}{x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*x^2 + 1)^(5/2)/(x^2*(a + b*asinh(c*x))),x)

[Out]

int((c^2*x^2 + 1)^(5/2)/(x^2*(a + b*asinh(c*x))), x)

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